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Some results on the upper bound of optimal values in interval convex quadratic programming
Journal of Computational and Applied Mathematics 302 (2016) 38–49
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Some results on the upper bound of optimal values in interval convex quadratic programming Wei Li a , Mengxue Xia a , Haohao Li b,∗ a
Institute of Operational Research & Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, China
b
School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
article
info
Article history: Received 11 May 2015 Received in revised form 10 January 2016 MSC: 65G40 Keywords: Interval quadratic programming Optimal values range Duality gap
abstract One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. For minimized problem with equality constraint, computing the upper bound of the optimal values is known to be NP-hard. One kind of the effective methods for computing the upper bound of interval quadratic programming is so called dual method, based on the dual property of the problem. To obtain the exact upper bound, the dual methods require that the duality gap is zero. However, it is not an easy task to check whether this condition is true when the data may vary inside intervals. In this paper, we first present an easy and efficient method for checking the zero duality gap. Then some relations between the exact upper bound and the optimal value of the dual model considered in dual methods are discussed in detail. We also report some numerical results and remarks to give an insight into the dual method’s behavior. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The interval systems and interval mathematical programming (IvMP)1 have been studied by many authors, see, e.g., [1–15] and a survey paper [16], since intervals naturally appear in many situations when handle inexact data. One of frequent problems in IvMP is to compute the range of optimal values [17,4,18–23]. Some authors studied the problem of computing the range of optimal values of interval quadratic programs (IvQP) [18,19,21,24]. It is known that finding the upper bound of the optimal values in IvQP is a computationally hard problem when the constraint includes interval linear equalities. There have been developed diverse methods for computing the range of the optimal values in IvQP. Liu [18] and Li [19] described some methods to compute the lower and upper bounds of IvQP with inequality and nonnegative constraints. Hladík [21] studied the more general IvQP and proposed an effective method to compute the bounds of IvQP with both equality and inequality constraints. For computing the upper bound, these methods described in [18,19,21] are based on the dual problem of IvQP (dual method for short), under the condition that the zero duality gap of a pair of primal and dual IvQP is specified. Recently, Li et al. [24] proposed a new method to compute the upper bound of optimal values of IvQP. In this method, only primal program is taken into consideration (primal method for short). The dual problem is not required and thus the condition that the duality gap is zero is also removed. Just like the primal and dual simplex algorithm for linear programming, both the primal method and the dual method for computing the upper bound of IvQP are very useful and they are suitable for different situations respectively, depending
∗
Corresponding author. E-mail address: [email protected] (H. Li).
1 We use abbreviation IvMP, instead of IMP, for interval mathematical programming, to avoid confusion with the abbreviation for integer mathematical programming. http://dx.doi.org/10.1016/j.cam.2016.01.044 0377-0427/© 2016 Elsevier B.V. All rights reserved.
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on the structure of IvQP. To obtain the exact upper bound f¯ of IvQP, the dual methods first compute the optimal value ψ of the Dorn dual, then one has f¯ = ψ if the zero duality gap is assured, otherwise one can only obtain f¯ ≥ ψ . When using dual method, the difficulty is that it is not an easy task to check whether there is a zero duality gap. In this paper, we first present an easy and efficient method for checking the zero duality gap. Then some relations between f¯ and ψ in dual methods are discussed in detail to give an insight into the dual method’s behavior. 2. Preliminaries Following notations from [8], an interval matrix is defined as A = [A, A] = {A ∈ Rm×n |A ≤ A ≤ A}, where A, A ∈ Rm×n , A ≤ A, and ‘‘≤’’ is understood componentwise. By Ac =
1 2
(A + A), A∆ =
1 2
(A − A),
we denote the center and the radius of A, respectively. Then A = [Ac − A∆ , Ac + A∆ ]. An interval vector b = [b, b] = {b ∈ Rm |b ≤ b ≤ b} is understood as one-column interval matrix. Let {±1}m be the set of all {−1, 1} m-dimensional vectors, i.e.
{±1}m = {y ∈ Rm | | y |= e}, where e = (1, . . . , 1)T is the m-dimensional vector of all 1, s and the absolute value of a matrix A = (aij ) is defined by |A| = (|aij |). For a given y ∈ {±1}m , let Ty = diag (y1 , . . . , ym ) denote the corresponding diagonal matrix. For each x ∈ Rn , we define its sign vector sgn x by
(sgn x)i =
1
−1
if xi ≥ 0, if xi < 0,
where i = 1, . . . , n. Then we have |x| = Tz x, where z = sgn x ∈ {±1}n . Given an interval matrix A = [Ac − A∆ , Ac + A∆ ], for each y ∈ {±1}m and z ∈ {±1}n , we define matrices Ayz = Ac − Ty A∆ Tz . Similarly, for an interval vector b = [bc − b∆ , bc + b∆ ] and for each y ∈ {±1}m , we define vectors by = bc + Ty b∆ . Let A ∈ Rm×n , B ∈ Rk×n , b ∈ Rm , c ∈ Rn , d ∈ Rk and Q ∈ Rn×n be given, consider the quadratic programming problem min
1
xT Qx + c T x subject to Ax ≤ b, Bx = d, x ≥ 0,
2
where Q is positive semidefinite. Briefly, we rewrite the problem as
Min
1 2
xT Qx + c T x|Ax ≤ b, Bx = d, x ≥ 0 .
(1)
The Dorn dual problem [25,26] of the quadratic program (1) is
1 Max − uT Qu − bT v − dT w|Qu + AT v + BT w + c ≥ 0, v ≥ 0 . 2
(2)
Let f (A, B, b, c , d, Q ) = inf
1 2
xT Qx + c T x|Ax ≤ b, Bx = d, x ≥ 0
and
1 g (A, B, b, c , d, Q ) = sup − uT Qu − bT v − dT w|Qu + AT v + BT w + c ≥ 0, v ≥ 0 2 denote the optimal values of (1) and (2), respectively.
(3)
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Lemma 2.1 ([25]). (i) If x = x0 is an optimal solution to problem (1) then an optimal solution (u, v, w)T = (u0 , v0 , w0 )T exists to problem (2). (ii) Conversely, if an optimal solution (u, v, w)T = (u0 , v0 , w0 )T to problem (2) exists then an optimal solution x = x0 to problem (1) also exists. In either case, f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ). From the weak and strong duality conditions we can get the following theorem, which describes the characteristics of the duality gap in the quadratic program (1). Theorem 2.1. The duality gap of the quadratic program is zero as long as either the primal or the dual problem is feasible. Proof. We consider separately three cases below. (a) Both problems (1) and (2) are feasible. Clearly, the duality gap is zero in this case. (b) Problem (1) is feasible and problem (2) is infeasible. If the problem (2) is infeasible, that is g (A, B, b, c , d, Q ) = −∞, we can get f (A, B, b, c , d, Q ) = −∞, since otherwise, we have −∞ < f (A, B, b, c , d, Q ) < ∞, then the primal problem has optimal solutions, and from Lemma 2.1 we know the dual problem (2) also has optimal solutions, which is a contradiction. Thus we have f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ) = −∞, the duality gap is zero. (c) Problem (1) is infeasible and problem (2) is feasible. Analogous to the case (b), we are easy to obtain f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ) = ∞, the duality gap is zero. The theorem is proved combining the discussion above.
The set of all m-by-n interval matrices will be denoted by IRm×n and the set of all m-dimensional interval vectors by IRm . Given A ∈ IRm×n , B ∈ IRk×n , b ∈ IRm , c ∈ IRn , d ∈ IRk and Q ∈ IRn×n , the interval convex quadratic program
Min
1 2
xT Qx + cT x|Ax ≤ b, Bx = d, x ≥ 0
(4)
is the family of convex quadratic programs (1) with data satisfying A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q, where Q is positive semidefinite for all Q ∈ Q. The lower and upper bounds of the optimal values are respectively defined as f (A, B, b, c, d, Q) = inf{f (A, B, b, c , d, Q )|A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q}, f (A, B, b, c, d, Q) = sup{f (A, B, b, c , d, Q )|A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q}. For given A, B, b, c, d, Q denote f = f (A, B, b, c, d, Q), f = f (A, B, b, c, d, Q). The Dorn duality gap for interval quadratic program (4) is zero means that for each scenario of problem (4) satisfies the zero duality gap. The following lemmas will be used in the proof of our results. Lemma 2.2 ([8]). The system Ax ≤ b is solvable if and only if for each p ≥ 0 satisfies AT p = 0 then bT p ≥ 0. Lemma 2.3 ([8]). The system Ax ≤ b is solvable for each A ∈ A, b ∈ b if and only if the system Ax1 − Ax2 ≤ b,
x1 , x2 ≥ 0
is solvable. Lemma 2.4 ([24]). We have
f = inf
1 2
x Q x + c x|Ax ≤ b, Bx ≤ d, Bx ≥ d, x ≥ 0 T
T
and f = f (A, B, b, c , d, Q ) = sup f (A, Bye , b, c , dy , Q ). y∈{±1}k
Let us denote
ψ = sup
1 T − uT Q u − bT v − dTc w + dT∆ |w||Q u + A v + BTc w + BT∆ |w| + c ≥ 0, v ≥ 0 . 2
Lemma 2.5 ([24]). If ψ = ∞, then f = ∞.
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Consider the interval quadratic program
Min
1 2
xT Q x + c x|Ax ≤ b, Bx = d, x ≥ 0 . T
(5)
Lemma 2.6 ([21]). If the Dorn duality gap of problem (5) is zero, then we have f = ψ. 3. A sufficient condition for zero Dorn duality gap As we known, zero duality gap plays an important role in the dual methods for computing the upper bound of IvQP. So far, however, it is not an easy task to verify whether this condition is satisfied when the data vary inside intervals. For convenient practice of theory and calculation, we give a sufficient condition for the Dorn duality gap of problem (5). It will be also used to discuss some properties of the dual methods in the next section. Theorem 3.1. If f is finite, then the Dorn duality gap of problem (5) is zero. Proof. If f is finite, then the quadratic program
Min
1 2
x Q x + c x|Ax ≤ b, Bx ≤ d, Bx ≥ d, x ≥ 0 T
T
(6)
has a finite optimal value by Lemma 2.4. From the strong duality theorem Lemma 2.1 we know that an optimal solution (u, v1 , v2 , v3 )T = (u∗ , v1∗ , v2∗ , v3∗ )T exists to its Dorn dual problem
1
T
T
T
Max − u Q u − b v1 − d v2 + d v3 |Q u + A v1 + B v2 − B v3 + c ≥ 0, vi ≥ 0, i = 1, 2, 3 . 2 T
T
T
T
Thus the system T
Q u + AT v1 + BT v2 − B v3 + c ≥ 0,
vi ≥ 0, i = 1, 2, 3
(7)
is solvable and hence from the Farkas Lemma 2.2 we get
−Q T −A (∀p ≥ 0) −B
B Let p = (
,
,
,
0 −E m 0 0
0 0 −Ek 0
0 0 T p = 0 ⇒ (c , 0, 0, 0)p ≥ 0 . 0 −E k
) then p1 , p2 , p3 , p4 ≥ 0 and Q T = Q due to the symmetry of Q, thus (8) can be rewritten as Qp = 0 1 Ap + p = 0 (∀p1 , p2 , p3 , p4 ≥ 0) Bp1 + p2 = 0 ⇒ c T p1 ≥ 0 . 1 3 Bp1 − p4 = 0 pT1
pT2
pT3
(8)
pT4 T ,
(9)
Now, we first prove the following results
Q p1 = 0 Ap1 + p2 = 0 T (∀p1 , p2 , p3 , p4 ≥ 0) Bp1 + p3 = 0 ⇒ c p1 ≥ 0 . Bp1 − p4 = 0
(10)
Indeed, if for each p1 , p2 , p3 , p4 ≥ 0 there hold
Q p = 0, 1 Ap1 + p2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0,
(11)
obviously we have Q p1 ≤ Q p1 = 0 due to the nonnegativity of vector p1 , and hence we get pT1 Q p1 ≤ 0. On the other hand, for each p1 there holds pT1 Q p1 ≥ 0, since Q is positive semidefinite, thus pT1 Q p1 = 0. Further, there exists a real matrix F
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such that Q = FTF, due to the positive semidefiniteness of the matrix Q . Thus, pT1 Q p1 = 0 leads to (Fp1 )T (Fp1 ) = 0 and hence we have Fp1 = 0. So we have Q p = F T Fp = 0. Then from (11) we can get
Q p = 0, 1 (A + 2A∆ )p1 + p2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0,
(12)
let p′2 = 2A∆ p1 + p2 , obviously, p′2 ≥ 0, then (12) can be rewritten as
Q p = 0, 1 Ap1 + p′2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0, T
By (9), we obtain c T p1 ≥ 0 and thus c p1 ≥ c T p1 ≥ 0, which proves (10). Similarly, the condition (10) can be rewritten as
T −Q −A (∀p ≥ 0) −B
0 −E m 0 0
B
0 0 −E k 0
0 0 p = 0 ⇒ (c T , 0, 0, 0)p ≥ 0 . 0 −E k
(13)
Hence the system T
T
Q u + A v + BT w 1 − B w 2 + c ≥ 0,
v, w1 , w2 ≥ 0
(14)
is solvable by Farkas Lemma 2.2. From Lemma 2.3, we can get for each B ∈ B the following system T
Q u + A v + BT w + c ≥ 0,
v≥0
(15)
is solvable. In fact, the system (15) can be rewritten as
T
Q 0
A Em
BT 0
u −c v ≥ . 0 w
(16)
Let
Q 0
H=
T
A Em
BT 0
,
H =
Q 0
T
BT 0
A Em
,
h=
−c 0
u
,
x=
v , w
then the solvability of system (16) is equivalent to Hx ≥ h. By Lemma 2.3 we have the system Hx ≥ h is solvable for each H ∈ H if and only if the following system
T T T T Q u1 + A v 1 + BT w1 − (Q u2 + A v 2 + B w 2 ) ≥ −c , v 1 − v 2 ≥ 0, 1 2 1 2 1 2 u ,u ,v ,v ,w ,w ≥ 0 is solvable. Put u = u1 − u2 , v = v 1 − v 2 , we obtain system (14). Meanwhile, for each B ∈ B, d ∈ d, consider the scenario of interval quadratic program (5)
Min
1 2
xT Q x + c x|Ax ≤ b, Bx = d, x ≥ 0 T
(17)
and its Dorn dual problem
1 T Max − uT Q u − bT v − dT w|Q u + A v + BT w + c ≥ 0, v ≥ 0 . 2
(18)
For all B ∈ B, the problem (18) is feasible due to the solvability of system (15), hence the zero duality gap is ensured by Theorem 2.1. Then by Theorem 3.1, together with Lemma 2.5 we have
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Theorem 3.2. Let f be finite or let the ψ be positively infinite. Then f = ψ . If we put Q = 0, A = 0 and b = 0 in (4), then we get a simple IvLP Min{cT x|Bx = d, x ≥ 0}.
(19)
As a special case of (4), from Lemma 2.4 we obtain f (B, c, d) = inf {c T x|Bx ≤ d, Bx ≥ d, x ≥ 0} and f (B, c, d) = sup f (Bye , c , dy ), y∈{±1}k
which has also been discussed in [8]. Analogous to Theorem 3.1, we can get a similar sufficient condition to interval linear program (19). Corollary 3.1. If f (B, c, d) is finite, then the duality gap of the following IvLP Min{c x|Bx = d, x ≥ 0} T
(20)
is zero. Remark. The conditions of Theorem 3.1 and Corollary 3.1 are sufficient but not necessary. See Examples 1 and 2 in Section 5. 4. The relationship between f and ψ From Section 3 of Hladík [21] and Section 4 of Li [24], we can see there exists some relations between f and ψ , and also we know the values of f and ψ are equal under the zero duality gap. In order to have a clear understanding of the relations between two formulas of the upper bound, we will make a deeper discussion in this section. Theorem 4.1. We have
f
if ψ = −∞; if ψ is finite; if ψ = ∞.
∈ {−∞, ∞}, > −∞, = ∞,
(a) (b) (c)
(21)
Proof. Part (21)(a). If ψ = −∞, which means the problem
Max
1
− u Qu − b v − 2
T
T
dTc
T
w + d∆ |w||Q u + A v + T
BTc
w + B∆ |w| + c ≥ 0, v ≥ 0 T
(22)
is infeasible. Now, we prove that the system T
Q u + A v + BT w + c ≥ 0,
v≥0
(23)
is not solvable for each B ∈ B. In fact, if for some B ∈ B (23) is solvable and note that we can get the system T
Q u + A v + BTc w + BT∆ |w| + c ≥ 0,
BTc
w + B∆ |w| ≥ B w , then from (23) T
T
v≥0
is solvable, which is a contradiction to the infeasibility of problem (22). Hence, for each B ∈ B, d ∈ d the Dorn dual problem (18) is infeasible, which means that each primal problem (17) is either infeasible or unbounded, so that f (A, B, b, c , d.Q ) ∈ {−∞, ∞} for each B ∈ B, d ∈ d. By Lemma 2.4 we have f = sup {f (A, B, b, c , d, Q )|B ∈ B, d ∈ d}, consequently, f ∈ {−∞, ∞}. Part (21)(b). If ψ is finite, then the feasible region of (22) is not empty. Let (u0 , v0 , w0 )T be an arbitrary feasible solution to problem (22), then it satisfies T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0.
Let y0 = −sgn w0 , since y0 ∈ {±1} and from (24) we obtain k
T
T
Q u0 + A v0 + (By0 e )T w0 + c = Q u0 + A v0 + BTc w0 − BT∆ Ty0 w0 + c T
= Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
(24)
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W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
which means (u0 , v0 , w0 )T is a feasible solution to problem
1 T T T T T Max − u Q u − b v − d w|Q u + A v + By0 e w + c ≥ 0, v ≥ 0 . 2
Thus, for each d ∈ d, from (3) there holds g (A, By0 e , b, c , d, Q ) > −∞, then from weak duality theorem [21,24,25] we have f (A, By0 e , b, c , d, Q ) ≥ g (A, By0 e , b, c , d, Q ) > −∞. Hence, from the definition of f we get f > −∞. Part (21)(c). See Lemma 2.5, which has been proved in [24].
The following lemma describes the relations between ψ and the interval quadratic program (5) under the assumption that ψ is finite, which will be used to obtain the next theorem. Lemma 4.1. If ψ is finite, then
ψ = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d}. Proof. Denote
α = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d}, thus we only need to prove ψ = α . (i) First, we prove α ≤ ψ by showing that f (A, B, b, c , d, Q ) ≤ ψ
(25)
holds for each B ∈ B, d ∈ d. Note that for each B ∈ B, d ∈ d the value of f (A, B, b, c , d, Q ) < ∞ can be divided into two cases, they are −∞ or finite value. We discuss them separately. If f (A, B, b, c , d, Q ) = −∞, then (25) holds clearly. If f (A, B, b, c , d, Q ) is finite. Then from Lemma 2.1 we have 1 f (A, B, b, c , d, Q ) = − (u∗ )T Q u∗ − bT v ∗ − dT w ∗ , 2 where (u∗ , v ∗ , w ∗ )T is an optimal solution of the Dorn dual problem (18). Since BTc w ∗ + BT∆ |w ∗ | ≥ BT w ∗ , then we know (u∗ , v ∗ , w∗ )T satisfies T
Q u∗ + A v ∗ + BTc w ∗ + BT∆ |w ∗ | + c ≥ 0, v ∗ ≥ 0, which means (u∗ , v ∗ , w ∗ )T is an feasible solution of the problem
1 T T T T T T Max − u Q u − b v − d w|Q u + A v + Bc w + B∆ |w| + c ≥ 0, v ≥ 0 . 2
(26)
Let T
S = {(u, v, w)T |Q u + A v + BTc w + BT∆ |w| + c ≥ 0, v ≥ 0}. Thus, 1 f (A, B, b, c , d, Q ) = − (u∗ )T Q u∗ − bT v ∗ − dT w ∗ 2
1 ≤ sup − uT Q u − bT v − dT w|(u, v, w)T ∈ S 2
(27)
meanwhile, as dT w ≥ dTc w − dT∆ |w| for each w , so we have 1
1
2
2
− uT Q u − bT v − dT w ≤ − uT Q u − bT v − dTc w + dT∆ |w|, which implies
sup
1 1 − uT Q u − bT v − dT w|(u, v, w)T ∈ S ≤ sup − uT Q u − bT v − dTc w + dT∆ |w||(u, v, w)T ∈ S . 2
2
(28)
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From (27) and (28) we have
1 T T T T T f (A, B, b, c , d, Q ) ≤ sup − u Q u − b v − dc w + d∆ |w||(u, v, w) ∈ S 2
= ψ. (ii) Second, we prove ψ ≤ α . Let (ˆu, vˆ , w) ˆ T be an optimal solution of the problem (22). Hence, we have 1
ˆ + dT∆ |w| ˆ ψ = − uˆ T Q uˆ − bT vˆ − dTc w
(29)
2
and T
Q uˆ + A vˆ + BTc w ˆ + BT∆ |w| ˆ + c ≥ 0,
vˆ ≥ 0,
which implies T
Q uˆ + A vˆ + (Byˆ e )T w ˆ + c ≥ 0,
vˆ ≥ 0,
where yˆ = −sgn w ˆ . Thus, (ˆu, vˆ , w) ˆ T is an feasible solution of problem
1
T
Max − u Q u − b v − (dyˆ ) w|Q u + A v + (Byˆ e ) w + c ≥ 0, v ≥ 0 . 2 T
T
T
T
From (29), (30) and (3) we have 1
ˆ + dT∆ |w| ˆ ψ = − uˆ T Q uˆ − bT vˆ − dTc w 2 1
ˆ = − uˆ T Q uˆ − bT vˆ − (dyˆ )T w 2 1 T T T T T ≤ sup − u Q u − b v − (dyˆ ) w|Q u + A v + (Byˆ e ) w + c ≥ 0, v ≥ 0 2
= g (A, Byˆ e , b, c , dyˆ , Q ). Thus g (A, Byˆ e , b, c , dyˆ , Q ) = ∞ or finite, since the ψ is finite. If g (A, Byˆ e , b, c , dyˆ , Q ) = ∞, which implies the problem (30) is unbounded. Let T
F = {(u, v, w)T |Q u + A v + (Byˆ e )T w + c ≥ 0, v ≥ 0}, then for each M > 0, there exists (u0 , v0 , w0 )T ∈ F such that 1
− uT0 Q u0 − bT v0 − (dyˆ )T w0 ≥ M . 2
Note that BTc w0 + BT∆ |w0 | ≥ (Byˆ e )T w0 and hence T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0,
which means (u0 , v0 , w0 )T is an feasible solution of problem (22). Thus, 1
ψ ≥ − uT0 Q u0 − bT v0 − dTc w0 + dT∆ |w0 | 2 1
≥ − uT0 Q u0 − bT v0 − (dyˆ )T w0 2
≥ M,
which is a contradiction to the finiteness of value ψ . If g (A, Byˆ e , b, c , dyˆ , Q ) is finite. From Lemma 2.1 we have f (A, Byˆ e , b, c , dyˆ , Q ) = g (A, Byˆ e , b, c , dyˆ , Q ) < ∞. Thus, f (A, Byˆ e , b, c , dyˆ , Q ) is finite and there holds
ψ ≤ f (A, Byˆ e , b, c , dyˆ , Q ). From the definition of α we obtain f (A, Byˆ e , b, c , dyˆ , Q ) ≤ α , since Byˆ e ∈ B, dyˆ ∈ d. Hence, we can get ψ ≤ α . Combining the results of (i) and (ii) we can get ψ = α .
(30)
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Analogous to Theorem 4.1, the following result describes the relations between ψ and f under the different values of f . Theorem 4.2. We have
−∞, ψ = f,
(a) (b) (c)
if f = −∞; if f is finite; if f = ∞.
cannot determine,
(31)
Proof. Part (31)(a). Since f = −∞, then f (A, B, b, c , d, Q ) = −∞ for each B ∈ B, d ∈ d. From the weak duality theorem, we have g (A, B, b, c , d, Q ) = −∞, that is, for each B ∈ B the inequality T
Q u + A v + BT w + c ≥ 0,
v≥0
is not solvable. Thus, we can get the following system T
Q u + A v + BTc w + BT∆ |w| + c ≥ 0,
v≥0
(32)
is not solvable. To prove this, assume that there exists (u0 , v0 , w0 )T such that T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0,
which implies T
Q u0 + A v0 + (Bye )T w0 + c ≥ 0,
v0 ≥ 0,
where y = −sgn w0 , a contradiction. The unsolvability of system (32) implies the infeasibility of problem (22), then ψ = −∞. Part (31)(b). Apparently ψ is finite, since the possibilities ψ = −∞ and ψ = ∞ are precluded by (21)(a) and (c) in Theorem 4.1. Meanwhile, Lemma 4.1 together with Lemma 2.4 gives
ψ = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d} = max {f (A, B, b, c , d, Q )|B ∈ B, d ∈ d} = f (A, B, b, c , d, Q ) = f . Part (31)(c). If f = ∞ we cannot determine the feature of ψ , which means all these three different cases for ψ may happen (see Examples 2–4 in Section 5). 5. Illustrative examples and remarks In this section, we give some illustrative examples for Theorems 4.1 and 4.2, and make some remarks. We refer the reader to [18,19,21,24] and the references therein for more examples, which can help us to understand Theorems 4.1 and 4.2. Example 1. Consider the interval linear program min [−5, 1]x1 + [−4, 0]x2 s.t. [−1, 2]x1 − 2x2 = [−1, 2],
− 3x1 + [−1, 4]x2 = [1, 1.4], x1 , x2 ≥ 0. The corresponding interval matrices and vectors are
c=
[−5, 1] , [−4, 0]
B=
[−1, 2] −3
−2 , [−1, 4]
d=
[−1, 2] . [1, 1.4]
According to model (20) we consider the IvLP
Min
1
[−1, 2] 0 x| −3
−2 [−1, 2] x= ,x ≥ 0 [−1, 4] [1, 1.4]
(33)
and its dual problem
Max [−1, 2]
[−1, 2] [1, 1.4] y| −3
−2 [−1, 4]
T y≤
1 0
.
(34)
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47
Now we check that whether the duality gap of problem (33) is zero or not. The feasible region of problem (34) is
[−1, 2] −3
T −2 1 y≤ . [−1, 4] 0
Clearly, for each matrix
[−1, 2] C ∈ −3
−2 [−1, 4]
the vector y = (0, 0)T satisfies CT y ≤
1 , 0
which implies that each scenario of dual problem (34) is feasible, then by Theorem 2.1 we know the duality gap of program (33) is zero. However, the lower bound of the optimal values can be determined by linear program min − 5x1 − 4x2 s.t. − x1 − 2x2 ≤ 2, 2x1 − 2x2 ≥ −1,
− 3x1 − x2 ≤ 1.4, − 3x1 + 4x2 ≥ 1, x1 , x2 ≥ 0. It can be shown that f = −∞. This example shows that the converse of Corollary 3.1 is not true.
Example 2. Consider the interval quadratic program min [2, 3]x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. − 2x1 + [−2, 1]x2 ≤ [1, 4],
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. According to model (5) we consider the following IvQP min 3x21 +
1 2
x22 − 2x1 x2 + x1 − 3x2
s.t. − 2x1 + x2 ≤ 1,
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. Clearly, the Dorn duality gap of (35) is zero, since the matrix Q =
(35)
6 −2
−2 1
is positive definite (see Section 3.2 in
Hladik [21]). However, the lower bound of the optimal values can be determined by quadratic program min 2x21 +
1 2
x22 − 2x1 x2 − 5x1 − 4x2
s.t. − 2x1 − 2x2 ≤ 4, 4x1 − 2x2 ≤ 2, 5x1 − 2x2 ≥ 2, 6x1 − 4x2 ≤ 1.4, 6x1 − 3x2 ≥ 1.4, x1 , x2 ≥ 0. It is easy to show that f = −∞.
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W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
Now put y0 = (1, −1)T , the f (A, By0 e , b, c , dy0 , Q ) can be calculated by solving the following problem min 3x21 +
1 2
x22 − 2x1 x2 + x1 − 3x2
s.t. − 2x1 + x2 ≤ 1, 4x1 − 2x2 = 2,
(36)
6x1 − 3x2 = 1.4, x1 , x2 ≥ 0. It can be easily verified that the problem (36) is infeasible. Thus, f (A, By0 e , b, c , dy0 , Q ) = ∞ and hence we can get f = ∞ by Lemma 2.4. Thus by Lemma 2.6 we have ψ = f = ∞, since the duality gap is zero. This example shows that the converse of Theorem 3.1 is not true, and confirm the possibility of f = ∞ and ψ = ∞. Example 3. Consider the interval quadratic program min 2x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. − 2x1 + [−2, 1]x2 ≤ [1, 4],
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. This is the Example 2 in [24]. It can be shown that the f = ∞. Whereas ψ = 40.6848, which is a finite value. This example shows that the possibility that f = ∞ but ψ is finite. Example 4. Consider the interval quadratic program min 2x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. [1, 2]x1 + [−2, −1]x2 ≤ [−1, 4], 6x1 − 3x2 = [1, 1.4], x1 , x2 ≥ 0. Consider the quadratic program 1 2 x2 − 2x1 x2 + x1 − 3x2 2 s.t. 2x1 − x2 ≤ −1,
min 2x21 +
6x1 − 3x2 = 1.4, x1 , x2 ≥ 0, which is the corresponding quadratic program for computing f (A, By0 e , b, c , dy0 , Q ) with y0 = 1. We are easy to check that there is no feasible solution to this problem, then f (A, By0 e , b, c , dy0 , Q ) = ∞ and hence f = ∞ by Lemma 2.4. Now, by the method described in [21], we can compute the value of ψ by considering two problems f1 = sup − 2u21 −
1 2
u22 + 2u1 u2 + v − w
s.t. 4u1 − 2u2 + 2v + 6w + 1 ≥ 0,
− 2u1 + u2 − v − 3w − 3 ≥ 0, v, w ≥ 0. and f2 = sup − 2u21 −
1 2
u22 + 2u1 u2 + v − 1.4w
s.t. 4u1 − 2u2 + 2v + 6w + 1 ≥ 0,
− 2u1 + u2 − v − 3w − 3 ≥ 0, v, −w ≥ 0. It can be shown that the two problems are both infeasible, then ψ = −∞.
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49
This example shows the possibility of f = ∞ but ψ = −∞. Moreover, the zero Dorn duality gap in this example is not guaranteed due to f ̸= ψ , then from Theorem 3.1 we can get f ∈ {−∞, ∞}. So far, the only possibility to compute the upper bound in general was to reduce the problem to solving 2m subproblems, where m is the number of the equality constraints, see, [18,19,21,24]. The contribution of these results shows that the problem can be handled by finite means. It is known that even determining the upper bound of interval linear program – a special case of interval quadratic program when the coefficient matrix of the quadratic term Q = 0 – is NP-hard [10]. So, the computing difficulty of primal or dual methods does not lie with the inadequateness of the proposed methods, but that it is inherently present in the problem itself. An interesting further research direction is to derive approximations of the upper bound. In fact, the recently published paper [23] presented approximation methods for the upper bound of the equality constrained interval linear program. It is interesting to establish approximation methods for the upper bound of the interval quadratic program. Acknowledgments The authors would like to thank anonymous referees for their comments and suggestions that helped to improve the paper. The authors were partially supported by the NSF of Zhejiang Province (Grant No. LY14A010028), NNSF of China (Grant Nos. 11526184, 71471051) and the Joint Key Grant of NNSF of China and Zhejiang Province (No. U1509217). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990. H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European J. Oper. Res. 48 (1990) 219–225. J.W. Chinneck, K. Ramadan, Linear programming with interval coefficients, J. Oper. Res. Soc. 51 (2) (2000) 209–220. S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity, Reliab. Comput. 8 (5) (2002) 321–418. X.Y. Wu, G.H. Huang, L. Liu, J.B. Li, An interval nonlinear program for the planning of waste management systems with economies-of-scale effects a case study for the region of Hamilton, Ontario, Canada, European J. Oper. Res. 171 (2) (2006) 349–372. M. Inuiguchi, J. Ramík, T. Tanino, M. Vlach, Satisficing solutions and duality in interval and fuzzy linear programming, Fuzzy Sets and Systems 135 (1) (2003) 151–177. M. Fiedler, J. Nedoma, J. Ramik, J. Rohn, K. Zimmermann, Linear Optimization Problems With Inexact Data, Springer-Verlag, New York, 2006. O.A. Prokopyev, S. Butenko, A.C. Trapp, Checking solvability of systems of interval linear equations and inequalities via mixed integer programming, European J. Oper. Res. 199 (1) (2009) 117–121. V. Gabrel, C. Murat, N. Remli, Linear programming with interval right hand sides, Int. Trans. Oper. Res. 17 (3) (2010) 397–408. W. Li, H. Wang, Q. Wang, Localized solutions to interval linear equations, J. Comput. Appl. Math. 238 (15) (2013) 29–38. M. Hladík, How to determine basis stability in interval linear programming, Optim. Lett. 8 (1) (2014) 375–389. W. Li, J. Luo, Q. Wang, Y. Li, Checking weak optimality of the solution to linear programming with interval right-hand side, Optim. Lett. 8 (4) (2014) 1287–1299. J. Luo, W. Li, Strong optimal solutions of interval linear programming, Linear Algebra Appl. 439 (2013) 2479–2493. W. Li, X. Liu, H. Li, Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions, Optim. Methods Softw. 30 (3) (2015) 516–530. M. Hladík, Interval Linear Programming: A Survey, in: Zoltan Adam Mann (Ed.), Linear Programming New Frontiers, Nova Science Publishers Inc., New York, 2012, pp. 1–46. F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res. 81 (1998) 51–62. S.T. Liu, R.T. Wang, A numerical solution method to interval quadratic programming, Appl. Math. Comput. 189 (2007) 1274–1281. W. Li, X. Tian, Numerical solution method for general interval quadratic programming, Appl. Math. Comput. 202 (2008) 589–595. M. Hladík, Optimal value range in interval linear programming, Fuzzy Optim. Decis. Mak. 8 (3) (2009) 283–294. M. Hladík, Optimal value bounds in nonlinear programming with interval data, Top 19 (2011) 93–106. M. Hladík, Weak and strong solvability of interval linear systems of equations and inequalities, Linear Algebra Appl. 438 (2013) 4156–4165. M. Hladík, On approximation of the best case optimal value in interval linear programming, Optim. Lett. 8 (2014) 1985–1997. W. Li, M. Xia, H. Li, New method for computing the upper bound of optimal value in interval quadratic program, J. Comput. Appl. Math. 288 (2015) 70–80. W.S. Dorn, Duality in quadratic programming, Quart. Appl. Math. 18 (2) (1960) 155–162. M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming, Theory and Algorithms, second ed., Wiley, New York, 1993.
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Some results on the upper bound of optimal values in interval convex quadratic programming Wei Li a , Mengxue Xia a , Haohao Li b,∗ a
Institute of Operational Research & Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, China
b
School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
article
info
Article history: Received 11 May 2015 Received in revised form 10 January 2016 MSC: 65G40 Keywords: Interval quadratic programming Optimal values range Duality gap
abstract One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. For minimized problem with equality constraint, computing the upper bound of the optimal values is known to be NP-hard. One kind of the effective methods for computing the upper bound of interval quadratic programming is so called dual method, based on the dual property of the problem. To obtain the exact upper bound, the dual methods require that the duality gap is zero. However, it is not an easy task to check whether this condition is true when the data may vary inside intervals. In this paper, we first present an easy and efficient method for checking the zero duality gap. Then some relations between the exact upper bound and the optimal value of the dual model considered in dual methods are discussed in detail. We also report some numerical results and remarks to give an insight into the dual method’s behavior. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The interval systems and interval mathematical programming (IvMP)1 have been studied by many authors, see, e.g., [1–15] and a survey paper [16], since intervals naturally appear in many situations when handle inexact data. One of frequent problems in IvMP is to compute the range of optimal values [17,4,18–23]. Some authors studied the problem of computing the range of optimal values of interval quadratic programs (IvQP) [18,19,21,24]. It is known that finding the upper bound of the optimal values in IvQP is a computationally hard problem when the constraint includes interval linear equalities. There have been developed diverse methods for computing the range of the optimal values in IvQP. Liu [18] and Li [19] described some methods to compute the lower and upper bounds of IvQP with inequality and nonnegative constraints. Hladík [21] studied the more general IvQP and proposed an effective method to compute the bounds of IvQP with both equality and inequality constraints. For computing the upper bound, these methods described in [18,19,21] are based on the dual problem of IvQP (dual method for short), under the condition that the zero duality gap of a pair of primal and dual IvQP is specified. Recently, Li et al. [24] proposed a new method to compute the upper bound of optimal values of IvQP. In this method, only primal program is taken into consideration (primal method for short). The dual problem is not required and thus the condition that the duality gap is zero is also removed. Just like the primal and dual simplex algorithm for linear programming, both the primal method and the dual method for computing the upper bound of IvQP are very useful and they are suitable for different situations respectively, depending
∗
Corresponding author. E-mail address: [email protected] (H. Li).
1 We use abbreviation IvMP, instead of IMP, for interval mathematical programming, to avoid confusion with the abbreviation for integer mathematical programming. http://dx.doi.org/10.1016/j.cam.2016.01.044 0377-0427/© 2016 Elsevier B.V. All rights reserved.
W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
39
on the structure of IvQP. To obtain the exact upper bound f¯ of IvQP, the dual methods first compute the optimal value ψ of the Dorn dual, then one has f¯ = ψ if the zero duality gap is assured, otherwise one can only obtain f¯ ≥ ψ . When using dual method, the difficulty is that it is not an easy task to check whether there is a zero duality gap. In this paper, we first present an easy and efficient method for checking the zero duality gap. Then some relations between f¯ and ψ in dual methods are discussed in detail to give an insight into the dual method’s behavior. 2. Preliminaries Following notations from [8], an interval matrix is defined as A = [A, A] = {A ∈ Rm×n |A ≤ A ≤ A}, where A, A ∈ Rm×n , A ≤ A, and ‘‘≤’’ is understood componentwise. By Ac =
1 2
(A + A), A∆ =
1 2
(A − A),
we denote the center and the radius of A, respectively. Then A = [Ac − A∆ , Ac + A∆ ]. An interval vector b = [b, b] = {b ∈ Rm |b ≤ b ≤ b} is understood as one-column interval matrix. Let {±1}m be the set of all {−1, 1} m-dimensional vectors, i.e.
{±1}m = {y ∈ Rm | | y |= e}, where e = (1, . . . , 1)T is the m-dimensional vector of all 1, s and the absolute value of a matrix A = (aij ) is defined by |A| = (|aij |). For a given y ∈ {±1}m , let Ty = diag (y1 , . . . , ym ) denote the corresponding diagonal matrix. For each x ∈ Rn , we define its sign vector sgn x by
(sgn x)i =
1
−1
if xi ≥ 0, if xi < 0,
where i = 1, . . . , n. Then we have |x| = Tz x, where z = sgn x ∈ {±1}n . Given an interval matrix A = [Ac − A∆ , Ac + A∆ ], for each y ∈ {±1}m and z ∈ {±1}n , we define matrices Ayz = Ac − Ty A∆ Tz . Similarly, for an interval vector b = [bc − b∆ , bc + b∆ ] and for each y ∈ {±1}m , we define vectors by = bc + Ty b∆ . Let A ∈ Rm×n , B ∈ Rk×n , b ∈ Rm , c ∈ Rn , d ∈ Rk and Q ∈ Rn×n be given, consider the quadratic programming problem min
1
xT Qx + c T x subject to Ax ≤ b, Bx = d, x ≥ 0,
2
where Q is positive semidefinite. Briefly, we rewrite the problem as
Min
1 2
xT Qx + c T x|Ax ≤ b, Bx = d, x ≥ 0 .
(1)
The Dorn dual problem [25,26] of the quadratic program (1) is
1 Max − uT Qu − bT v − dT w|Qu + AT v + BT w + c ≥ 0, v ≥ 0 . 2
(2)
Let f (A, B, b, c , d, Q ) = inf
1 2
xT Qx + c T x|Ax ≤ b, Bx = d, x ≥ 0
and
1 g (A, B, b, c , d, Q ) = sup − uT Qu − bT v − dT w|Qu + AT v + BT w + c ≥ 0, v ≥ 0 2 denote the optimal values of (1) and (2), respectively.
(3)
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W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
Lemma 2.1 ([25]). (i) If x = x0 is an optimal solution to problem (1) then an optimal solution (u, v, w)T = (u0 , v0 , w0 )T exists to problem (2). (ii) Conversely, if an optimal solution (u, v, w)T = (u0 , v0 , w0 )T to problem (2) exists then an optimal solution x = x0 to problem (1) also exists. In either case, f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ). From the weak and strong duality conditions we can get the following theorem, which describes the characteristics of the duality gap in the quadratic program (1). Theorem 2.1. The duality gap of the quadratic program is zero as long as either the primal or the dual problem is feasible. Proof. We consider separately three cases below. (a) Both problems (1) and (2) are feasible. Clearly, the duality gap is zero in this case. (b) Problem (1) is feasible and problem (2) is infeasible. If the problem (2) is infeasible, that is g (A, B, b, c , d, Q ) = −∞, we can get f (A, B, b, c , d, Q ) = −∞, since otherwise, we have −∞ < f (A, B, b, c , d, Q ) < ∞, then the primal problem has optimal solutions, and from Lemma 2.1 we know the dual problem (2) also has optimal solutions, which is a contradiction. Thus we have f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ) = −∞, the duality gap is zero. (c) Problem (1) is infeasible and problem (2) is feasible. Analogous to the case (b), we are easy to obtain f (A, B, b, c , d, Q ) = g (A, B, b, c , d, Q ) = ∞, the duality gap is zero. The theorem is proved combining the discussion above.
The set of all m-by-n interval matrices will be denoted by IRm×n and the set of all m-dimensional interval vectors by IRm . Given A ∈ IRm×n , B ∈ IRk×n , b ∈ IRm , c ∈ IRn , d ∈ IRk and Q ∈ IRn×n , the interval convex quadratic program
Min
1 2
xT Qx + cT x|Ax ≤ b, Bx = d, x ≥ 0
(4)
is the family of convex quadratic programs (1) with data satisfying A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q, where Q is positive semidefinite for all Q ∈ Q. The lower and upper bounds of the optimal values are respectively defined as f (A, B, b, c, d, Q) = inf{f (A, B, b, c , d, Q )|A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q}, f (A, B, b, c, d, Q) = sup{f (A, B, b, c , d, Q )|A ∈ A, B ∈ B, b ∈ b, c ∈ c, d ∈ d, Q ∈ Q}. For given A, B, b, c, d, Q denote f = f (A, B, b, c, d, Q), f = f (A, B, b, c, d, Q). The Dorn duality gap for interval quadratic program (4) is zero means that for each scenario of problem (4) satisfies the zero duality gap. The following lemmas will be used in the proof of our results. Lemma 2.2 ([8]). The system Ax ≤ b is solvable if and only if for each p ≥ 0 satisfies AT p = 0 then bT p ≥ 0. Lemma 2.3 ([8]). The system Ax ≤ b is solvable for each A ∈ A, b ∈ b if and only if the system Ax1 − Ax2 ≤ b,
x1 , x2 ≥ 0
is solvable. Lemma 2.4 ([24]). We have
f = inf
1 2
x Q x + c x|Ax ≤ b, Bx ≤ d, Bx ≥ d, x ≥ 0 T
T
and f = f (A, B, b, c , d, Q ) = sup f (A, Bye , b, c , dy , Q ). y∈{±1}k
Let us denote
ψ = sup
1 T − uT Q u − bT v − dTc w + dT∆ |w||Q u + A v + BTc w + BT∆ |w| + c ≥ 0, v ≥ 0 . 2
Lemma 2.5 ([24]). If ψ = ∞, then f = ∞.
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41
Consider the interval quadratic program
Min
1 2
xT Q x + c x|Ax ≤ b, Bx = d, x ≥ 0 . T
(5)
Lemma 2.6 ([21]). If the Dorn duality gap of problem (5) is zero, then we have f = ψ. 3. A sufficient condition for zero Dorn duality gap As we known, zero duality gap plays an important role in the dual methods for computing the upper bound of IvQP. So far, however, it is not an easy task to verify whether this condition is satisfied when the data vary inside intervals. For convenient practice of theory and calculation, we give a sufficient condition for the Dorn duality gap of problem (5). It will be also used to discuss some properties of the dual methods in the next section. Theorem 3.1. If f is finite, then the Dorn duality gap of problem (5) is zero. Proof. If f is finite, then the quadratic program
Min
1 2
x Q x + c x|Ax ≤ b, Bx ≤ d, Bx ≥ d, x ≥ 0 T
T
(6)
has a finite optimal value by Lemma 2.4. From the strong duality theorem Lemma 2.1 we know that an optimal solution (u, v1 , v2 , v3 )T = (u∗ , v1∗ , v2∗ , v3∗ )T exists to its Dorn dual problem
1
T
T
T
Max − u Q u − b v1 − d v2 + d v3 |Q u + A v1 + B v2 − B v3 + c ≥ 0, vi ≥ 0, i = 1, 2, 3 . 2 T
T
T
T
Thus the system T
Q u + AT v1 + BT v2 − B v3 + c ≥ 0,
vi ≥ 0, i = 1, 2, 3
(7)
is solvable and hence from the Farkas Lemma 2.2 we get
−Q T −A (∀p ≥ 0) −B
B Let p = (
,
,
,
0 −E m 0 0
0 0 −Ek 0
0 0 T p = 0 ⇒ (c , 0, 0, 0)p ≥ 0 . 0 −E k
) then p1 , p2 , p3 , p4 ≥ 0 and Q T = Q due to the symmetry of Q, thus (8) can be rewritten as Qp = 0 1 Ap + p = 0 (∀p1 , p2 , p3 , p4 ≥ 0) Bp1 + p2 = 0 ⇒ c T p1 ≥ 0 . 1 3 Bp1 − p4 = 0 pT1
pT2
pT3
(8)
pT4 T ,
(9)
Now, we first prove the following results
Q p1 = 0 Ap1 + p2 = 0 T (∀p1 , p2 , p3 , p4 ≥ 0) Bp1 + p3 = 0 ⇒ c p1 ≥ 0 . Bp1 − p4 = 0
(10)
Indeed, if for each p1 , p2 , p3 , p4 ≥ 0 there hold
Q p = 0, 1 Ap1 + p2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0,
(11)
obviously we have Q p1 ≤ Q p1 = 0 due to the nonnegativity of vector p1 , and hence we get pT1 Q p1 ≤ 0. On the other hand, for each p1 there holds pT1 Q p1 ≥ 0, since Q is positive semidefinite, thus pT1 Q p1 = 0. Further, there exists a real matrix F
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W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
such that Q = FTF, due to the positive semidefiniteness of the matrix Q . Thus, pT1 Q p1 = 0 leads to (Fp1 )T (Fp1 ) = 0 and hence we have Fp1 = 0. So we have Q p = F T Fp = 0. Then from (11) we can get
Q p = 0, 1 (A + 2A∆ )p1 + p2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0,
(12)
let p′2 = 2A∆ p1 + p2 , obviously, p′2 ≥ 0, then (12) can be rewritten as
Q p = 0, 1 Ap1 + p′2 = 0, Bp1 + p3 = 0, Bp1 − p4 = 0, T
By (9), we obtain c T p1 ≥ 0 and thus c p1 ≥ c T p1 ≥ 0, which proves (10). Similarly, the condition (10) can be rewritten as
T −Q −A (∀p ≥ 0) −B
0 −E m 0 0
B
0 0 −E k 0
0 0 p = 0 ⇒ (c T , 0, 0, 0)p ≥ 0 . 0 −E k
(13)
Hence the system T
T
Q u + A v + BT w 1 − B w 2 + c ≥ 0,
v, w1 , w2 ≥ 0
(14)
is solvable by Farkas Lemma 2.2. From Lemma 2.3, we can get for each B ∈ B the following system T
Q u + A v + BT w + c ≥ 0,
v≥0
(15)
is solvable. In fact, the system (15) can be rewritten as
T
Q 0
A Em
BT 0
u −c v ≥ . 0 w
(16)
Let
Q 0
H=
T
A Em
BT 0
,
H =
Q 0
T
BT 0
A Em
,
h=
−c 0
u
,
x=
v , w
then the solvability of system (16) is equivalent to Hx ≥ h. By Lemma 2.3 we have the system Hx ≥ h is solvable for each H ∈ H if and only if the following system
T T T T Q u1 + A v 1 + BT w1 − (Q u2 + A v 2 + B w 2 ) ≥ −c , v 1 − v 2 ≥ 0, 1 2 1 2 1 2 u ,u ,v ,v ,w ,w ≥ 0 is solvable. Put u = u1 − u2 , v = v 1 − v 2 , we obtain system (14). Meanwhile, for each B ∈ B, d ∈ d, consider the scenario of interval quadratic program (5)
Min
1 2
xT Q x + c x|Ax ≤ b, Bx = d, x ≥ 0 T
(17)
and its Dorn dual problem
1 T Max − uT Q u − bT v − dT w|Q u + A v + BT w + c ≥ 0, v ≥ 0 . 2
(18)
For all B ∈ B, the problem (18) is feasible due to the solvability of system (15), hence the zero duality gap is ensured by Theorem 2.1. Then by Theorem 3.1, together with Lemma 2.5 we have
W. Li et al. / Journal of Computational and Applied Mathematics 302 (2016) 38–49
43
Theorem 3.2. Let f be finite or let the ψ be positively infinite. Then f = ψ . If we put Q = 0, A = 0 and b = 0 in (4), then we get a simple IvLP Min{cT x|Bx = d, x ≥ 0}.
(19)
As a special case of (4), from Lemma 2.4 we obtain f (B, c, d) = inf {c T x|Bx ≤ d, Bx ≥ d, x ≥ 0} and f (B, c, d) = sup f (Bye , c , dy ), y∈{±1}k
which has also been discussed in [8]. Analogous to Theorem 3.1, we can get a similar sufficient condition to interval linear program (19). Corollary 3.1. If f (B, c, d) is finite, then the duality gap of the following IvLP Min{c x|Bx = d, x ≥ 0} T
(20)
is zero. Remark. The conditions of Theorem 3.1 and Corollary 3.1 are sufficient but not necessary. See Examples 1 and 2 in Section 5. 4. The relationship between f and ψ From Section 3 of Hladík [21] and Section 4 of Li [24], we can see there exists some relations between f and ψ , and also we know the values of f and ψ are equal under the zero duality gap. In order to have a clear understanding of the relations between two formulas of the upper bound, we will make a deeper discussion in this section. Theorem 4.1. We have
f
if ψ = −∞; if ψ is finite; if ψ = ∞.
∈ {−∞, ∞}, > −∞, = ∞,
(a) (b) (c)
(21)
Proof. Part (21)(a). If ψ = −∞, which means the problem
Max
1
− u Qu − b v − 2
T
T
dTc
T
w + d∆ |w||Q u + A v + T
BTc
w + B∆ |w| + c ≥ 0, v ≥ 0 T
(22)
is infeasible. Now, we prove that the system T
Q u + A v + BT w + c ≥ 0,
v≥0
(23)
is not solvable for each B ∈ B. In fact, if for some B ∈ B (23) is solvable and note that we can get the system T
Q u + A v + BTc w + BT∆ |w| + c ≥ 0,
BTc
w + B∆ |w| ≥ B w , then from (23) T
T
v≥0
is solvable, which is a contradiction to the infeasibility of problem (22). Hence, for each B ∈ B, d ∈ d the Dorn dual problem (18) is infeasible, which means that each primal problem (17) is either infeasible or unbounded, so that f (A, B, b, c , d.Q ) ∈ {−∞, ∞} for each B ∈ B, d ∈ d. By Lemma 2.4 we have f = sup {f (A, B, b, c , d, Q )|B ∈ B, d ∈ d}, consequently, f ∈ {−∞, ∞}. Part (21)(b). If ψ is finite, then the feasible region of (22) is not empty. Let (u0 , v0 , w0 )T be an arbitrary feasible solution to problem (22), then it satisfies T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0.
Let y0 = −sgn w0 , since y0 ∈ {±1} and from (24) we obtain k
T
T
Q u0 + A v0 + (By0 e )T w0 + c = Q u0 + A v0 + BTc w0 − BT∆ Ty0 w0 + c T
= Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
(24)
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which means (u0 , v0 , w0 )T is a feasible solution to problem
1 T T T T T Max − u Q u − b v − d w|Q u + A v + By0 e w + c ≥ 0, v ≥ 0 . 2
Thus, for each d ∈ d, from (3) there holds g (A, By0 e , b, c , d, Q ) > −∞, then from weak duality theorem [21,24,25] we have f (A, By0 e , b, c , d, Q ) ≥ g (A, By0 e , b, c , d, Q ) > −∞. Hence, from the definition of f we get f > −∞. Part (21)(c). See Lemma 2.5, which has been proved in [24].
The following lemma describes the relations between ψ and the interval quadratic program (5) under the assumption that ψ is finite, which will be used to obtain the next theorem. Lemma 4.1. If ψ is finite, then
ψ = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d}. Proof. Denote
α = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d}, thus we only need to prove ψ = α . (i) First, we prove α ≤ ψ by showing that f (A, B, b, c , d, Q ) ≤ ψ
(25)
holds for each B ∈ B, d ∈ d. Note that for each B ∈ B, d ∈ d the value of f (A, B, b, c , d, Q ) < ∞ can be divided into two cases, they are −∞ or finite value. We discuss them separately. If f (A, B, b, c , d, Q ) = −∞, then (25) holds clearly. If f (A, B, b, c , d, Q ) is finite. Then from Lemma 2.1 we have 1 f (A, B, b, c , d, Q ) = − (u∗ )T Q u∗ − bT v ∗ − dT w ∗ , 2 where (u∗ , v ∗ , w ∗ )T is an optimal solution of the Dorn dual problem (18). Since BTc w ∗ + BT∆ |w ∗ | ≥ BT w ∗ , then we know (u∗ , v ∗ , w∗ )T satisfies T
Q u∗ + A v ∗ + BTc w ∗ + BT∆ |w ∗ | + c ≥ 0, v ∗ ≥ 0, which means (u∗ , v ∗ , w ∗ )T is an feasible solution of the problem
1 T T T T T T Max − u Q u − b v − d w|Q u + A v + Bc w + B∆ |w| + c ≥ 0, v ≥ 0 . 2
(26)
Let T
S = {(u, v, w)T |Q u + A v + BTc w + BT∆ |w| + c ≥ 0, v ≥ 0}. Thus, 1 f (A, B, b, c , d, Q ) = − (u∗ )T Q u∗ − bT v ∗ − dT w ∗ 2
1 ≤ sup − uT Q u − bT v − dT w|(u, v, w)T ∈ S 2
(27)
meanwhile, as dT w ≥ dTc w − dT∆ |w| for each w , so we have 1
1
2
2
− uT Q u − bT v − dT w ≤ − uT Q u − bT v − dTc w + dT∆ |w|, which implies
sup
1 1 − uT Q u − bT v − dT w|(u, v, w)T ∈ S ≤ sup − uT Q u − bT v − dTc w + dT∆ |w||(u, v, w)T ∈ S . 2
2
(28)
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45
From (27) and (28) we have
1 T T T T T f (A, B, b, c , d, Q ) ≤ sup − u Q u − b v − dc w + d∆ |w||(u, v, w) ∈ S 2
= ψ. (ii) Second, we prove ψ ≤ α . Let (ˆu, vˆ , w) ˆ T be an optimal solution of the problem (22). Hence, we have 1
ˆ + dT∆ |w| ˆ ψ = − uˆ T Q uˆ − bT vˆ − dTc w
(29)
2
and T
Q uˆ + A vˆ + BTc w ˆ + BT∆ |w| ˆ + c ≥ 0,
vˆ ≥ 0,
which implies T
Q uˆ + A vˆ + (Byˆ e )T w ˆ + c ≥ 0,
vˆ ≥ 0,
where yˆ = −sgn w ˆ . Thus, (ˆu, vˆ , w) ˆ T is an feasible solution of problem
1
T
Max − u Q u − b v − (dyˆ ) w|Q u + A v + (Byˆ e ) w + c ≥ 0, v ≥ 0 . 2 T
T
T
T
From (29), (30) and (3) we have 1
ˆ + dT∆ |w| ˆ ψ = − uˆ T Q uˆ − bT vˆ − dTc w 2 1
ˆ = − uˆ T Q uˆ − bT vˆ − (dyˆ )T w 2 1 T T T T T ≤ sup − u Q u − b v − (dyˆ ) w|Q u + A v + (Byˆ e ) w + c ≥ 0, v ≥ 0 2
= g (A, Byˆ e , b, c , dyˆ , Q ). Thus g (A, Byˆ e , b, c , dyˆ , Q ) = ∞ or finite, since the ψ is finite. If g (A, Byˆ e , b, c , dyˆ , Q ) = ∞, which implies the problem (30) is unbounded. Let T
F = {(u, v, w)T |Q u + A v + (Byˆ e )T w + c ≥ 0, v ≥ 0}, then for each M > 0, there exists (u0 , v0 , w0 )T ∈ F such that 1
− uT0 Q u0 − bT v0 − (dyˆ )T w0 ≥ M . 2
Note that BTc w0 + BT∆ |w0 | ≥ (Byˆ e )T w0 and hence T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0,
which means (u0 , v0 , w0 )T is an feasible solution of problem (22). Thus, 1
ψ ≥ − uT0 Q u0 − bT v0 − dTc w0 + dT∆ |w0 | 2 1
≥ − uT0 Q u0 − bT v0 − (dyˆ )T w0 2
≥ M,
which is a contradiction to the finiteness of value ψ . If g (A, Byˆ e , b, c , dyˆ , Q ) is finite. From Lemma 2.1 we have f (A, Byˆ e , b, c , dyˆ , Q ) = g (A, Byˆ e , b, c , dyˆ , Q ) < ∞. Thus, f (A, Byˆ e , b, c , dyˆ , Q ) is finite and there holds
ψ ≤ f (A, Byˆ e , b, c , dyˆ , Q ). From the definition of α we obtain f (A, Byˆ e , b, c , dyˆ , Q ) ≤ α , since Byˆ e ∈ B, dyˆ ∈ d. Hence, we can get ψ ≤ α . Combining the results of (i) and (ii) we can get ψ = α .
(30)
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Analogous to Theorem 4.1, the following result describes the relations between ψ and f under the different values of f . Theorem 4.2. We have
−∞, ψ = f,
(a) (b) (c)
if f = −∞; if f is finite; if f = ∞.
cannot determine,
(31)
Proof. Part (31)(a). Since f = −∞, then f (A, B, b, c , d, Q ) = −∞ for each B ∈ B, d ∈ d. From the weak duality theorem, we have g (A, B, b, c , d, Q ) = −∞, that is, for each B ∈ B the inequality T
Q u + A v + BT w + c ≥ 0,
v≥0
is not solvable. Thus, we can get the following system T
Q u + A v + BTc w + BT∆ |w| + c ≥ 0,
v≥0
(32)
is not solvable. To prove this, assume that there exists (u0 , v0 , w0 )T such that T
Q u0 + A v0 + BTc w0 + BT∆ |w0 | + c ≥ 0,
v0 ≥ 0,
which implies T
Q u0 + A v0 + (Bye )T w0 + c ≥ 0,
v0 ≥ 0,
where y = −sgn w0 , a contradiction. The unsolvability of system (32) implies the infeasibility of problem (22), then ψ = −∞. Part (31)(b). Apparently ψ is finite, since the possibilities ψ = −∞ and ψ = ∞ are precluded by (21)(a) and (c) in Theorem 4.1. Meanwhile, Lemma 4.1 together with Lemma 2.4 gives
ψ = max {f (A, B, b, c , d, Q )|f (A, B, b, c , d, Q ) < ∞, B ∈ B, d ∈ d} = max {f (A, B, b, c , d, Q )|B ∈ B, d ∈ d} = f (A, B, b, c , d, Q ) = f . Part (31)(c). If f = ∞ we cannot determine the feature of ψ , which means all these three different cases for ψ may happen (see Examples 2–4 in Section 5). 5. Illustrative examples and remarks In this section, we give some illustrative examples for Theorems 4.1 and 4.2, and make some remarks. We refer the reader to [18,19,21,24] and the references therein for more examples, which can help us to understand Theorems 4.1 and 4.2. Example 1. Consider the interval linear program min [−5, 1]x1 + [−4, 0]x2 s.t. [−1, 2]x1 − 2x2 = [−1, 2],
− 3x1 + [−1, 4]x2 = [1, 1.4], x1 , x2 ≥ 0. The corresponding interval matrices and vectors are
c=
[−5, 1] , [−4, 0]
B=
[−1, 2] −3
−2 , [−1, 4]
d=
[−1, 2] . [1, 1.4]
According to model (20) we consider the IvLP
Min
1
[−1, 2] 0 x| −3
−2 [−1, 2] x= ,x ≥ 0 [−1, 4] [1, 1.4]
(33)
and its dual problem
Max [−1, 2]
[−1, 2] [1, 1.4] y| −3
−2 [−1, 4]
T y≤
1 0
.
(34)
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Now we check that whether the duality gap of problem (33) is zero or not. The feasible region of problem (34) is
[−1, 2] −3
T −2 1 y≤ . [−1, 4] 0
Clearly, for each matrix
[−1, 2] C ∈ −3
−2 [−1, 4]
the vector y = (0, 0)T satisfies CT y ≤
1 , 0
which implies that each scenario of dual problem (34) is feasible, then by Theorem 2.1 we know the duality gap of program (33) is zero. However, the lower bound of the optimal values can be determined by linear program min − 5x1 − 4x2 s.t. − x1 − 2x2 ≤ 2, 2x1 − 2x2 ≥ −1,
− 3x1 − x2 ≤ 1.4, − 3x1 + 4x2 ≥ 1, x1 , x2 ≥ 0. It can be shown that f = −∞. This example shows that the converse of Corollary 3.1 is not true.
Example 2. Consider the interval quadratic program min [2, 3]x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. − 2x1 + [−2, 1]x2 ≤ [1, 4],
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. According to model (5) we consider the following IvQP min 3x21 +
1 2
x22 − 2x1 x2 + x1 − 3x2
s.t. − 2x1 + x2 ≤ 1,
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. Clearly, the Dorn duality gap of (35) is zero, since the matrix Q =
(35)
6 −2
−2 1
is positive definite (see Section 3.2 in
Hladik [21]). However, the lower bound of the optimal values can be determined by quadratic program min 2x21 +
1 2
x22 − 2x1 x2 − 5x1 − 4x2
s.t. − 2x1 − 2x2 ≤ 4, 4x1 − 2x2 ≤ 2, 5x1 − 2x2 ≥ 2, 6x1 − 4x2 ≤ 1.4, 6x1 − 3x2 ≥ 1.4, x1 , x2 ≥ 0. It is easy to show that f = −∞.
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Now put y0 = (1, −1)T , the f (A, By0 e , b, c , dy0 , Q ) can be calculated by solving the following problem min 3x21 +
1 2
x22 − 2x1 x2 + x1 − 3x2
s.t. − 2x1 + x2 ≤ 1, 4x1 − 2x2 = 2,
(36)
6x1 − 3x2 = 1.4, x1 , x2 ≥ 0. It can be easily verified that the problem (36) is infeasible. Thus, f (A, By0 e , b, c , dy0 , Q ) = ∞ and hence we can get f = ∞ by Lemma 2.4. Thus by Lemma 2.6 we have ψ = f = ∞, since the duality gap is zero. This example shows that the converse of Theorem 3.1 is not true, and confirm the possibility of f = ∞ and ψ = ∞. Example 3. Consider the interval quadratic program min 2x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. − 2x1 + [−2, 1]x2 ≤ [1, 4],
[4, 5]x1 − 2x2 = 2, 6x1 + [−4, −3]x2 = 1.4, x1 , x2 ≥ 0. This is the Example 2 in [24]. It can be shown that the f = ∞. Whereas ψ = 40.6848, which is a finite value. This example shows that the possibility that f = ∞ but ψ is finite. Example 4. Consider the interval quadratic program min 2x21 +
1 2
x22 − 2x1 x2 + [−5, 1]x1 + [−4, −3]x2
s.t. [1, 2]x1 + [−2, −1]x2 ≤ [−1, 4], 6x1 − 3x2 = [1, 1.4], x1 , x2 ≥ 0. Consider the quadratic program 1 2 x2 − 2x1 x2 + x1 − 3x2 2 s.t. 2x1 − x2 ≤ −1,
min 2x21 +
6x1 − 3x2 = 1.4, x1 , x2 ≥ 0, which is the corresponding quadratic program for computing f (A, By0 e , b, c , dy0 , Q ) with y0 = 1. We are easy to check that there is no feasible solution to this problem, then f (A, By0 e , b, c , dy0 , Q ) = ∞ and hence f = ∞ by Lemma 2.4. Now, by the method described in [21], we can compute the value of ψ by considering two problems f1 = sup − 2u21 −
1 2
u22 + 2u1 u2 + v − w
s.t. 4u1 − 2u2 + 2v + 6w + 1 ≥ 0,
− 2u1 + u2 − v − 3w − 3 ≥ 0, v, w ≥ 0. and f2 = sup − 2u21 −
1 2
u22 + 2u1 u2 + v − 1.4w
s.t. 4u1 − 2u2 + 2v + 6w + 1 ≥ 0,
− 2u1 + u2 − v − 3w − 3 ≥ 0, v, −w ≥ 0. It can be shown that the two problems are both infeasible, then ψ = −∞.
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49
This example shows the possibility of f = ∞ but ψ = −∞. Moreover, the zero Dorn duality gap in this example is not guaranteed due to f ̸= ψ , then from Theorem 3.1 we can get f ∈ {−∞, ∞}. So far, the only possibility to compute the upper bound in general was to reduce the problem to solving 2m subproblems, where m is the number of the equality constraints, see, [18,19,21,24]. The contribution of these results shows that the problem can be handled by finite means. It is known that even determining the upper bound of interval linear program – a special case of interval quadratic program when the coefficient matrix of the quadratic term Q = 0 – is NP-hard [10]. So, the computing difficulty of primal or dual methods does not lie with the inadequateness of the proposed methods, but that it is inherently present in the problem itself. An interesting further research direction is to derive approximations of the upper bound. In fact, the recently published paper [23] presented approximation methods for the upper bound of the equality constrained interval linear program. It is interesting to establish approximation methods for the upper bound of the interval quadratic program. Acknowledgments The authors would like to thank anonymous referees for their comments and suggestions that helped to improve the paper. The authors were partially supported by the NSF of Zhejiang Province (Grant No. LY14A010028), NNSF of China (Grant Nos. 11526184, 71471051) and the Joint Key Grant of NNSF of China and Zhejiang Province (No. U1509217). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990. H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European J. Oper. Res. 48 (1990) 219–225. J.W. Chinneck, K. Ramadan, Linear programming with interval coefficients, J. Oper. Res. Soc. 51 (2) (2000) 209–220. S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity, Reliab. Comput. 8 (5) (2002) 321–418. X.Y. Wu, G.H. Huang, L. Liu, J.B. Li, An interval nonlinear program for the planning of waste management systems with economies-of-scale effects a case study for the region of Hamilton, Ontario, Canada, European J. Oper. Res. 171 (2) (2006) 349–372. M. Inuiguchi, J. Ramík, T. Tanino, M. Vlach, Satisficing solutions and duality in interval and fuzzy linear programming, Fuzzy Sets and Systems 135 (1) (2003) 151–177. M. Fiedler, J. Nedoma, J. Ramik, J. Rohn, K. Zimmermann, Linear Optimization Problems With Inexact Data, Springer-Verlag, New York, 2006. O.A. Prokopyev, S. Butenko, A.C. Trapp, Checking solvability of systems of interval linear equations and inequalities via mixed integer programming, European J. Oper. Res. 199 (1) (2009) 117–121. V. Gabrel, C. Murat, N. Remli, Linear programming with interval right hand sides, Int. Trans. Oper. Res. 17 (3) (2010) 397–408. W. Li, H. Wang, Q. Wang, Localized solutions to interval linear equations, J. Comput. Appl. Math. 238 (15) (2013) 29–38. M. Hladík, How to determine basis stability in interval linear programming, Optim. Lett. 8 (1) (2014) 375–389. W. Li, J. Luo, Q. Wang, Y. Li, Checking weak optimality of the solution to linear programming with interval right-hand side, Optim. Lett. 8 (4) (2014) 1287–1299. J. Luo, W. Li, Strong optimal solutions of interval linear programming, Linear Algebra Appl. 439 (2013) 2479–2493. W. Li, X. Liu, H. Li, Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions, Optim. Methods Softw. 30 (3) (2015) 516–530. M. Hladík, Interval Linear Programming: A Survey, in: Zoltan Adam Mann (Ed.), Linear Programming New Frontiers, Nova Science Publishers Inc., New York, 2012, pp. 1–46. F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res. 81 (1998) 51–62. S.T. Liu, R.T. Wang, A numerical solution method to interval quadratic programming, Appl. Math. Comput. 189 (2007) 1274–1281. W. Li, X. Tian, Numerical solution method for general interval quadratic programming, Appl. Math. Comput. 202 (2008) 589–595. M. Hladík, Optimal value range in interval linear programming, Fuzzy Optim. Decis. Mak. 8 (3) (2009) 283–294. M. Hladík, Optimal value bounds in nonlinear programming with interval data, Top 19 (2011) 93–106. M. Hladík, Weak and strong solvability of interval linear systems of equations and inequalities, Linear Algebra Appl. 438 (2013) 4156–4165. M. Hladík, On approximation of the best case optimal value in interval linear programming, Optim. Lett. 8 (2014) 1985–1997. W. Li, M. Xia, H. Li, New method for computing the upper bound of optimal value in interval quadratic program, J. Comput. Appl. Math. 288 (2015) 70–80. W.S. Dorn, Duality in quadratic programming, Quart. Appl. Math. 18 (2) (1960) 155–162. M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming, Theory and Algorithms, second ed., Wiley, New York, 1993.