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The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function
European Journal of Operational Research 176 (2007) 46–59 www.elsevier.com/locate/ejor
Continuous Optimization
The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function Hsien-Chung Wu Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan Received 26 November 2004; accepted 2 September 2005 Available online 6 December 2005
Abstract The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Hausdorff metric; Hukuhara difference; H-differentiability; Interval-valued function; KKT optimality conditions
1. Introduction The occurrence of randomness and imprecision in the real world is inevitable owing to some unexpected situations. Therefore, imposing the uncertainty upon the conventional optimization problems becomes an interesting research topic. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. The books written by Birge and Louveaux [5], Kall [10], Pre´kopa [15], Stancu-Minasian [18] and Vajda [20] give many useful techniques for solving the stochastic optimization problems. The collection of papers on fuzzy optimization edited by Słowin´ski [16]
E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.09.007
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and Delgado et al. [6] gives the main stream of this topic. Lai and Hwang [11,12] also give an insightful survey. On the other hand, the book edited by Słowin´ski and Teghem [17] gives the comparisons between fuzzy optimization and stochastic optimization for the multiobjective programming problems. Inuiguchi and Ramı´k [8] also gives a brief review of fuzzy optimization and a comparison with stochastic optimization in portfolio selection problem. In this paper, we consider an alternative choice for considering the uncertainty into the conventional optimization problems, that is, the interval-valued optimization problems. Stancu-Minasian and Tigan [19] also give the related subject with the interval-valued optimization problem. As we have known, the usual ordering ‘‘6’’ is a total ordering on R; that is, for any two real numbers in R, we can determine their order without difficulty. However, for any two closed intervals in R, there is not a natural ordering among the set of all closed intervals in R, and we have to define an ordering to determine their order. In this paper, we provide two solution concepts for the optimization problem (nonlinear programming problem) with interval-valued objective function. One of the solution concepts follows from Ishibuchi and Tanaka [9]. For these two solution concepts, two ordering relationships between two closed intervals in R will be proposed. Under these settings, we derive the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. In Section 2, we introduce some basic properties and arithmetics of intervals. In Section 3, we invoke the well-known Hausdorff metric to define the distance between any two closed intervals. Using this metric, we can consider the continuity and limit of an interval-valued function (an interval-valued function means that its function value is a closed interval). In Section 4, we invoke the Hukuhara difference to define the difference of any two closed intervals. Using this Hukuhara difference and the concept of limit in interval-valued function which is defined in Section 3, we are capable of proposing the differentiation of an interval-valued function. In Section 5, we formulate an optimization problem with interval-valued objective function and provide two solution concepts for this problem. In the final Section 6, we derive the KKT conditions for our problem.
2. Arithmetics of intervals Let Kc ðRÞ denote the class of all nonempty, compact and convex subsets of R. Let A; B 2 Kc ðRÞ. Then A + B is defined by A + B = {a + b : a 2 A and b 2 B} and A is defined by A = {a : a 2 A}. Therefore, A B = A + (B). Let us denote by I the class of all closed and bounded intervals in R. Throughout this paper, when we say that A is a closed interval, we implicitly means that A is also bounded in R. If A is a closed interval, we also adopt the notation A = [aL, aU], where aL and aU means the lower and upper bounds of A, respectively. Let A = [aL, aU] and B = [bL, bU] be in I. Then, by definition, we have (i) A + B = {a + b : a 2 A and b 2 B} = [aL + bL,aU + bU]; (ii) A = {a : a 2 A} = [aU, aL]. Therefore, we see that A B = A + (B) = [aL bU, aU bL]. We also see that ( ½kaL ; kaU if k P 0 kA ¼ fka : a 2 Ag ¼ ½kaU ; kaL if k < 0; where k is a real number. For the more details on the topic of interval analysis, we refer to Moore [13,14] and Alefeld and Herzberger [1].
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3. Limit and continuity of interval-valued function Let A Rn and B Rn . Then the Hausdorff metric between A and B is defined by d H ðA; BÞ ¼ max sup inf ka bk; sup inf ka bk ; a2A b2B
b2B a2A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kÆk is an Euclidean norm defined by kxk ¼ x21 þ x2n for x ¼ ðx1 ; . . . ; xn Þ 2 Rn . If A = [aL, aU] and B = [bL, bU] are two closed intervals in R, then it is not hard to see that d H ðA; BÞ ¼ maxfjaL bL j; jaU bU jg.
ð1Þ
Let {An} and A be closed intervals in R. We say that the sequence of closed intervals {An} converges to A, denoted by lim An ¼ A;
n!1
if, for every > 0, there exists a N > 0 such that, for n > N, we have dH(An, A) < . L U Proposition 3.1. Let fAn ¼ ½aLn ; aU n g and A = [a , a ] be closed intervals in R. Then limn!1An = A if and L L U U only if limn!1 an ¼ a and limn!1 an ¼ a . U Proof. From (1), we see that dH(An, A) < implies jaLn aL j < and jaU n a j < . The remaining proof follows from the routine arguments. h
The function f : Rn ! I defined on an Euclidean space Rn is called an interval-valued function, i.e., f(x) = f(x1, . . . , xn) is a closed interval in R for each x 2 Rn . The interval-valued function f can also be written as f(x) = [f L(x), f U(x)], where f L and f U are real-valued functions defined on Rn and satisfy f L(x) 6 f U(x) for every x 2 Rn . Let f be an interval-valued function defined on Rn and A = [aL, aU] be an interval in R. For c 2 Rn , we write lim f ðxÞ ¼ A; x!c
if, for every > 0, there exists a d > 0 such that, for kx ck < d, we have dH(f(x), A) < . Proposition 3.2. Let f be an interval-valued function defined on Rn and A = [aL, aU] be an interval in R. Then limx!cf(x) = A if and only if limx!cf L(x) = aL and limx!cf U(x) = aU. Proof. The proof is similar with that of Proposition 3.1.
h
Suppose now that the interval-valued function f is defined on R. Then we can similarly define the righthand limit and left-hand limit lim f ðxÞ and lim f ðxÞ;
x!cþ
x!c
respectively. We can also show that limx!c+f(x) = A if and only if limx!c+f L(x) = aL and limx!c+f U(x) = aU, and limx!cf(x) = A if and only if limx!cf L(x) = aL and limx!cf U(x) = aU. Let f be an interval-valued function defined on Rn . We say that f is continuous at c 2 Rn if lim f ðxÞ ¼ f ðcÞ. x!c
Then we have the following useful result.
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Proposition 3.3. Let f be an interval-valued function defined on Rn . Then f is continuous at c 2 Rn if and only if f L and f U are continuous at c. Proof. The proof is obvious by using the routine arguments and Proposition 3.2.
h
4. Differentiation of interval-valued functions We shall consider two kinds of differentiation of interval-valued function. The first one is very straightforward as we shall see below. Definition 4.1. Let X be an open set in R. An interval-valued function f : X ! I with f(x) = [f L(x), f U(x)] is called weakly differentiable at x0 if the real-valued functions f L and f U are differentiable at x0 (in the usual sense). Next we are going to propose another differentiation of interval-valued function. Let A; B 2 Kc ðRÞ. If there exists a C 2 Kc ðRÞ such that A = B + C, then C is called the Hukuhara difference. We also write C = A B (Ref. Banks and Jacobs [3]). Now we consider the Hukuhara difference in I. Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. If there exists a closed interval C = [cL, cU] such that A = B + C, then C is called the Hukuhara difference. Since A = B + C, it is easy to see that aL = bL + cL and aU = bU + cU, i.e., cL = aL bL and cU = aU bU. Therefore, this closed interval C exists if aL bL 6 aU bU. In this case, C = [aLbL, aUbU] and we also write C = A B. Therefore, when we say that the Hukuhara difference C = A B exists, we implicitly means that aL bL 6 aU bU. Now we have the following result. Proposition 4.1. Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. If aL bL 6 aUbU then the Hukuhara difference C = A B exits and C = [cL, cU] = [aL bL, aU bU]. We remark that if A = [aL, aU] is a closed interval in R and h is a positive real number, then A/h is read as A/h = [aL/h, aU/h]. Now we are going to propose the differentiation of interval-valued function by invoking the Hukuhara difference. Definition 4.2. Let X be an open set in R. An interval-valued function f : X ! I is called H-differentiable (or strongly differentiable) at x0 if there exists a closed interval A(x0) (note that this interval depends on x0) in R such that the limits lim
h!0þ
f ðx0 þ hÞ f ðx0 Þ f ðx0 Þ f ðx0 hÞ and lim h!0þ h h
both exists and are equal to A(x0). In this case, A(x0) is called the H-derivative of f at x0. Let us remark that, when we say that f is H-differentiable at x0, we implicitly means that f(x0 + h) f(x0) and f(x0) f(x0 h) exist for every h > 0. The next result will show that the strong differentiability implies the weak differentiability. Proposition 4.2. Let X be an open set in R. If an interval-valued function f : X ! I is H-differentiable (or strongly differentiable) at x0 with H-derivative A(x0) = [aL(x0), aU(x0)], then f is weakly differentiable at x0. Furthermore, we have (f L) 0 (x0) = aL(x0) and (f U) 0 (x0) = aU(x0). Proof. The result follows from Propositions 4.1 and 3.2 immediately. Conversely, we have the following result.
h
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Proposition 4.3. Let X be an open set in R and f : X ! I be an interval-valued function defined on X. Suppose that f is weakly differentiable at x0 with derivatives ðf L Þ0 ðx0 Þ ¼ ^aL ðx0 Þ and ðf U Þ0 ðx0 Þ ¼ ^aU ðx0 Þ. (i) If f L(x0 + h) f L(x0) 6 f U(x0 + h) f U(x0) and f L(x0) f L(x0 h) 6 f U(x0) f U(x0 h) for every h > 0, then f is H-differentiable (or strongly differentiable) at x0 with H-derivative Aðx0 Þ ¼ ½^ aL ðx0 Þ; ^ aU ðx0 Þ. L U (ii) If ^ a ðx0 Þ > ^ a ðx0 Þ, then f is H-nondifferentiable at x0.
Proof 0
0
(i) Since ðf L Þ ðx0 Þ ¼ ^ aL ðx0 Þ and ðf U Þ ðx0 Þ ¼ ^ aU ðx0 Þ, we have lim
h!0þ
f L ðx0 þ hÞ f L ðx0 Þ f L ðx0 Þ f L ðx0 hÞ ¼^ aL ðx0 Þ ¼ lim h!0þ h h
and lim
h!0þ
f U ðx0 þ hÞ f U ðx0 Þ f U ðx0 Þ f U ðx0 hÞ ¼^ aU ðx0 Þ ¼ lim . h!0þ h h
Since f L(x0 + h) f L(x0) 6 f U(x0 + h) f U(x0) and f L(x0) f L(x0 h) 6 f U(x0) f U(x0 h) for every h > 0, we have ^ aL ðx0 Þ 6 ^ aU ðx0 Þ. Therefore, we can form the interval Aðx0 Þ ¼ ½^aL ðx0 Þ; ^aU ðx0 Þ. From Propositions 4.1 and 3.2, we see that lim
h!0þ
f ðx0 þ hÞ f ðx0 Þ f ðx0 Þ f ðx0 hÞ ¼ Aðx0 Þ ¼ lim . h!0þ h h
This shows that f is H-differentiable at x0. (ii) Suppose that f is H-differentiable at x0 with H-derivative A(x0) = [aL(x0), aU(x0)]. Then, from Proposition 4.2, we have (f L) 0 (x0) = aL(x0) and (f U) 0 (x0) = aU(x0). By the uniqueness of derivatives, this shows that Aðx0 Þ ¼ ½^ aL ðx0 Þ; ^ aU ðx0 Þ, i.e., ^aL ðx0 Þ 6 ^aU ðx0 Þ, which contradicts the hypothesis L U ^ a ðx0 Þ > ^ a ðx0 Þ. h 2
Example 4.1. Let f1 : R ! ½x2 þ 1; ðx þ 1Þ þ 3 be an interval-valued function defined on R. Using Proposition 4.3 (i), we see that f1 is H-differentiable at any x0 2 R with H-derivative [2x0, 2x0 + 2]. Now let f2 : [0, 2] ! [x2 + x + 1, x2 + 3] be an interval-valued function defined on [0, 2]. Then f2 is weakly differentiable on the open interval (0, 2). However, using Proposition 4.3 (ii), we see that f2 is H-nondifferentiable on (0, 2). Now we are going to consider the interval-valued function f defined on Rn , i.e., f(x) = f(x1, . . . , xn) is an interval in R for each x ¼ ðx1 ; . . . ; xn Þ 2 Rn . Therefore, we also have the corresponding real-valued functions f L(x) = f L(x1, . . . , xn) and f U(x) = f U(x1, . . . , xn) defined on Rn . Proposition 4.4. (Apostol [2, Theorem 12.11]) Let f be a real-valued function defined on Rn . Assume that one of the partial derivatives of/ox1, . . . , of/oxn exists at x0 and that the remaining n 1 partial derivatives exist on some neighborhoods of x0 and are continuous at x0. Then f is differentiable at x0. Inspired by the above Proposition 4.4, we propose the following definition. ð0Þ
ð0Þ
Definition 4.3. Let f be an interval-valued function defined on X Rn and x0 ¼ ðx1 ; . . . ; xn Þ 2 X be fixed.
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(i) We say that f is weakly differentiable at x0 if the real-valued functions f L and f U are differentiable at x0 (which imply that all of the partial derivatives of L/oxi and of U/oxi exist at x0 for i = 1, . . . , n). (ii) We say that f has the ith partial H-derivative Ai ðx0 Þ ¼ ½aLi ðx0 Þ; aU i ðx0 Þ at x0 if the interval-valued funcð0Þ ð0Þ ð0Þ ð0Þ ð0Þ tion gðxi Þ ¼ f ðx1 ; . . . ; xi1 ; xi ; xiþ1 ; . . . ; xn Þ is H-differentiable at xi with H-derivative Ai(x0). We also write Ai(x0) as (of/oxi)H(x0). (iii) We say that f is H-differentiable at x0 if one of the partial H-derivatives (of/o x1)H, . . . , (of/oxn)H exists at x0 and the remaining n 1 partial H-derivatives exist on some neighborhoods of x0 and are continuous at x0 (in the sense of interval-valued function). Proposition 4.5. Let f be an interval-valued function defined on X Rn . If f is H-differentiable at x0 2 X, then f is weakly differentiable at x0. Proof. The result follows from Propositions 3.3, 4.2 and 4.4 immediately.
h ð0Þ
Definition 4.4. Let f be an interval-valued function defined on X Rn and x0 ¼ ðx1 ; . . . ; xnð0Þ Þ 2 X be fixed. (i) We say that f is weakly continuously differentiable at x0 if the real-valued functions f L and f U are continuously differentiable at x0 (i.e., all of the partial derivatives of f L and f U exist on some neighborhoods of x0 and are continuous at x0). (ii) We say that f is continuously H-differentiable (or strongly continuously differentiable) at x0 if all of the partial H-derivatives (of/ox1)H, . . . , (of/oxn)H exist on some neighborhoods of x0 and are continuous at x0 (in the sense of interval-valued function). Using Propositions 3.3, 4.2 and 4.4 again, we also have the following result. Proposition 4.6. Let f be an interval-valued function defined on X Rn . If f is continuously H-differentiable at x0 2 X, then f is weakly continuously differentiable at x0. 5. Formulation and solution concept of interval-valued optimization problem Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. We write A LU B if and only if aL 6 bL and aU 6 bU . It is easy to see that ‘‘LU’’ is a partial ordering on I. Now we consider the following interval-valued optimization problem ðIVOP1Þ min f ðxÞ ¼ f ðx1 ; . . . ; xn Þ ¼ ½f L ðx1 ; . . . ; xn Þ; f U ðx1 ; . . . ; xn Þ ¼ ½f L ðxÞ; f U ðxÞ subject to x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 X Rn ;
ð2Þ
where the feasible set X is assumed as a convex subset of Rn . Since the objective value f(x) is a closed interval, we need to interprete the meaning of minimization in problem (IVOP1). The most direct way is to invoke the order relation ‘‘LU’’ that was defined in (2). However ‘‘LU’’ is a partial ordering, not a total ordering, on I, we may follow the similar solution concept (the Pareto optimal solution) used in multiobjective programming problem to interprete the meaning of minimization in problem (IVOP1). Now we write A LU B if and only if A LU B and A 5 B. Equivalently, A LU B if and only if ( ( ( aL < bL aL 6 bL aL < bL or or . ð3Þ aU 6 bU aU < bU aU < bU
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H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
Definition 5.1. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-I solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ LU f ðx Þ. Next we are going to introduce the second solution concept following from Ishibuchi and Tanaka [9]. Let A = [aL, aU] be a closed interval in R. Then we can calculate the center aC ¼ 12 ðaL þ aU Þ and the half-width aW ¼ 12 ðaU aL Þ of A. In this case, we can use the new notation haC, aWi to denote the closed interval A, i.e., we write A = haC, aWi. Let A = haC, aWi and B = hbC, bWi be two closed intervals. Ishibuchi and Tanaka [9] proposed the ordering relation between A and B by considering the maximization and minimization problems separately. (i) For maximization, we write A CW B if and only if aC 6 bC
and
aW P bW .
The half-width of interval can be regarded as uncertainty. Therefore, the interval with higher center and lower half-width (i.e., the less uncertainty) is preferred for maximization problem. (ii) For minimization, we write A CW B if and only if aC 6 bC
and
aW 6 bW .
The interval with lower center and lower half-width (i.e., the less uncertainty) is preferred for minimization problem. Also, we write A CW B if and only if A CW B and A 5 B. In this paper, we consider only the minimization problem without loss of generality. Definition 5.2. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-II solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ LU f ðx Þ or f ð xÞ CW f ðx Þ. Remark 5.1. Let x* be a feasible solution, i.e., x* 2 X. We see that if x* is a type-I solution of problem (IVOP1) then x* is also a type-II solution of problem (IVOP1). In order to make the discussions more convenient, we define another ordering relation on I. Let A = [aL, aU] = haC, aWi and B = [bL, bU] = hbC, bWi be two closed intervals in R. Then we write A UC B if and only if aU 6 bU
and
aC 6 bC .
We also write A UC B if and only if A UC B and A 5 B. Ishibuchi and Tanaka [9] proved that (i) A UC B if and only if A LU B or A CW B; (ii) A UC B if and only if A LU B or A CW B. Therefore, Definition 5.2 can be re-stated as follows. Definition 5.3. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-II solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ UC f ðx Þ. Example 5.1. Let f : [0, 2] ! [x2 + x + 1, x2 + 3] be an interval-valued function defined on [0, 2]. It is not hard to see that x* = 0 is a type-I solution. Let us write f C ðxÞ ¼ 12 ðf L ðxÞ þ f U ðxÞÞ ¼ x2 þ 2x þ 2 and f W ðxÞ ¼ 12 ðf U ðxÞ f L ðxÞÞ ¼ 1 2x. We now take x ¼ 12. Then we see that x ¼ 12 is not a type-I solution. However, we are going to show that x ¼ 12 is a type-II solution. Suppose that x ¼ 12 is not a type-II solution. Then there exists an x such that f ðxÞ
H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
(
f C ðxÞ < f C f W ðxÞ 6
(
1
2 f W 12
or
f C ðxÞ 6 f C f W ðxÞ <
(
1
2 f W 12
or
f C ðxÞ < f C
53
1
f W ðxÞ < f W
21
ð4Þ
2
Then we see that 6f C ðxÞ þ 2f W ðxÞ < 6f C ð12Þ þ 2f W ð12Þ. Under some algebraic calculations, we get 2 2ðx 12 Þ < 0. Therefore, a contradiction occurs. This shows that x ¼ 12 is a type-II solution. 6. The Karush–Kuhn–Tucker optimality conditions Let f and gi, i = 1, . . . , m, be real-valued functions defined on Rn . Then we consider the following optimization problem ðPÞ
min
f ðxÞ ¼ f ðx1 ; . . . ; xn Þ
subject to
gi ðxÞ 6 0;
i ¼ 1; . . . ; m.
Suppose that the constraint functions gi are convex on Rn for each i = 1, . . . , m. Then the feasible set X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg is a convex subset of Rn . The well-known Karush–Kuhn–Tucker condition for problem (P) (e.g., see Horst et al. [7] or Bazaraa et al. [4]) is stated as follows. Theorem 6.1. Assume that the constraint functions gi : Rn ! R are convex on Rn for i = 1, . . . , m. Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set and a point x* 2 X. Suppose that the objective function f : Rn ! R is convex at x*, and f, gi, i = 1, . . . , m, are continuously differentiable at x*. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is an optimal solution of problem (P). Now we consider the following interval-valued minimization problem ðIVOP2Þ
min
f ðxÞ ¼ ½f L ðxÞ; f U ðxÞ
subject to gi ðxÞ 6 0;
i ¼ 1; . . . ; m;
where the real-valued constraint functions gi : Rn ! R are convex on Rn for i = 1, . . . , m. We see that problem (IVOP2) follows from problem (IVOP1) by taking the convex set X as X = {x : gi(x) 6 0, i = 1, . . . , m}. 6.1. KKT conditions for weakly differentiable case In this subsection, the interval-valued objective function f is assumed as weakly (continuously) differentiable at a feasible solution x*. Therefore, from Propositions 4.5 and 4.6, we see that all of the results presented in this subsection also hold true for (continuously) H-differentiable case. However, we shall present the KKT conditions in the interval-valued form for (continuously) H-differentiable case in the next subsection. First of all, we propose the concept of convexity for interval-valued functions. Definition 6.1. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on a convex set X Rn . We say that f is LU-convex at x* if f ðkx þ ð1 kÞxÞ LU kf ðx Þ þ ð1 kÞf ðxÞ
ð5Þ
for each k 2 (0, 1) and each x 2 X. Moreover, we can similarly define the UC-convexity at x* by using the ordering relation ‘‘UC’’.
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H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
Proposition 6.1. Let X be a convex subset of Rn and f be an interval-valued function defined on X. Then we have the following properties. (i) f is LU-convex at x* if and only if f L and f U are convex at x*. (ii) f is UC-convex at x* if and only if f U and f C are convex at x*. (iii) If f is LU-convex at x*, then f is also UC-convex at x*.
Proof. (i) and (ii) follow from the definition, and (iii) follows from the fact that A LU B implies A UC B for A; B 2 I. h Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set of problem (IVOP2) and a point x* 2 X. We say that the real-valued constraint functions gi, i = 1, . . . , m, satisfy the KKT assumptions at x* if gi are convex on Rn and continuously differentiable at x* for i = 1, . . . , m. Now we are in a position to present the Karush–Kuhn–Tucker optimality conditions for problem (IVOP2). Theorem 6.2. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is LU-convex and weakly continuously differentiable at x*. If there exist (Lagrange) multipliers 0 < kL ; kU 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kL rf L ðx Þ þ kU rf U ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. Since f(x) = [f L(x), f U(x)], we can define a real-valued function f ðxÞ ¼ kL f L ðxÞ þ kU f U ðxÞ.
ð6Þ x*,
Since f is LU-convex and weakly continuously differentiable at by definition and Proposition 6.1 (i), we see that the real-valued functions f L and f U are convex and continuously differentiable at x*. Therefore, f is also convex and continuously differentiable at x*. Since rf ðx Þ ¼ kL rf L ðx Þ þ kU rf U ðx Þ; according to conditions (i) and (ii) of this theorem, we obtain the following two new conditions Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m. Using Theorem 6.1, we see that x* is an optimal solution of the real-valued objective function f subject to the same constraints of problem (IVOP2), i.e., f ðx Þ 6 f ð xÞ ð7Þ ð6¼ x Þ 2 X . We are going to prove this theorem by contradiction. Suppose that x* is not a type-I for any x 2 X such that solution of problem (IVOP2). Then, according to Definition 5.1, there exists an x f ð xÞ LU f ðx Þ, i.e., from (3), L L L f ð xÞ < f L ðx Þ xÞ 6 f L ðx Þ xÞ < f L ðx Þ f ð f ð or or . ð8Þ U U U U U f ð f ð f ð xÞ 6 f ðx Þ xÞ < f ðx Þ xÞ < f U ðx Þ
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55
Therefore, from expressions (6) and (8), we see that f ð xÞ < f ðx Þ (since kL > 0 and kU > 0) which contradicts (7). From Remark 5.1, it also shows that x* is a type-II solution of problem (IVOP2). This completes the proof. h Example 6.1. Let us consider the following interval-valued minimization problem min subject to
f ðxÞ ¼ ½x2 þ x þ 1; x2 þ 3 x260 x 6 0.
We write g1(x) = x 2 and g2(x) = x. Then the assumptions presented in Theorem 6.2 are satisfied, and the KKT conditions are given below: (i) kL Æ (2x* + 1) + kU Æ 2x* + l1l2 = 0; (ii) l1 Æ (x*2) = 0 = l2 Æ x*. Let us take x* = 0. Then condition (ii) implies l1 = 0 and condition (i) implies kL = l2. Let us take the multipliers kL = kU = l2 = 1 and l1 = 0. Then Theorem 6.2 shows that x* = 0 is a type-I and type-II solution. Theorem 6.3. Under the same assumptions of Theorem 6.2, let k be any integer with 1 < k < m. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pk (i) rf L ðx Þ þ Pi¼1 li rgi ðx Þ ¼ 0; m (ii) rf U ðx Þ þ i¼kþ1 li rgi ðx Þ ¼ 0; (iii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). U Li ¼ kL li for i = 1, . . . , k and l U Proof. Let 0 < kL , kU 2 R, l i ¼ k li for i = k + 1, . . . , m. Then conditions (i), (ii) and (iii) of this theorem can be rewritten as
Pk L P i rgi ðx Þ þ mi¼kþ1 l U (i) kL rf L ðx Þ þ kU rf U ðx Þ þ i¼1 l i rg i ðx Þ ¼ 0; L U i gi ðx Þ ¼ 0 for i = 1, . . . , k and l i gi ðx Þ ¼ 0 for i = k + 1, . . . , m. (ii) l Using Theorem 6.2, this completes the proof.
h
Theorem 6.4. Under the same assumptions of Theorem 6.2, let f C ¼ 12 ðf L þ f U Þ. If there exist (Lagrange) multipliers 0 < kU , kC 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kU rf U ðx Þ þ kC rf C ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. We define a real-valued function f ðxÞ ¼ kU f U ðxÞ þ kC f C ðxÞ. Since f is LU-convex at x* and f C ¼ 12 ðf L þ f U Þ, we see that f C is convex and continuously differentiable at x* by using Proposition 6.1.
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(i) This shows that f is also convex and continuously differentiable at x*. Now suppose that x* is not a typeII solution, using the similar arguments in the proof of Theorem 6.2, we can show that x* is a type-II solution of problem (SIVOP3). Since 1 kU rf U ðx Þ þ kC rf C ðx Þ ¼ kU rf U ðx Þ þ kC rf L ðx Þ þ rf U ðx Þ 2 1 C L 1 C U k þ k rf U ðx Þ; ¼ k rf ðx Þ þ 2 2 we conclude that x* is also a type-I solution by using Theorem 6.2.
h
In the sequel, we are going to relax the convexity assumption of interval-valued objective function by considering the pseudoconvexity. Let f be a differentiable real-valued function defined on a nonempty open convex subset X of Rn . Then f is convex at x* if and only if f(x) P f(x*) + $f(x*)T(xx*) for x 2 X (Ref. Bazaraa et al. [4, Theorem 3.3.3]), i.e., f(x)f(x*) P $f(x*)T(xx*). Therefore, we see that if f(x) 6 f(x*) then $f(x*)T(x x*) 6 0. Also, if f(x) < f(x*) then $f(x*)T(x x*) < 0. Let us recall that f is pseudoconvex at x* if f(x) < f(x*) then $f(x*)T(x x*) < 0 for x 2 X, and f is strictly pseudoconvex at x* if f(x) 6 f(x*) then $f(x*)T(x x*) < 0 for x 2 X (Bazaraa et al. [4, p. 114]). It is well-known that the strict convexity implies the strict pseudoconvexity. Inspired by Proposition 6.1, we propose the following definition. Definition 6.2. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on a convex set X Rn . We say that f is pseudoconvex at x* if the real-valued functions f L and f U are pseudoconvex at x*. Let X be a nonempty feasible set and x* 2 clX (the closure of X). The cone of feasible directions of X at x*, denoted by D, is defined by D ¼ fd 2 Rn : d 6¼ 0; there exists a d > 0 such that x þ gd 2 X for all g 2 ð0; dÞg. Each d of D is called a feasible direction of X. The following proposition, from Bazaraa et al. [4], is very useful. Proposition 6.2 (Bazaraa et al. [4, Lemma 4.2.4]). Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set and a point x* 2 X. Assume that gi are differentiable at x* for all i = 1, . . . , m. Let I = {i : gi(x*) = 0} be the index set for the active constraints. Then T
D fd 2 Rn : rgi ðx Þ d 6 0 for each i 2 Ig (note that this proposition still hold true if we just assume that gi are continuous at x* instead of differentiable at x* for i 62 I). The Tuckers theorem of the alternative states that, given matrices A and C, exactly one of the following system has a solution: System I: Ax 6 0, Ax 5 0, Cx 6 0 for some x 2 Rn ; System II: ATk + CTl = 0 for some (k, l), k > 0, l P 0. Theorem 6.5. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is weakly differentiable and pseudoconvex at x*. If there exist (Lagrange) multipliers 0 6 lLi , lU i 2 R, i = 1, . . . , m, such that
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Pm (i) rf L ðx Þ þ i¼1 lLi rgi ðx Þ ¼ 0; Pm (ii) rf U ðx Þ þ i¼1 lU i rg i ðx Þ ¼ 0; (iii) lLi gi ðx Þ ¼ 0 ¼ lU i g i ðx Þ for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. We are going to prove this result by contradiction. Suppose that x* is not a type-I solution. Then 6¼ x such that f ð there exists an x xÞ LU f ðx Þ, which implies that f L ð xÞ < f L ðx Þ or f U ð xÞ < f U ðx Þ from L U * (8). Since f is weakly differentiable and pseudoconvex at x , i.e., f and f are differentiable and pseudoT T convex at x* by definition, we have rf L ðx Þ ð x x Þ < 0 or rf U ðx Þ ð x x Þ < 0. First of all, we consider the case rf L ðx ÞT ð x x Þ < 0
ð9Þ
, x 2 X . x . Then x ¼ x þ gd ¼ g x þ ð1 gÞx 2 X for g 2 (0, 1) since X is a convex set and x Let d ¼ x This shows that d 2 D. From Proposition 6.2, we conclude that T
rgi ðx Þ d 6 0 for each i 2 I;
ð10Þ
where I is the index set for the active constraints. Let A = $f L(x*)T and C be the matrix whose rows are $gi(x*)T for i 2 I. According to the Tuckers theorem of the alternative, since system I has a solution x from d¼x P (9) and (10), there exist no multipliers 0 < k 2 R and 0 6 li 2 R, i 2 I, such that krf L ðx Þ þP i2I li rgi ðx Þ ¼ 0; or, equivalently, there exist no multipliers 0 6 lLi 2 R, i 2 I, such that L L rf rgi ðx Þ ¼ 0, where lLi ¼ li =k, which violates conditions (i) and (iii), since P ðxL Þ þ i2I li P m L L * i¼1 li rgi ðx Þ with li g i ðx Þ ¼ 0 for all i = 1, . . . , m (i.e., gi(x ) 5 0 for i 62 I). Similarly, i2I li rgi ðx Þ ¼ U T if rf ðx Þ ð x x Þ < 0, then conditions (ii) and (iii) will be violated. This shows that x* is a type-I solution. From Remark 5.1, the proof is complete. h Next we present the KKT conditions for type-II solution only. Theorem 6.6. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is UC-convex and weakly continuously differentiable at x*. If there exist (Lagrange) multipliers 0 < kU ; kC 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kU rf U ðx Þ þ kC rf C ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-II solution of problem (IVOP2). Proof. We define a real-valued function f ðxÞ ¼ kU f U ðxÞ þ kC f C ðxÞ. Since f is UC-convex at x*, we see that f is also convex and continuously differentiable at x* by using Proposition 6.1 (ii). Using the similar arguments in the proof of Theorem 6.4, we can show that x* is a type-II solution. However, although we still have 1 C L 1 C U C U U C k þ k rf U ðx Þ; k rf ðx Þ þ k rf ðx Þ ¼ k rf ðx Þ þ 2 2 we cannot conclude that x* is also a type-I solution by using Theorem 6.2, since the assumptions in this theorem is different from that of Theorem 6.2 and the UC-convexity does not necessarily imply LU-convexity in general. h
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6.2. KKT conditions for H-differentiable case In the sequel, we are going to present the KKT conditions in the interval-valued form for continuously H-differentiable case. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on Rn and be H-differentiable at x0 2 Rn . The gradient of f at x0, denoted by $f(x0), is defined by T of of rf ðx0 Þ ¼ ðx0 Þ; . . . ; ðx0 Þ ; ox1 oxn where (of/oxj)(x0) is the jth partial H-derivative of f at x0 by referring to Definition 4.3. Moreover, the partial H-derivative L
of of of U ðx0 Þ ¼ ðx0 Þ; ðx0 Þ oxj oxj oxj is a closed interval. Let gi be real-valued functions defined on Rn for i = 1, . . . , m. Then T ogi ogi ðx0 Þ; . . . ; ðx0 Þ ; rgi ðx0 Þ ¼ ox1 oxn where (ogi/oxj)(x0) can also be regarded as a closed interval
ogi ogi ðx0 Þ; ðx0 Þ . oxj oxj Then the following equation m X li rgi ðx0 Þ ¼ 0; rf ðx0 Þ þ i¼1
where li are nonnegative real numbers, can be interpreted as m m X X of L og of U og ðx0 Þ þ li i ðx0 Þ ¼ 0 ¼ ðx0 Þ þ li i ðx0 Þ oxj ox ox oxj j j i¼1 i¼1 for all j = 1, . . . , n. Equivalently, (11) can be rewritten as m m X X li rgi ðx0 Þ ¼ 0 ¼ rf U ðx0 Þ þ li rgi ðx0 Þ; rf L ðx0 Þ þ i¼1
ð11Þ
ð12Þ
i¼1
which also implies rf L ðx0 Þ þ rf U ðx0 Þ þ
m X
i rgi ðx0 Þ ¼ 0; l
ð13Þ
i¼1
i ¼ 2li for i = 1, . . . , m. where l Theorem 6.7. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is LU-convex and continuously H-differentiable at x*. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2).
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Proof. The result follows from Eq. (13) and Theorem 6.2 by taking kL = 1 = kU.
59
h
Although we just present one kind of KKT conditions in the interval-valued form as described in Theorem 6.7, in fact, we still can derive many kinds of KKT conditions in the interval-valued form, under the same assumptions as described in many of the above theorems, by using Eqs. (12) and (13). We omit the details in order to save space.
References [1] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, NY, 1983. [2] T.M. Apostol, Mathematical Analysis, second ed., Addison-Wesley Publishing Company, 1974. [3] H.T. Banks, M.Q. Jacobs, A differential calculus for multifunctions, Journal of Mathematical Analysis and Applications 29 (1970) 246–272. [4] M.S. Bazarra, H.D. Sherali, C.M. Shetty, Nonlinear Programming, Wiley, NY, 1993. [5] J.R. Birge, F. Louveaux, Introduction to Stochastic Programming, Physica-Verlag, NY, 1997. [6] M. Delgado, J. Kacprzyk, J.-L. Verdegay, M.A. Vila (Eds.), Fuzzy Optimization: Recent Advances, Physica-Verlag, NY, 1994. [7] R. Horst, P.M. Pardalos, N.V. Thoai, Introduction to Global Optimization, second ed., Kluwer Academic Publishers, Boston, 2000. [8] M. Inuiguchi, J. Ramı´k, Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems 111 (2000) 3–28. [9] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research 48 (1990) 219–225. [10] P. Kall, Stochastic Linear Programming, Springer-Verlag, NY, 1976. [11] Y.-J. Lai, C.-L. Hwang, Fuzzy mathematical programming: Methods and applications, Lecture Notes in Economics and Mathematical Systems, 394, Springer-Verlag, NY, 1992. [12] Y.-J. Lai, C.-L. Hwang, Fuzzy multiple objective decision making: Methods and applications, Lecture Notes in Economics and Mathematical Systems, 404, Springer-Verlag, NY, 1994. [13] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966. [14] R.E. Moore, Method and Applications of Interval Analysis, SIAM, Philadelphia, 1979. [15] A. Pre´kopa, Stochastic Programming, Kluwer Academic Publishers, Boston, 1995. [16] R. Słowin´ski (Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. [17] R. Słowin´ski, J. Teghem (Eds.), Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Prog, Kluwer Academic Publishers, Boston, 1990. [18] I.M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D. Reidel Publishing Company, Dordrecht, 1984. [19] I.M. Stancu-Minasian, St. Tigan, Multiobjective mathematical programming with inexact data, in: R. Słowin´ski, J. Teghem (Eds.), Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Boston, 1990, pp. 395–418. [20] S. Vajda, Probabilistic Programming, Academic Press, NY, 1972.
Continuous Optimization
The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function Hsien-Chung Wu Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan Received 26 November 2004; accepted 2 September 2005 Available online 6 December 2005
Abstract The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Hausdorff metric; Hukuhara difference; H-differentiability; Interval-valued function; KKT optimality conditions
1. Introduction The occurrence of randomness and imprecision in the real world is inevitable owing to some unexpected situations. Therefore, imposing the uncertainty upon the conventional optimization problems becomes an interesting research topic. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. The books written by Birge and Louveaux [5], Kall [10], Pre´kopa [15], Stancu-Minasian [18] and Vajda [20] give many useful techniques for solving the stochastic optimization problems. The collection of papers on fuzzy optimization edited by Słowin´ski [16]
E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.09.007
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and Delgado et al. [6] gives the main stream of this topic. Lai and Hwang [11,12] also give an insightful survey. On the other hand, the book edited by Słowin´ski and Teghem [17] gives the comparisons between fuzzy optimization and stochastic optimization for the multiobjective programming problems. Inuiguchi and Ramı´k [8] also gives a brief review of fuzzy optimization and a comparison with stochastic optimization in portfolio selection problem. In this paper, we consider an alternative choice for considering the uncertainty into the conventional optimization problems, that is, the interval-valued optimization problems. Stancu-Minasian and Tigan [19] also give the related subject with the interval-valued optimization problem. As we have known, the usual ordering ‘‘6’’ is a total ordering on R; that is, for any two real numbers in R, we can determine their order without difficulty. However, for any two closed intervals in R, there is not a natural ordering among the set of all closed intervals in R, and we have to define an ordering to determine their order. In this paper, we provide two solution concepts for the optimization problem (nonlinear programming problem) with interval-valued objective function. One of the solution concepts follows from Ishibuchi and Tanaka [9]. For these two solution concepts, two ordering relationships between two closed intervals in R will be proposed. Under these settings, we derive the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. In Section 2, we introduce some basic properties and arithmetics of intervals. In Section 3, we invoke the well-known Hausdorff metric to define the distance between any two closed intervals. Using this metric, we can consider the continuity and limit of an interval-valued function (an interval-valued function means that its function value is a closed interval). In Section 4, we invoke the Hukuhara difference to define the difference of any two closed intervals. Using this Hukuhara difference and the concept of limit in interval-valued function which is defined in Section 3, we are capable of proposing the differentiation of an interval-valued function. In Section 5, we formulate an optimization problem with interval-valued objective function and provide two solution concepts for this problem. In the final Section 6, we derive the KKT conditions for our problem.
2. Arithmetics of intervals Let Kc ðRÞ denote the class of all nonempty, compact and convex subsets of R. Let A; B 2 Kc ðRÞ. Then A + B is defined by A + B = {a + b : a 2 A and b 2 B} and A is defined by A = {a : a 2 A}. Therefore, A B = A + (B). Let us denote by I the class of all closed and bounded intervals in R. Throughout this paper, when we say that A is a closed interval, we implicitly means that A is also bounded in R. If A is a closed interval, we also adopt the notation A = [aL, aU], where aL and aU means the lower and upper bounds of A, respectively. Let A = [aL, aU] and B = [bL, bU] be in I. Then, by definition, we have (i) A + B = {a + b : a 2 A and b 2 B} = [aL + bL,aU + bU]; (ii) A = {a : a 2 A} = [aU, aL]. Therefore, we see that A B = A + (B) = [aL bU, aU bL]. We also see that ( ½kaL ; kaU if k P 0 kA ¼ fka : a 2 Ag ¼ ½kaU ; kaL if k < 0; where k is a real number. For the more details on the topic of interval analysis, we refer to Moore [13,14] and Alefeld and Herzberger [1].
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3. Limit and continuity of interval-valued function Let A Rn and B Rn . Then the Hausdorff metric between A and B is defined by d H ðA; BÞ ¼ max sup inf ka bk; sup inf ka bk ; a2A b2B
b2B a2A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kÆk is an Euclidean norm defined by kxk ¼ x21 þ x2n for x ¼ ðx1 ; . . . ; xn Þ 2 Rn . If A = [aL, aU] and B = [bL, bU] are two closed intervals in R, then it is not hard to see that d H ðA; BÞ ¼ maxfjaL bL j; jaU bU jg.
ð1Þ
Let {An} and A be closed intervals in R. We say that the sequence of closed intervals {An} converges to A, denoted by lim An ¼ A;
n!1
if, for every > 0, there exists a N > 0 such that, for n > N, we have dH(An, A) < . L U Proposition 3.1. Let fAn ¼ ½aLn ; aU n g and A = [a , a ] be closed intervals in R. Then limn!1An = A if and L L U U only if limn!1 an ¼ a and limn!1 an ¼ a . U Proof. From (1), we see that dH(An, A) < implies jaLn aL j < and jaU n a j < . The remaining proof follows from the routine arguments. h
The function f : Rn ! I defined on an Euclidean space Rn is called an interval-valued function, i.e., f(x) = f(x1, . . . , xn) is a closed interval in R for each x 2 Rn . The interval-valued function f can also be written as f(x) = [f L(x), f U(x)], where f L and f U are real-valued functions defined on Rn and satisfy f L(x) 6 f U(x) for every x 2 Rn . Let f be an interval-valued function defined on Rn and A = [aL, aU] be an interval in R. For c 2 Rn , we write lim f ðxÞ ¼ A; x!c
if, for every > 0, there exists a d > 0 such that, for kx ck < d, we have dH(f(x), A) < . Proposition 3.2. Let f be an interval-valued function defined on Rn and A = [aL, aU] be an interval in R. Then limx!cf(x) = A if and only if limx!cf L(x) = aL and limx!cf U(x) = aU. Proof. The proof is similar with that of Proposition 3.1.
h
Suppose now that the interval-valued function f is defined on R. Then we can similarly define the righthand limit and left-hand limit lim f ðxÞ and lim f ðxÞ;
x!cþ
x!c
respectively. We can also show that limx!c+f(x) = A if and only if limx!c+f L(x) = aL and limx!c+f U(x) = aU, and limx!cf(x) = A if and only if limx!cf L(x) = aL and limx!cf U(x) = aU. Let f be an interval-valued function defined on Rn . We say that f is continuous at c 2 Rn if lim f ðxÞ ¼ f ðcÞ. x!c
Then we have the following useful result.
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Proposition 3.3. Let f be an interval-valued function defined on Rn . Then f is continuous at c 2 Rn if and only if f L and f U are continuous at c. Proof. The proof is obvious by using the routine arguments and Proposition 3.2.
h
4. Differentiation of interval-valued functions We shall consider two kinds of differentiation of interval-valued function. The first one is very straightforward as we shall see below. Definition 4.1. Let X be an open set in R. An interval-valued function f : X ! I with f(x) = [f L(x), f U(x)] is called weakly differentiable at x0 if the real-valued functions f L and f U are differentiable at x0 (in the usual sense). Next we are going to propose another differentiation of interval-valued function. Let A; B 2 Kc ðRÞ. If there exists a C 2 Kc ðRÞ such that A = B + C, then C is called the Hukuhara difference. We also write C = A B (Ref. Banks and Jacobs [3]). Now we consider the Hukuhara difference in I. Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. If there exists a closed interval C = [cL, cU] such that A = B + C, then C is called the Hukuhara difference. Since A = B + C, it is easy to see that aL = bL + cL and aU = bU + cU, i.e., cL = aL bL and cU = aU bU. Therefore, this closed interval C exists if aL bL 6 aU bU. In this case, C = [aLbL, aUbU] and we also write C = A B. Therefore, when we say that the Hukuhara difference C = A B exists, we implicitly means that aL bL 6 aU bU. Now we have the following result. Proposition 4.1. Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. If aL bL 6 aUbU then the Hukuhara difference C = A B exits and C = [cL, cU] = [aL bL, aU bU]. We remark that if A = [aL, aU] is a closed interval in R and h is a positive real number, then A/h is read as A/h = [aL/h, aU/h]. Now we are going to propose the differentiation of interval-valued function by invoking the Hukuhara difference. Definition 4.2. Let X be an open set in R. An interval-valued function f : X ! I is called H-differentiable (or strongly differentiable) at x0 if there exists a closed interval A(x0) (note that this interval depends on x0) in R such that the limits lim
h!0þ
f ðx0 þ hÞ f ðx0 Þ f ðx0 Þ f ðx0 hÞ and lim h!0þ h h
both exists and are equal to A(x0). In this case, A(x0) is called the H-derivative of f at x0. Let us remark that, when we say that f is H-differentiable at x0, we implicitly means that f(x0 + h) f(x0) and f(x0) f(x0 h) exist for every h > 0. The next result will show that the strong differentiability implies the weak differentiability. Proposition 4.2. Let X be an open set in R. If an interval-valued function f : X ! I is H-differentiable (or strongly differentiable) at x0 with H-derivative A(x0) = [aL(x0), aU(x0)], then f is weakly differentiable at x0. Furthermore, we have (f L) 0 (x0) = aL(x0) and (f U) 0 (x0) = aU(x0). Proof. The result follows from Propositions 4.1 and 3.2 immediately. Conversely, we have the following result.
h
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Proposition 4.3. Let X be an open set in R and f : X ! I be an interval-valued function defined on X. Suppose that f is weakly differentiable at x0 with derivatives ðf L Þ0 ðx0 Þ ¼ ^aL ðx0 Þ and ðf U Þ0 ðx0 Þ ¼ ^aU ðx0 Þ. (i) If f L(x0 + h) f L(x0) 6 f U(x0 + h) f U(x0) and f L(x0) f L(x0 h) 6 f U(x0) f U(x0 h) for every h > 0, then f is H-differentiable (or strongly differentiable) at x0 with H-derivative Aðx0 Þ ¼ ½^ aL ðx0 Þ; ^ aU ðx0 Þ. L U (ii) If ^ a ðx0 Þ > ^ a ðx0 Þ, then f is H-nondifferentiable at x0.
Proof 0
0
(i) Since ðf L Þ ðx0 Þ ¼ ^ aL ðx0 Þ and ðf U Þ ðx0 Þ ¼ ^ aU ðx0 Þ, we have lim
h!0þ
f L ðx0 þ hÞ f L ðx0 Þ f L ðx0 Þ f L ðx0 hÞ ¼^ aL ðx0 Þ ¼ lim h!0þ h h
and lim
h!0þ
f U ðx0 þ hÞ f U ðx0 Þ f U ðx0 Þ f U ðx0 hÞ ¼^ aU ðx0 Þ ¼ lim . h!0þ h h
Since f L(x0 + h) f L(x0) 6 f U(x0 + h) f U(x0) and f L(x0) f L(x0 h) 6 f U(x0) f U(x0 h) for every h > 0, we have ^ aL ðx0 Þ 6 ^ aU ðx0 Þ. Therefore, we can form the interval Aðx0 Þ ¼ ½^aL ðx0 Þ; ^aU ðx0 Þ. From Propositions 4.1 and 3.2, we see that lim
h!0þ
f ðx0 þ hÞ f ðx0 Þ f ðx0 Þ f ðx0 hÞ ¼ Aðx0 Þ ¼ lim . h!0þ h h
This shows that f is H-differentiable at x0. (ii) Suppose that f is H-differentiable at x0 with H-derivative A(x0) = [aL(x0), aU(x0)]. Then, from Proposition 4.2, we have (f L) 0 (x0) = aL(x0) and (f U) 0 (x0) = aU(x0). By the uniqueness of derivatives, this shows that Aðx0 Þ ¼ ½^ aL ðx0 Þ; ^ aU ðx0 Þ, i.e., ^aL ðx0 Þ 6 ^aU ðx0 Þ, which contradicts the hypothesis L U ^ a ðx0 Þ > ^ a ðx0 Þ. h 2
Example 4.1. Let f1 : R ! ½x2 þ 1; ðx þ 1Þ þ 3 be an interval-valued function defined on R. Using Proposition 4.3 (i), we see that f1 is H-differentiable at any x0 2 R with H-derivative [2x0, 2x0 + 2]. Now let f2 : [0, 2] ! [x2 + x + 1, x2 + 3] be an interval-valued function defined on [0, 2]. Then f2 is weakly differentiable on the open interval (0, 2). However, using Proposition 4.3 (ii), we see that f2 is H-nondifferentiable on (0, 2). Now we are going to consider the interval-valued function f defined on Rn , i.e., f(x) = f(x1, . . . , xn) is an interval in R for each x ¼ ðx1 ; . . . ; xn Þ 2 Rn . Therefore, we also have the corresponding real-valued functions f L(x) = f L(x1, . . . , xn) and f U(x) = f U(x1, . . . , xn) defined on Rn . Proposition 4.4. (Apostol [2, Theorem 12.11]) Let f be a real-valued function defined on Rn . Assume that one of the partial derivatives of/ox1, . . . , of/oxn exists at x0 and that the remaining n 1 partial derivatives exist on some neighborhoods of x0 and are continuous at x0. Then f is differentiable at x0. Inspired by the above Proposition 4.4, we propose the following definition. ð0Þ
ð0Þ
Definition 4.3. Let f be an interval-valued function defined on X Rn and x0 ¼ ðx1 ; . . . ; xn Þ 2 X be fixed.
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(i) We say that f is weakly differentiable at x0 if the real-valued functions f L and f U are differentiable at x0 (which imply that all of the partial derivatives of L/oxi and of U/oxi exist at x0 for i = 1, . . . , n). (ii) We say that f has the ith partial H-derivative Ai ðx0 Þ ¼ ½aLi ðx0 Þ; aU i ðx0 Þ at x0 if the interval-valued funcð0Þ ð0Þ ð0Þ ð0Þ ð0Þ tion gðxi Þ ¼ f ðx1 ; . . . ; xi1 ; xi ; xiþ1 ; . . . ; xn Þ is H-differentiable at xi with H-derivative Ai(x0). We also write Ai(x0) as (of/oxi)H(x0). (iii) We say that f is H-differentiable at x0 if one of the partial H-derivatives (of/o x1)H, . . . , (of/oxn)H exists at x0 and the remaining n 1 partial H-derivatives exist on some neighborhoods of x0 and are continuous at x0 (in the sense of interval-valued function). Proposition 4.5. Let f be an interval-valued function defined on X Rn . If f is H-differentiable at x0 2 X, then f is weakly differentiable at x0. Proof. The result follows from Propositions 3.3, 4.2 and 4.4 immediately.
h ð0Þ
Definition 4.4. Let f be an interval-valued function defined on X Rn and x0 ¼ ðx1 ; . . . ; xnð0Þ Þ 2 X be fixed. (i) We say that f is weakly continuously differentiable at x0 if the real-valued functions f L and f U are continuously differentiable at x0 (i.e., all of the partial derivatives of f L and f U exist on some neighborhoods of x0 and are continuous at x0). (ii) We say that f is continuously H-differentiable (or strongly continuously differentiable) at x0 if all of the partial H-derivatives (of/ox1)H, . . . , (of/oxn)H exist on some neighborhoods of x0 and are continuous at x0 (in the sense of interval-valued function). Using Propositions 3.3, 4.2 and 4.4 again, we also have the following result. Proposition 4.6. Let f be an interval-valued function defined on X Rn . If f is continuously H-differentiable at x0 2 X, then f is weakly continuously differentiable at x0. 5. Formulation and solution concept of interval-valued optimization problem Let A = [aL, aU] and B = [bL, bU] be two closed intervals in R. We write A LU B if and only if aL 6 bL and aU 6 bU . It is easy to see that ‘‘LU’’ is a partial ordering on I. Now we consider the following interval-valued optimization problem ðIVOP1Þ min f ðxÞ ¼ f ðx1 ; . . . ; xn Þ ¼ ½f L ðx1 ; . . . ; xn Þ; f U ðx1 ; . . . ; xn Þ ¼ ½f L ðxÞ; f U ðxÞ subject to x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 X Rn ;
ð2Þ
where the feasible set X is assumed as a convex subset of Rn . Since the objective value f(x) is a closed interval, we need to interprete the meaning of minimization in problem (IVOP1). The most direct way is to invoke the order relation ‘‘LU’’ that was defined in (2). However ‘‘LU’’ is a partial ordering, not a total ordering, on I, we may follow the similar solution concept (the Pareto optimal solution) used in multiobjective programming problem to interprete the meaning of minimization in problem (IVOP1). Now we write A LU B if and only if A LU B and A 5 B. Equivalently, A LU B if and only if ( ( ( aL < bL aL 6 bL aL < bL or or . ð3Þ aU 6 bU aU < bU aU < bU
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Definition 5.1. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-I solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ LU f ðx Þ. Next we are going to introduce the second solution concept following from Ishibuchi and Tanaka [9]. Let A = [aL, aU] be a closed interval in R. Then we can calculate the center aC ¼ 12 ðaL þ aU Þ and the half-width aW ¼ 12 ðaU aL Þ of A. In this case, we can use the new notation haC, aWi to denote the closed interval A, i.e., we write A = haC, aWi. Let A = haC, aWi and B = hbC, bWi be two closed intervals. Ishibuchi and Tanaka [9] proposed the ordering relation between A and B by considering the maximization and minimization problems separately. (i) For maximization, we write A CW B if and only if aC 6 bC
and
aW P bW .
The half-width of interval can be regarded as uncertainty. Therefore, the interval with higher center and lower half-width (i.e., the less uncertainty) is preferred for maximization problem. (ii) For minimization, we write A CW B if and only if aC 6 bC
and
aW 6 bW .
The interval with lower center and lower half-width (i.e., the less uncertainty) is preferred for minimization problem. Also, we write A CW B if and only if A CW B and A 5 B. In this paper, we consider only the minimization problem without loss of generality. Definition 5.2. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-II solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ LU f ðx Þ or f ð xÞ CW f ðx Þ. Remark 5.1. Let x* be a feasible solution, i.e., x* 2 X. We see that if x* is a type-I solution of problem (IVOP1) then x* is also a type-II solution of problem (IVOP1). In order to make the discussions more convenient, we define another ordering relation on I. Let A = [aL, aU] = haC, aWi and B = [bL, bU] = hbC, bWi be two closed intervals in R. Then we write A UC B if and only if aU 6 bU
and
aC 6 bC .
We also write A UC B if and only if A UC B and A 5 B. Ishibuchi and Tanaka [9] proved that (i) A UC B if and only if A LU B or A CW B; (ii) A UC B if and only if A LU B or A CW B. Therefore, Definition 5.2 can be re-stated as follows. Definition 5.3. Let x* be a feasible solution, i.e., x* 2 X. We say that x* is a type-II solution of problem 2 X such that f ð (IVOP1) if there exists no x xÞ UC f ðx Þ. Example 5.1. Let f : [0, 2] ! [x2 + x + 1, x2 + 3] be an interval-valued function defined on [0, 2]. It is not hard to see that x* = 0 is a type-I solution. Let us write f C ðxÞ ¼ 12 ðf L ðxÞ þ f U ðxÞÞ ¼ x2 þ 2x þ 2 and f W ðxÞ ¼ 12 ðf U ðxÞ f L ðxÞÞ ¼ 1 2x. We now take x ¼ 12. Then we see that x ¼ 12 is not a type-I solution. However, we are going to show that x ¼ 12 is a type-II solution. Suppose that x ¼ 12 is not a type-II solution. Then there exists an x such that f ðxÞ
H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
(
f C ðxÞ < f C f W ðxÞ 6
(
1
2 f W 12
or
f C ðxÞ 6 f C f W ðxÞ <
(
1
2 f W 12
or
f C ðxÞ < f C
53
1
f W ðxÞ < f W
21
ð4Þ
2
Then we see that 6f C ðxÞ þ 2f W ðxÞ < 6f C ð12Þ þ 2f W ð12Þ. Under some algebraic calculations, we get 2 2ðx 12 Þ < 0. Therefore, a contradiction occurs. This shows that x ¼ 12 is a type-II solution. 6. The Karush–Kuhn–Tucker optimality conditions Let f and gi, i = 1, . . . , m, be real-valued functions defined on Rn . Then we consider the following optimization problem ðPÞ
min
f ðxÞ ¼ f ðx1 ; . . . ; xn Þ
subject to
gi ðxÞ 6 0;
i ¼ 1; . . . ; m.
Suppose that the constraint functions gi are convex on Rn for each i = 1, . . . , m. Then the feasible set X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg is a convex subset of Rn . The well-known Karush–Kuhn–Tucker condition for problem (P) (e.g., see Horst et al. [7] or Bazaraa et al. [4]) is stated as follows. Theorem 6.1. Assume that the constraint functions gi : Rn ! R are convex on Rn for i = 1, . . . , m. Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set and a point x* 2 X. Suppose that the objective function f : Rn ! R is convex at x*, and f, gi, i = 1, . . . , m, are continuously differentiable at x*. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is an optimal solution of problem (P). Now we consider the following interval-valued minimization problem ðIVOP2Þ
min
f ðxÞ ¼ ½f L ðxÞ; f U ðxÞ
subject to gi ðxÞ 6 0;
i ¼ 1; . . . ; m;
where the real-valued constraint functions gi : Rn ! R are convex on Rn for i = 1, . . . , m. We see that problem (IVOP2) follows from problem (IVOP1) by taking the convex set X as X = {x : gi(x) 6 0, i = 1, . . . , m}. 6.1. KKT conditions for weakly differentiable case In this subsection, the interval-valued objective function f is assumed as weakly (continuously) differentiable at a feasible solution x*. Therefore, from Propositions 4.5 and 4.6, we see that all of the results presented in this subsection also hold true for (continuously) H-differentiable case. However, we shall present the KKT conditions in the interval-valued form for (continuously) H-differentiable case in the next subsection. First of all, we propose the concept of convexity for interval-valued functions. Definition 6.1. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on a convex set X Rn . We say that f is LU-convex at x* if f ðkx þ ð1 kÞxÞ LU kf ðx Þ þ ð1 kÞf ðxÞ
ð5Þ
for each k 2 (0, 1) and each x 2 X. Moreover, we can similarly define the UC-convexity at x* by using the ordering relation ‘‘UC’’.
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Proposition 6.1. Let X be a convex subset of Rn and f be an interval-valued function defined on X. Then we have the following properties. (i) f is LU-convex at x* if and only if f L and f U are convex at x*. (ii) f is UC-convex at x* if and only if f U and f C are convex at x*. (iii) If f is LU-convex at x*, then f is also UC-convex at x*.
Proof. (i) and (ii) follow from the definition, and (iii) follows from the fact that A LU B implies A UC B for A; B 2 I. h Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set of problem (IVOP2) and a point x* 2 X. We say that the real-valued constraint functions gi, i = 1, . . . , m, satisfy the KKT assumptions at x* if gi are convex on Rn and continuously differentiable at x* for i = 1, . . . , m. Now we are in a position to present the Karush–Kuhn–Tucker optimality conditions for problem (IVOP2). Theorem 6.2. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is LU-convex and weakly continuously differentiable at x*. If there exist (Lagrange) multipliers 0 < kL ; kU 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kL rf L ðx Þ þ kU rf U ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. Since f(x) = [f L(x), f U(x)], we can define a real-valued function f ðxÞ ¼ kL f L ðxÞ þ kU f U ðxÞ.
ð6Þ x*,
Since f is LU-convex and weakly continuously differentiable at by definition and Proposition 6.1 (i), we see that the real-valued functions f L and f U are convex and continuously differentiable at x*. Therefore, f is also convex and continuously differentiable at x*. Since rf ðx Þ ¼ kL rf L ðx Þ þ kU rf U ðx Þ; according to conditions (i) and (ii) of this theorem, we obtain the following two new conditions Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m. Using Theorem 6.1, we see that x* is an optimal solution of the real-valued objective function f subject to the same constraints of problem (IVOP2), i.e., f ðx Þ 6 f ð xÞ ð7Þ ð6¼ x Þ 2 X . We are going to prove this theorem by contradiction. Suppose that x* is not a type-I for any x 2 X such that solution of problem (IVOP2). Then, according to Definition 5.1, there exists an x f ð xÞ LU f ðx Þ, i.e., from (3), L L L f ð xÞ < f L ðx Þ xÞ 6 f L ðx Þ xÞ < f L ðx Þ f ð f ð or or . ð8Þ U U U U U f ð f ð f ð xÞ 6 f ðx Þ xÞ < f ðx Þ xÞ < f U ðx Þ
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55
Therefore, from expressions (6) and (8), we see that f ð xÞ < f ðx Þ (since kL > 0 and kU > 0) which contradicts (7). From Remark 5.1, it also shows that x* is a type-II solution of problem (IVOP2). This completes the proof. h Example 6.1. Let us consider the following interval-valued minimization problem min subject to
f ðxÞ ¼ ½x2 þ x þ 1; x2 þ 3 x260 x 6 0.
We write g1(x) = x 2 and g2(x) = x. Then the assumptions presented in Theorem 6.2 are satisfied, and the KKT conditions are given below: (i) kL Æ (2x* + 1) + kU Æ 2x* + l1l2 = 0; (ii) l1 Æ (x*2) = 0 = l2 Æ x*. Let us take x* = 0. Then condition (ii) implies l1 = 0 and condition (i) implies kL = l2. Let us take the multipliers kL = kU = l2 = 1 and l1 = 0. Then Theorem 6.2 shows that x* = 0 is a type-I and type-II solution. Theorem 6.3. Under the same assumptions of Theorem 6.2, let k be any integer with 1 < k < m. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pk (i) rf L ðx Þ þ Pi¼1 li rgi ðx Þ ¼ 0; m (ii) rf U ðx Þ þ i¼kþ1 li rgi ðx Þ ¼ 0; (iii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). U Li ¼ kL li for i = 1, . . . , k and l U Proof. Let 0 < kL , kU 2 R, l i ¼ k li for i = k + 1, . . . , m. Then conditions (i), (ii) and (iii) of this theorem can be rewritten as
Pk L P i rgi ðx Þ þ mi¼kþ1 l U (i) kL rf L ðx Þ þ kU rf U ðx Þ þ i¼1 l i rg i ðx Þ ¼ 0; L U i gi ðx Þ ¼ 0 for i = 1, . . . , k and l i gi ðx Þ ¼ 0 for i = k + 1, . . . , m. (ii) l Using Theorem 6.2, this completes the proof.
h
Theorem 6.4. Under the same assumptions of Theorem 6.2, let f C ¼ 12 ðf L þ f U Þ. If there exist (Lagrange) multipliers 0 < kU , kC 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kU rf U ðx Þ þ kC rf C ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. We define a real-valued function f ðxÞ ¼ kU f U ðxÞ þ kC f C ðxÞ. Since f is LU-convex at x* and f C ¼ 12 ðf L þ f U Þ, we see that f C is convex and continuously differentiable at x* by using Proposition 6.1.
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(i) This shows that f is also convex and continuously differentiable at x*. Now suppose that x* is not a typeII solution, using the similar arguments in the proof of Theorem 6.2, we can show that x* is a type-II solution of problem (SIVOP3). Since 1 kU rf U ðx Þ þ kC rf C ðx Þ ¼ kU rf U ðx Þ þ kC rf L ðx Þ þ rf U ðx Þ 2 1 C L 1 C U k þ k rf U ðx Þ; ¼ k rf ðx Þ þ 2 2 we conclude that x* is also a type-I solution by using Theorem 6.2.
h
In the sequel, we are going to relax the convexity assumption of interval-valued objective function by considering the pseudoconvexity. Let f be a differentiable real-valued function defined on a nonempty open convex subset X of Rn . Then f is convex at x* if and only if f(x) P f(x*) + $f(x*)T(xx*) for x 2 X (Ref. Bazaraa et al. [4, Theorem 3.3.3]), i.e., f(x)f(x*) P $f(x*)T(xx*). Therefore, we see that if f(x) 6 f(x*) then $f(x*)T(x x*) 6 0. Also, if f(x) < f(x*) then $f(x*)T(x x*) < 0. Let us recall that f is pseudoconvex at x* if f(x) < f(x*) then $f(x*)T(x x*) < 0 for x 2 X, and f is strictly pseudoconvex at x* if f(x) 6 f(x*) then $f(x*)T(x x*) < 0 for x 2 X (Bazaraa et al. [4, p. 114]). It is well-known that the strict convexity implies the strict pseudoconvexity. Inspired by Proposition 6.1, we propose the following definition. Definition 6.2. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on a convex set X Rn . We say that f is pseudoconvex at x* if the real-valued functions f L and f U are pseudoconvex at x*. Let X be a nonempty feasible set and x* 2 clX (the closure of X). The cone of feasible directions of X at x*, denoted by D, is defined by D ¼ fd 2 Rn : d 6¼ 0; there exists a d > 0 such that x þ gd 2 X for all g 2 ð0; dÞg. Each d of D is called a feasible direction of X. The following proposition, from Bazaraa et al. [4], is very useful. Proposition 6.2 (Bazaraa et al. [4, Lemma 4.2.4]). Let X ¼ fx 2 Rn : gi ðxÞ 6 0; i ¼ 1; . . . ; mg be a feasible set and a point x* 2 X. Assume that gi are differentiable at x* for all i = 1, . . . , m. Let I = {i : gi(x*) = 0} be the index set for the active constraints. Then T
D fd 2 Rn : rgi ðx Þ d 6 0 for each i 2 Ig (note that this proposition still hold true if we just assume that gi are continuous at x* instead of differentiable at x* for i 62 I). The Tuckers theorem of the alternative states that, given matrices A and C, exactly one of the following system has a solution: System I: Ax 6 0, Ax 5 0, Cx 6 0 for some x 2 Rn ; System II: ATk + CTl = 0 for some (k, l), k > 0, l P 0. Theorem 6.5. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is weakly differentiable and pseudoconvex at x*. If there exist (Lagrange) multipliers 0 6 lLi , lU i 2 R, i = 1, . . . , m, such that
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57
Pm (i) rf L ðx Þ þ i¼1 lLi rgi ðx Þ ¼ 0; Pm (ii) rf U ðx Þ þ i¼1 lU i rg i ðx Þ ¼ 0; (iii) lLi gi ðx Þ ¼ 0 ¼ lU i g i ðx Þ for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2). Proof. We are going to prove this result by contradiction. Suppose that x* is not a type-I solution. Then 6¼ x such that f ð there exists an x xÞ LU f ðx Þ, which implies that f L ð xÞ < f L ðx Þ or f U ð xÞ < f U ðx Þ from L U * (8). Since f is weakly differentiable and pseudoconvex at x , i.e., f and f are differentiable and pseudoT T convex at x* by definition, we have rf L ðx Þ ð x x Þ < 0 or rf U ðx Þ ð x x Þ < 0. First of all, we consider the case rf L ðx ÞT ð x x Þ < 0
ð9Þ
, x 2 X . x . Then x ¼ x þ gd ¼ g x þ ð1 gÞx 2 X for g 2 (0, 1) since X is a convex set and x Let d ¼ x This shows that d 2 D. From Proposition 6.2, we conclude that T
rgi ðx Þ d 6 0 for each i 2 I;
ð10Þ
where I is the index set for the active constraints. Let A = $f L(x*)T and C be the matrix whose rows are $gi(x*)T for i 2 I. According to the Tuckers theorem of the alternative, since system I has a solution x from d¼x P (9) and (10), there exist no multipliers 0 < k 2 R and 0 6 li 2 R, i 2 I, such that krf L ðx Þ þP i2I li rgi ðx Þ ¼ 0; or, equivalently, there exist no multipliers 0 6 lLi 2 R, i 2 I, such that L L rf rgi ðx Þ ¼ 0, where lLi ¼ li =k, which violates conditions (i) and (iii), since P ðxL Þ þ i2I li P m L L * i¼1 li rgi ðx Þ with li g i ðx Þ ¼ 0 for all i = 1, . . . , m (i.e., gi(x ) 5 0 for i 62 I). Similarly, i2I li rgi ðx Þ ¼ U T if rf ðx Þ ð x x Þ < 0, then conditions (ii) and (iii) will be violated. This shows that x* is a type-I solution. From Remark 5.1, the proof is complete. h Next we present the KKT conditions for type-II solution only. Theorem 6.6. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is UC-convex and weakly continuously differentiable at x*. If there exist (Lagrange) multipliers 0 < kU ; kC 2 R and 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) kU rf U ðx Þ þ kC rf C ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-II solution of problem (IVOP2). Proof. We define a real-valued function f ðxÞ ¼ kU f U ðxÞ þ kC f C ðxÞ. Since f is UC-convex at x*, we see that f is also convex and continuously differentiable at x* by using Proposition 6.1 (ii). Using the similar arguments in the proof of Theorem 6.4, we can show that x* is a type-II solution. However, although we still have 1 C L 1 C U C U U C k þ k rf U ðx Þ; k rf ðx Þ þ k rf ðx Þ ¼ k rf ðx Þ þ 2 2 we cannot conclude that x* is also a type-I solution by using Theorem 6.2, since the assumptions in this theorem is different from that of Theorem 6.2 and the UC-convexity does not necessarily imply LU-convexity in general. h
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H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
6.2. KKT conditions for H-differentiable case In the sequel, we are going to present the KKT conditions in the interval-valued form for continuously H-differentiable case. Let f(x) = [f L(x), f U(x)] be an interval-valued function defined on Rn and be H-differentiable at x0 2 Rn . The gradient of f at x0, denoted by $f(x0), is defined by T of of rf ðx0 Þ ¼ ðx0 Þ; . . . ; ðx0 Þ ; ox1 oxn where (of/oxj)(x0) is the jth partial H-derivative of f at x0 by referring to Definition 4.3. Moreover, the partial H-derivative L
of of of U ðx0 Þ ¼ ðx0 Þ; ðx0 Þ oxj oxj oxj is a closed interval. Let gi be real-valued functions defined on Rn for i = 1, . . . , m. Then T ogi ogi ðx0 Þ; . . . ; ðx0 Þ ; rgi ðx0 Þ ¼ ox1 oxn where (ogi/oxj)(x0) can also be regarded as a closed interval
ogi ogi ðx0 Þ; ðx0 Þ . oxj oxj Then the following equation m X li rgi ðx0 Þ ¼ 0; rf ðx0 Þ þ i¼1
where li are nonnegative real numbers, can be interpreted as m m X X of L og of U og ðx0 Þ þ li i ðx0 Þ ¼ 0 ¼ ðx0 Þ þ li i ðx0 Þ oxj ox ox oxj j j i¼1 i¼1 for all j = 1, . . . , n. Equivalently, (11) can be rewritten as m m X X li rgi ðx0 Þ ¼ 0 ¼ rf U ðx0 Þ þ li rgi ðx0 Þ; rf L ðx0 Þ þ i¼1
ð11Þ
ð12Þ
i¼1
which also implies rf L ðx0 Þ þ rf U ðx0 Þ þ
m X
i rgi ðx0 Þ ¼ 0; l
ð13Þ
i¼1
i ¼ 2li for i = 1, . . . , m. where l Theorem 6.7. Suppose that the real-valued constraint functions gi, i = 1, . . . , m, of problem (IVOP2) satisfy the KKT assumptions at x* and the interval-valued objective function f : Rn ! I is LU-convex and continuously H-differentiable at x*. If there exist (Lagrange) multipliers 0 6 li 2 R, i = 1, . . . , m, such that Pm (i) rf ðx Þ þ i¼1 li rgi ðx Þ ¼ 0; (ii) ligi(x*) = 0 for all i = 1, . . . , m, then x* is a type-I and type-II solution of problem (IVOP2).
H.-C. Wu / European Journal of Operational Research 176 (2007) 46–59
Proof. The result follows from Eq. (13) and Theorem 6.2 by taking kL = 1 = kU.
59
h
Although we just present one kind of KKT conditions in the interval-valued form as described in Theorem 6.7, in fact, we still can derive many kinds of KKT conditions in the interval-valued form, under the same assumptions as described in many of the above theorems, by using Eqs. (12) and (13). We omit the details in order to save space.
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