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Nonsmooth Invex Functions and Sufficient Optimality Conditions
Journal of Mathematical Analysis and Applications 255, 319᎐332 Ž2001. doi:10.1006rjmaa.2000.7263, available online at http:rrwww.idealibrary.com on
Nonsmooth Invex Functions and Sufficient Optimality Conditions Marco Castellani Dipartimento di Sistemi ed Istituzioni per L’Economia, Piazza del Santuario, 19 67040 L’Aquila, Italy E-mail: [email protected] Submitted by H. P. Benson Received December 23, 1997
In this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of K-directional derivative. Characterizations of nonsmooth K-invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the K-directional derivatives. 䊚 2001 Academic Press
1. INTRODUCTION AND NOTATIONS In w15x, Hanson presented a weakened concept of convexity for differentiable functions: a differentiable function f : X ª ⺢ is said to be in¨ ex, if there exists a function : X = X ª ⺢ n such that 䢇
f Ž x 2 . y f Ž x 1 . G ² ⵜf Ž x 1 . , Ž x 1 , x 2 . : ,
᭙ x1 , x 2 g X .
The name invex descends from a contraction of ‘‘invariant convex’’ and it was proposed by Craven w6x. In w7x, Craven and Glover showed that the class of invex functions is equivalent to the class of functions whose stationary points are global minima. In w18x, Kaul and Kaur considered the following generalizations: f is said to be pseudoin¨ ex if there exists a function : X = X ª ⺢ n such that 䢇
² ⵜf Ž x 1 . , Ž x 1 , x 2 . : G 0 « f Ž x 2 . G f Ž x 1 . ,
᭙ x1 , x 2 g X ;
319 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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MARCO CASTELLANI
f is said to be quasi-in¨ ex if there exists a function : X = X ª ⺢ n such that 䢇
f Ž x 2 . F f Ž x 1 . « ² ⵜf Ž x 1 . , Ž x 1 , x 2 . : F 0,
᭙ x1 , x 2 g X .
By means of these concepts they established sufficient optimality conditions for a nonlinear programming problem with inequality constraints. In w1, 11x relations among convex and invex functions and their generalizations were studied. Recently Reiland w22x extended the concept of invexity to the class of locally Lipschitz functions using the generalized gradient of Clarke. An analogous extension was made by Jeyakumar w17x by means of the notion of approximate quasidifferentiability for nonsmooth functions. Different definitions of nonsmooth invexity were introduced by Ye w26x and Giorgi and Guerraggio w12x who studied the relations among all these classes of functions. In this paper we propose a unifying definition of invexity for nonsmooth functions exploiting the concept of local cone approximation introduced in w9x. Moreover, via such an approach, we give sufficient optimality conditions for inequality constrained extremum problems without requiring the convexity of the directional derivatives. In the sequel X : ⺢ n will be an open set. Given the function f : X ª ⺢, the epigraph of f is epi f [ Ž x, y . g X = ⺢ : f Ž x . F y 4 . The set epi f will be locally approximated at the point Ž x, f Ž x .. by a local cone approximation K and a positively homogeneous function f K Ž x, ⭈ . will be uniquely determined. DEFINITION 1.1. Let f : X ª ⺢, x g X and K be a local cone approximation; the positively homogeneous function f K Ž x, ⭈ .: ⺢ n ª wy⬁, q⬁x defined by f K Ž x, y . [ inf  g ⺢ : Ž y,  . g K Ž epi f , Ž x, f Ž x . . . 4 is called the K-directional deri¨ ati¨ e of f at x. By means of Definition 1.1 we can recover most of the generalized directional derivatives used in literature; for instance 䢇
the upper Dini directional deri¨ ati¨ e of f at x Dq f Ž x, y . [ lim sup tª0 q
f Ž x q ty . y f Ž x . t
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is associated to the cone of the feasible directions F Ž Q, x . [ y g ⺢ n : ᭙ t k 4 ª 0q, x q t k y g Q 4 ; 䢇
the lower Dini directional deri¨ ati¨ e of f at x Dy f Ž x, y . [ lim inf q
f Ž x q ty . y f Ž x . t
tª0
is associated to the cone of the weak feasible directions WF Ž Q, x . [ y g ⺢ n : ᭚ t k 4 ª 0q s.t. x q t k y g Q 4 ; 䢇
if f is locally Lipschitz, the Clarke directional deri¨ ati¨ e of f at x f⬚ Ž x, y . [
f Ž x⬘ q ty . y f Ž x⬘ .
lim sup
t
Ž x ⬘, t .ª Ž x , 0 q.
is associated to Clarke’s tangent cone TC l Ž Q, x . [ y g ⺢ n : ᭙ x k 4 ª x s.t. x k g Q, ᭙ t k 4 ª 0q, ᭚ y k 4 ª y s.t. x k q t k y k g Q 4 . It is well known that f⬚Ž x, ⭈ . G Dq f Ž x, ⭈ . G Dy f Ž x, ⭈ .. For a more detailed review about the local cone approximations we refer to w10x. DEFINITION 1.2. Let f : X ª ⺢, x g X and K be a local cone approximation; f is said to be K-subdifferentiable at x if there exists a convex compact set ⭸ K f Ž x . such that 䢇
f K Ž x, y . s
max
x*g ⭸ f Ž x .
² x*, y : ,
K
᭙ y g ⺢n;
the set ⭸ K f Ž x . is called the K-subdifferential of f at x. f is said K-quasidifferentiable at x if there exist two convex compact sets ⭸ K f Ž x . and ⭸ K f Ž x . such that 䢇
f K Ž x, y . s
max
x*g ⭸ f Ž x . K
² x*, y : y
max
x*g ⭸ f Ž x . K
² x*, y : ,
᭙ y g ⺢n;
the sets ⭸ K f Ž x . and ⭸ K f Ž x . are called the K-subdifferential and K-superdifferential of f at x, respectively.
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Remark 1.1. It is immediate to observe that every K-subdifferentiable function is K-quasidifferentiable with ⭸ K f Ž x . s 04 and ⭸ K f Ž x . s ⭸ K f Ž x .. It is well known that the Clarke derivative is bounded and convex; therefore there exists the Clarke subdifferential ⭸ ⬚ f [ ⭸ T C l f. If f is directionally differentiable Ži.e., Dy f Ž x, ⭈ . s Dq f Ž x, ⭈ . [ f ⬘Ž x, ⭈ .. and it is F-subdifferentiable we say that f is quasidifferentiable in the sense of Pshenichnyi w21x, while if f is F-quasidifferentiable we say that f is quasidifferentiable in the sense of Demyanov and Rubinov w8x. DEFINITION 1.3. Let f : X ª ⺢ and K be a local cone approximation; x g X is said to be a K-inf-stationary point for f if f K Ž x, y . G 0 for each y g ⺢ n. The following result gives the characterization of a K-inf-stationary point for K-quasidifferentiable functions. THEOREM 1.1. Let f : X ª ⺢ and K be a local cone approximation. If f is K-quasidifferentiable, then x g X is a K-inf-stationary point for f if and only if ⭸ K f Ž x . : ⭸ K f Ž x .. Proof. Let x be a K-inf-stationary point and let us suppose by contradiction that there exists x* g ⭸ K f Ž x . such that x* f ⭸ K f Ž x .. Then there exist y g ⺢ n and ) 0 such that ² x*, y : G ² x*, y : q ,
᭙ x* g ⭸ K f Ž x . ;
hence ² x*, y : G
max
x*g ⭸ K f Ž x .
² x*, y : q ,
and then y G
max
x*g ⭸ K f Ž x .
² x*, y : y ² x*, y : G f K Ž x, y .
that contradicts the assumption. The converse implication is immediate. Remark 1.2. In particular, if f is K-subdifferentiable, then x g X is a K-inf-stationary point for f if and only if 0 g ⭸ K f Ž x .. In w3, 4x it was proved that if K is an isotone local approximation Ži.e., K Ž A, x . : K Ž B, x . for each A : B ., then every local minimizer of f over ⺢ n is a K-inf-stationary point for f. Unfortunately, in general it is not possible to deduce that a K-inf-stationary point is a local optimal solution. For this reason, in Section 2, we will introduce the concept of K-invexity.
NONSMOOTH INVEX FUNCTIONS
323
2. K-INVEXITY In this section we propose a unifying definition of invexity for nonsmooth functions. DEFINITION 2.1. Let K be a local cone approximation; the function f : X ª ⺢ is said to be K-in¨ ex if there exists a function : X = X ª ⺢ n such that f Ž x 2 . y f Ž x1 . G f K Ž x1 , Ž x1 , x 2 . . ,
᭙ x1 , x 2 g X .
The function is said to be the kernel of the K-invexity. By means of Definition 2.1, we can obtain all the definitions of invexity for nonsmooth functions. For instance if we use Clarke’s tangent cone for locally Lipschitz functions, we recover the concept of invexity introduced by Reiland w22x; if we consider the class of the directionally differentiable functions and we take K s F, we get the d-invexity given by Ye w26x. Moreover if f is also F-subdifferentiable or F-quasidifferentiable, we obtain the P-invexity and the DR-invexity, respectively, studied in w12x. Finally, we observe that if f K 1 Ž x, ⭈ . G f K 2 Ž x, ⭈ . and f is K 1-invex then f is K 2-invex with respect to the same kernel; in particular every locally Lipschitz TC l-invex function w22x is also K-invex for K s F, WF. The following result shows the characterization of K-invexity for K-subŽor quasi-. differentiable functions. THEOREM 2.1. Let f : X ª ⺢ and K be a local cone approximation. If f is K-quasidifferentiable, then f is K-in¨ ex with respect to the kernel if and only if for each x 1 , x 2 g X and for each x* g ⭸ K f Ž x 1 . there exists x*Ž x 1 , x 2 . g ⭸ K f Ž x 1 . such that f Ž x 2 . y f Ž x 1 . G ² x* y x* Ž x 1 , x 2 . , Ž x 1 , x 2 . : . Proof. Let f be K-invex; then for each x 1 , x 2 g X and for each x* g ⭸ K f Ž x 1 . f Ž x 2 . y f Ž x1 . G f K Ž x1 , Ž x1 , x 2 . . G ² x*, Ž x 1 , x 2 . : y
max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . : .
Since ⭸ K f Ž x 1 . is a compact set, there exists x*Ž x 1 , x 2 . g ⭸ K f Ž x 1 . such that max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . : s ² x* Ž x 1 , x 2 . , Ž x 1 , x 2 . :
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and the thesis is achieved. For the converse, by assumption f Ž x 2 . y f Ž x1 . G G
max
² x*, Ž x 1 , x 2 . : y ² x* Ž x 1 , x 2 . , Ž x 1 , x 2 . :
max
² x*, Ž x 1 , x 2 . : y
x*g ⭸ K f Ž x 1 .
x*g ⭸ K f Ž x 1 .
max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . :
s f K Ž x1 , Ž x1 , x 2 . . .
Remark 2.1. In particular, if f is K-subdifferentiable, then f is K-invex with respect to the kernel if and only if for each x 1 , x 2 g X f Ž x 2 . y f Ž x 1 . G ² x*, Ž x 1 , x 2 . : ,
᭙ x* g ⭸ K f Ž x 1 . .
To deepen the analysis of the structure of this class of functions, we consider a generalization of the result given by Craven and Glover w7x and Ben-Israel and Mond w1x and proved for differentiable functions and adapted by Giorgi and Guerraggio w12x for Lipschitz nonsmooth functions. THEOREM 2.2. Let f : X ª ⺢ and K be a local cone approximation; f is K-in¨ ex if and only if e¨ ery K-inf-stationary point is a global minimum point. Proof. Consider the following two cases. Ža. If x 1 , x 2 g X are such that f Ž x 2 . G f Ž x 1 . we take Ž x 1 , x 2 . [ 0. Žb. If x 1 , x 2 g X are such that f Ž x 2 . - f Ž x 1 ., then x 1 cannot be a K-inf-stationary point and therefore there exists a direction y g ⺢ n such that f K Ž x 1 , y . - 0. If we consider the function
Ž x1 , x 2 . [
f Ž x 2 . y f Ž x1 . f K Ž x1 , y .
y
then f K Ž x1 , Ž x1 , x 2 . . s
f Ž x 2 . y f Ž x1 . f K Ž x1 , y .
f K Ž x1 , y . s f Ž x 2 . y f Ž x1 . ,
hence f is K-invex with respect to . The converse implication is immediate.
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NONSMOOTH INVEX FUNCTIONS
EXAMPLE 2.1. Given f Ž x . s y5 x 5, where 5 ⭈ 5 is the Euclidean norm in ⺢ n, then
¡ 1 ¢5 y 5 , ¡y 1 ² x, y :, ~ 5 5 f ⬘ Ž x, y . s ¢y5 yx5, ² x, y : , y f⬚ Ž x, y . s ~ 5 x 5
if x / 0, if x s 0, if x / 0, if x s 0.
Since x s 0 is a TC l-inf-stationary point but it is not a global minimum, for Theorem 2.2, f is not TC l-invex; on the contrary f is F ŽWF .-invex with respect to the kernel
¡5 x 5 x , Ž x , x . s~ 5 x 5 ¢x , 2
if x 1 / 0,
1
1
1
2
if x 1 s 0.
2
The function f : ⺢ ª ⺢ defined by f Ž x. s
½
< x <, y< x < ,
if x g ⺡, if x f ⺡
is not F-invex because x s 0 is a F-inf-stationary point Ž Dq f Ž0, y . s < y <. but it is not a global minimum; on the contrary it is WF-invex with respect to the kernel
¡ x Ž < x < y < x <. , Ž x , x . s~ < x < ¢< x < , 1
1
1
2
2
if x 1 / 0,
1
2
if x 1 s 0.
Straightforward extensions of K-invexity can be made as follows. DEFINITION 2.2. Let K be a local cone approximation; the function f : X ª ⺢ is said to be 䢇
K-pseudoin¨ ex if there exists a function : X = X ª ⺢ n such that f K Ž x1 , Ž x1 , x 2 . . G 0 « f Ž x 2 . G f Ž x1 . ,
䢇
᭙ x1 , x 2 g X ;
K-quasi-in¨ ex if there exists a function : X = X ª ⺢ n such that f Ž x 2 . F f Ž x 1 . « f K Ž x 1 , Ž x 1 , x 2 . . F 0,
᭙ x1 , x 2 g X ;
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MARCO CASTELLANI
strictly K-quasi-in¨ ex if there exists a functional : X = X ª ⺢ n such that 䢇
f Ž x 2 . F f Ž x 1 . « f K Ž x 1 , Ž x 1 , x 2 . . - 0,
᭙ x1 / x 2 g X .
We observe that every K-invex function is both K-pseudoinvex and K-quasi-invex with respect to the same kernel. In w19x it was shown the existence of a K-pseudoinvex function with respect to the kernel which is not K-invex with respect to the same . Nevertheless, from the characterization given in Theorem 2.2 for K-invex functions, it is immediate to deduce that the two definitions coincide. In other words a K-pseudoinvex function may be not K-invex for the same but will be K-invex for some . With the next proof we emphasize this fact and we show how the kernel of invexity can change. THEOREM 2.3. Let f : X ª ⺢ and K be a local cone approximation; if f is K-pseudoin¨ ex then f is K-in¨ ex. Proof. Let f be K-pseudoinvex with respect to and, in order to show that f is K-invex, we consider the following two cases. Ža. If x 1 , x 2 g X are such that f Ž x 2 . G f Ž x 1 . it is sufficient to take Ž x 1 , x 2 . [ 0. Žb. If x 1 , x 2 g X are such that f Ž x 2 . - f Ž x 1 ., then f K Ž x 1 , Ž x 1 , x 2 .. - 0. If we consider the function
Ž x1 , x 2 . [
f Ž x 2 . y f Ž x1 . f
K
f
K
Ž x1 , Ž x1 , x 2 . .
Ž x1 , x 2 .
then f K Ž x1 , Ž x1 , x 2 . . s
f Ž x 2 . y f Ž x1 .
Ž x1 , Ž x1 , x 2 . .
f K Ž x1 , Ž x1 , x 2 . .
s f Ž x 2 . y f Ž x1 . , hence f is K-invex with respect to . Remark 2.2. We observe that no assumption is required for proving Theorem 2.3. The same result was established by Giorgi and Guerraggio w12x for the case of locally Lipschitz functions and via Clarke’s tangent cone. Moreover, the same result was obtained by Reiland w22x but under the unnecessary condition that the cone
D Ž ⭸ ⬚ f Ž x 1 . = Ž f Ž x 2 . y f Ž x 1 . . 4 .
G0
be closed.
NONSMOOTH INVEX FUNCTIONS
327
3. SUFFICIENT OPTIMALITY CONDITIONS In this section we study the extremum problem min f 0 Ž x . : f i Ž x . F 0, i g I 4 ,
Ž P.
where X : ⺢ n is an open set, f 0 , f i : X ª ⺢, and I [ 1, . . . , m4 . For every feasible point x we denote I Ž x . [ i g I: f i Ž x . s 04 ; moreover I0 [ I j 04 and I0 Ž x . [ I Ž x . j 04 . In the last two decades many generalizations of the Kuhn᎐Tucker necessary optimality condition for the problem Ž P . have been stated without assuming the differentiability of the functions f i Žsee w5, 16, 23, 24x and references therein .. It has been proved Žsee for instance w3, 4x. that these necessary optimality conditions are equivalent to the impossibility of suitable systems of sublinear functions. DEFINITION 3.1. Let x be a feasible point for Ž P . and K i , with i g I0 Ž x ., be local cone approximations; the point x is said to be: a weakly stationary point for the problem Ž P . with respect to K i if the following system is impossible 䢇
½
f 0K 0 Ž x, y . - 0, f iK i Ž x, y . - 0,
i g IŽ x. ;
Ž S1 .
a strongly stationary point for the problem Ž P . with respect to K i if the following system is impossible 䢇
½
f K 0 Ž x, y . - 0, f iK i Ž x, y . F 0,
i g IŽ x. .
Ž S2 .
In w3, 4, 23, 25x it was shown that it is always possible to choose suitable local cone approximations K i , with i g I0 Ž x ., which do not depend on the functions f i and such that every local optimal solution x is a weakly stationary point with respect to K i . Such local cone approximations are called admissible. For instance, K 0 s WF and K i s F, or, if f i are locally Lipschitz functions, K 0 s K i s T, are admissible. Moreover, if some regularity condition holds, it is possible to prove w2x that every weakly stationary point is a strongly stationary point. Finally, if f iK i Ž x, ⭈ ., with i g I0 Ž x ., are convex Žor difference of convex functions or, more generally, pointwise minimum of sublinear functions., it is possible to prove, through theorems of the alternative w13, 14x, that the impossibility of the systems Ž S1 . and Ž S2 . are equivalent to the generaliza-
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tions of the John and Kuhn᎐Tucker necessary optimality conditions, respectively. In other words if, for instance, f iK i Ž x, ⭈ ., with i g I0 Ž x ., are convex, the impossibility of Ž S1 . is equivalent to the existence of the John multipliers i G 0, with i g I0 Ž x ., not all zero, such that 0g
Ý
i ⭸ K i fi Ž x . ;
Ž J.
igI 0 Ž x .
while the impossibility of Ž S2 ., under a suitable regularity condition, is equivalent to the existence of the Kuhn᎐Tucker multipliers i G 0, with i g I Ž x ., such that 0 g ⭸ K 0 f0 Ž x . q
Ý
i ⭸ K i fi Ž x . .
Ž KT .
igI Ž x .
All the sufficiency results stated for nonsmooth invex functions are deduced from the necessary optimality conditions Ž J . or Ž KT .; hence they require the convexity of the directional derivatives w19, 22, 26x. The results of this section show that, under suitable assumptions of invexity, it is possible to deduce sufficient optimality conditions directly from the impossibility of the systems Ž S1 . and Ž S2 .. THEOREM 3.1. Let x be a strongly stationary point for the problem Ž P . with respect to K i , i g I0 Ž x .. If f 0 is K 0-pseudoin¨ ex and f i are K i-quasi-in¨ ex with respect to the same kernel then x is a global optimal solution for Ž P .. Proof. Let x be any feasible point for Ž P .. Then fi Ž x . F 0 s fi Ž x . ,
᭙i g IŽ x. .
By the K i-quasiinvexity of f i , we have f iK i Ž x, Ž x, x . . F 0,
᭙i g IŽ x. .
Since Ž S2 . is impossible f 0K 0 Ž x, Ž x, x . . G 0, and, by the K 0-pseudoinvexity of f 0 , we conclude f 0 Ž x . G f 0 Ž x .. EXAMPLE 3.1. Given the problem min f 0 Ž x 1 , x 2 . s x 12 y 2 < x 1 < q < x 2 < , f 1 Ž x 1 , x 2 . s x 14 q 5 < x 1 < y 2 < x 2 < F 0,
NONSMOOTH INVEX FUNCTIONS
329
if we consider the admissible pair ŽWF, F ., we have f 0X Ž Ž x 1 , x 2 . , Ž 1 , 2 . . s 2 x 11 y 2 Ž x 1 , 1 . q Ž x 2 , 2 . , f 1X Ž Ž x 1 , x 2 . , Ž 1 , 2 . . s 4 x 131 q 5 Ž x 1 , 1 . y 2 Ž x 2 , 2 . , where : ⺢ 2 ª ⺢ is defined by
¡ ¢
, Ž x, . s < < , y ,
~
if x ) 0, if x s 0, if x - 0.
It is immediate to see that x s Ž0, 0. is a strongly stationary point, that is, the following system is impossible
½
f 0X Ž Ž 0, 0 . , Ž 1 , 2 . . s y2 <1 < q < 2 < - 0, f 1X Ž Ž 0, 0 . , Ž 1 , 2 . . s 5 <1 < y 2 < 2 < F 0.
Since f 0 is WF-pseudoinvex and f 1 is F-quasi-invex with respect to the same kernel
Ž Ž x 1 , x 2 . , Ž y 1 , y 2 . . s Ž yx 1 q Ž x 1 , < y 1 < . , yx 2 q Ž x 2 , < y 2 < . . , the conditions of Theorem 3.1 are satisfied and x s Ž0, 0. is an optimal solution of Ž P .. Some remarks are needed: first of all we observe that f iX ŽŽ0, 0., Ž1 , 2 .. are not sublinear and therefore we cannot consider a necessary optimality condition of Kuhn᎐Tucker-type with subdifferentials w26x. Nevertheless the functions are quasidifferentiable w8x and therefore it is possible to have a necessary optimality condition expressed via quasidifferentials w8x. Moreover, since f i are Lipschitzian, we can exploit the Clarke’s directional derivative. In such a way we get the impossibility of the system
½
< < < < fT 0 Ž Ž 0, 0 . , Ž 1 , 2 . . s 2 1 q 2 - 0, < < < < fT 0 Ž Ž 0, 0 . , Ž 1 , 2 . . s 5 1 q 2 2 F 0.
We observe that x s Ž0, 0. is a TC l-inf-stationary point for f 0 but it is not a global minimum; therefore f 0 is not TC l-pseudoinvex. Hence we cannot apply Theorem 3.1 and related theory w19, 22x. We have noted that the impossibility of Ž S2 . descends from the impossibility of Ž S1 . in the presence of a regularity condition. Nevertheless, even if we do not have regularity but we strengthen the hypothesis of invexity of the constraint functions, the impossibility of the system Ž S1 . implies the optimality of x.
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MARCO CASTELLANI
THEOREM 3.2. Let x be a weakly stationary point for the problem Ž P . with respect to K i , i g I0 Ž x .. If f 0 is K 0-pseudoin¨ ex and the f i are strictly K i-quasi-in¨ ex with respect to the same kernel then x is a global optimal solution for Ž P .. Proof. The proof is the same as that of Theorem 3.1 except that, by the strict K i-quasi-invexity of f i we can deduce f iK i Ž x, Ž x, x .. - 0. 4. DUALITY We conclude giving weak and strong duality results for problems with K i-subdifferentiable functions. Consider the following modified Mond᎐ Weir w20x dual problem:
¡max f Ž u. ~
0 g ⭸ K 0 f0 Ž u. q
Ý i ⭸ K fi Ž u . , i
igI
i f i Ž u . G 0, i G 0.
Ž D.
¢
The following duality results are established for Ž P . and Ž D .. THEOREM 4.1 ŽWeak Duality.. Let x be feasible for Ž P . and Ž u, . be feasible for Ž D .. If f 0 is K 0-pseudoin¨ ex and f i is K i-quasi-in¨ ex with respect to the same kernel , then f 0 Ž x . G f 0 Ž u.. Proof. Since i G 0 and f i Ž x . F 0 we have i f i Ž x . F i f i Ž u.. By K-quasi-invexity of f i we obtain ² uUi , Ž x, u . : F 0,
᭙ uUi g ⭸ K i f i Ž u . .
Moreover, by feasibility of Ž u, ., there exists uUi g ⭸ K i f i Ž u. such that uU0 q
Ý i uUi s 0. igI
Therefore 0G
Ý i ² uUi , Ž x, u . : s ² y uU0 , Ž x, u . : . igI
Since f 0 is K-pseudoinvex we achieve the thesis. THEOREM 4.2 ŽStrong Duality.. Let x be a local minimum of Ž P . and assume that a regularity condition holds at x. Then there exists such that
NONSMOOTH INVEX FUNCTIONS
331
Ž x, . is feasible for Ž D .. Moreo¨ er if f 0 is K 0-pseudoin¨ ex and f i is K i-quasi-in¨ ex with respect to the same kernel , then x and Ž x, . are global minima of Ž P . and Ž D ., respecti¨ ely. Proof. By assumption there exists such that Ž x, . is feasible for Ž D .. From Theorem 4.1 we achieve the thesis. When the f i are locally Lipschitz, choosing K i s TC l , we get a result similar to one expressed in w19, 22x.
REFERENCES 1. A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 Ž1986., 1᎐9. 2. M. Castellani and F. Romeo, On Constraint Qualifications in Nonlinear Programming, Technical Report, University of Pisa, 1998. 3. M. Castellani and M. Pappalardo, First order cone approximations and necessary optimality conditions, Optimization 35 Ž1995., 113᎐126. 4. M. Castellani and M. Pappalardo, Local second-order approximations and applications in optimization, Optimization, 37 Ž1996., 305᎐321. 5. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Wiley, New York, 1984. 6. B. D. Craven, Invex functions and constrained local minima, Bull. Austral. Math. Soc. 24 Ž1981., 357᎐366. 7. B. D. Craven and B. M. Glover, Invex functions and duality, J. Austral. Math. Soc. Ser. A 39 Ž1985., 1᎐20. 8. V. F. Demyanov and M. A. Rubinov, On quasidifferentiable functions, So¨ iet Math. Dokl. 21 Ž1980., 14᎐17. 9. K. H. Elster and J. Thierfelder, On cone approximations and generalized directional derivatives, in ‘‘Nonsmooth Optimization and Related Topics’’ ŽF. H. Clarke, V. F. Demyanov, and F. Giannessi, Eds.., pp. 133᎐159, 1989. 10. K. H. Elster and J. Thierfelder, On cone approximations and generalized directional derivatives, in ‘‘Nonsmooth Optimization and Related Topics’’ ŽF. H. Clarke, V. F. Demyanov, and F. Giannessi, Eds.., pp. 133᎐159, 1989. 11. G. Giorgi, A note on the relationship between convexity and invexity, J. Austral. Math. Soc. Ser. B 32 Ž1990., 97᎐99. 12. G. Giorgi and A. Guerraggio, Various types of nonsmooth invex functions, J. Inform. Optim. Sci. 17 Ž1996., 137᎐150. 13. B. M. Glover, Y. Ishizuka, V. Jeyakumar, and H. D. Tuan, Complete characterizations of global optimality for problems involving the pointwise minimum of sublinear functions, SIAM J. Optim. 6 Ž1996., 362᎐372. 14. B. M. Glover, V. Jeyakumar, and W. Oettli, A Farkas lemma for difference sublinear systems and quasidifferentiable programming, Math. Programming 63 Ž1994., 109᎐125. 15. M. A. Hanson, On sufficiency of the Kuhn᎐Tucker conditions, J. Math. Anal. Appl. 80 Ž1981., 545᎐550. 16. J. B. Hiriart-Urruty, On optimality conditions in nondifferentiable programming, Math. Programming 14 Ž1978., 73᎐86. 17. V. Jeyakumar, On optimality conditions in nonsmooth inequality constrained minimization, Numer. Funct. Anal. Optim. 9 Ž1987., 535᎐546.
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18. R. N. Kaul and S. Kaur, Optimality criteria in nonlinear programming involving nonconvex functions, J. Math. Anal. Appl. 105 Ž1985., 104᎐112. 19. R. N. Kaul, S. K. Suneja, and C. S. Lalitha, Generalized nonsmooth invexity, J. Inform. Optim. Sci. 15 Ž1994., 1᎐17. 20. B. Mond and T. Weir, Generalized convexity and duality, in ‘‘Generalized Concavity in Optimization and Economics’’ ŽS. Schaible and W. T. Ziemba, Eds.., pp. 263᎐280, 1981. 21. B. N. Pshenichnyi, ‘‘Necessary Conditions for an Extremum,’’ Dekker, New York, 1971. 22. T. W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc. 42 Ž1990., 437᎐446. 23. D. E. Ward, Isotone tangent cones and nonsmooth optimization, Optimization 18 Ž1987., 769᎐783. 24. D. E. Ward, Directional derivative calculus and optimality conditions in nonsmooth mathematical programming, J. Inform. Optim. Sci. 10 Ž1989., 81᎐96. 25. D. E. Ward, Convex directional derivatives in optimization, in Lecture Notes in Econom. and Math. Systems, Vol. 345, pp. 36᎐51, Springer-Verlag, New YorkrBerlin, 1990. 26. Y. L. Ye, d-invexity and optimality conditions, J. Math. Anal. Appl. 162 Ž1991., 242᎐249.
Nonsmooth Invex Functions and Sufficient Optimality Conditions Marco Castellani Dipartimento di Sistemi ed Istituzioni per L’Economia, Piazza del Santuario, 19 67040 L’Aquila, Italy E-mail: [email protected] Submitted by H. P. Benson Received December 23, 1997
In this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of K-directional derivative. Characterizations of nonsmooth K-invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the K-directional derivatives. 䊚 2001 Academic Press
1. INTRODUCTION AND NOTATIONS In w15x, Hanson presented a weakened concept of convexity for differentiable functions: a differentiable function f : X ª ⺢ is said to be in¨ ex, if there exists a function : X = X ª ⺢ n such that 䢇
f Ž x 2 . y f Ž x 1 . G ² ⵜf Ž x 1 . , Ž x 1 , x 2 . : ,
᭙ x1 , x 2 g X .
The name invex descends from a contraction of ‘‘invariant convex’’ and it was proposed by Craven w6x. In w7x, Craven and Glover showed that the class of invex functions is equivalent to the class of functions whose stationary points are global minima. In w18x, Kaul and Kaur considered the following generalizations: f is said to be pseudoin¨ ex if there exists a function : X = X ª ⺢ n such that 䢇
² ⵜf Ž x 1 . , Ž x 1 , x 2 . : G 0 « f Ž x 2 . G f Ž x 1 . ,
᭙ x1 , x 2 g X ;
319 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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MARCO CASTELLANI
f is said to be quasi-in¨ ex if there exists a function : X = X ª ⺢ n such that 䢇
f Ž x 2 . F f Ž x 1 . « ² ⵜf Ž x 1 . , Ž x 1 , x 2 . : F 0,
᭙ x1 , x 2 g X .
By means of these concepts they established sufficient optimality conditions for a nonlinear programming problem with inequality constraints. In w1, 11x relations among convex and invex functions and their generalizations were studied. Recently Reiland w22x extended the concept of invexity to the class of locally Lipschitz functions using the generalized gradient of Clarke. An analogous extension was made by Jeyakumar w17x by means of the notion of approximate quasidifferentiability for nonsmooth functions. Different definitions of nonsmooth invexity were introduced by Ye w26x and Giorgi and Guerraggio w12x who studied the relations among all these classes of functions. In this paper we propose a unifying definition of invexity for nonsmooth functions exploiting the concept of local cone approximation introduced in w9x. Moreover, via such an approach, we give sufficient optimality conditions for inequality constrained extremum problems without requiring the convexity of the directional derivatives. In the sequel X : ⺢ n will be an open set. Given the function f : X ª ⺢, the epigraph of f is epi f [ Ž x, y . g X = ⺢ : f Ž x . F y 4 . The set epi f will be locally approximated at the point Ž x, f Ž x .. by a local cone approximation K and a positively homogeneous function f K Ž x, ⭈ . will be uniquely determined. DEFINITION 1.1. Let f : X ª ⺢, x g X and K be a local cone approximation; the positively homogeneous function f K Ž x, ⭈ .: ⺢ n ª wy⬁, q⬁x defined by f K Ž x, y . [ inf  g ⺢ : Ž y,  . g K Ž epi f , Ž x, f Ž x . . . 4 is called the K-directional deri¨ ati¨ e of f at x. By means of Definition 1.1 we can recover most of the generalized directional derivatives used in literature; for instance 䢇
the upper Dini directional deri¨ ati¨ e of f at x Dq f Ž x, y . [ lim sup tª0 q
f Ž x q ty . y f Ž x . t
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is associated to the cone of the feasible directions F Ž Q, x . [ y g ⺢ n : ᭙ t k 4 ª 0q, x q t k y g Q 4 ; 䢇
the lower Dini directional deri¨ ati¨ e of f at x Dy f Ž x, y . [ lim inf q
f Ž x q ty . y f Ž x . t
tª0
is associated to the cone of the weak feasible directions WF Ž Q, x . [ y g ⺢ n : ᭚ t k 4 ª 0q s.t. x q t k y g Q 4 ; 䢇
if f is locally Lipschitz, the Clarke directional deri¨ ati¨ e of f at x f⬚ Ž x, y . [
f Ž x⬘ q ty . y f Ž x⬘ .
lim sup
t
Ž x ⬘, t .ª Ž x , 0 q.
is associated to Clarke’s tangent cone TC l Ž Q, x . [ y g ⺢ n : ᭙ x k 4 ª x s.t. x k g Q, ᭙ t k 4 ª 0q, ᭚ y k 4 ª y s.t. x k q t k y k g Q 4 . It is well known that f⬚Ž x, ⭈ . G Dq f Ž x, ⭈ . G Dy f Ž x, ⭈ .. For a more detailed review about the local cone approximations we refer to w10x. DEFINITION 1.2. Let f : X ª ⺢, x g X and K be a local cone approximation; f is said to be K-subdifferentiable at x if there exists a convex compact set ⭸ K f Ž x . such that 䢇
f K Ž x, y . s
max
x*g ⭸ f Ž x .
² x*, y : ,
K
᭙ y g ⺢n;
the set ⭸ K f Ž x . is called the K-subdifferential of f at x. f is said K-quasidifferentiable at x if there exist two convex compact sets ⭸ K f Ž x . and ⭸ K f Ž x . such that 䢇
f K Ž x, y . s
max
x*g ⭸ f Ž x . K
² x*, y : y
max
x*g ⭸ f Ž x . K
² x*, y : ,
᭙ y g ⺢n;
the sets ⭸ K f Ž x . and ⭸ K f Ž x . are called the K-subdifferential and K-superdifferential of f at x, respectively.
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Remark 1.1. It is immediate to observe that every K-subdifferentiable function is K-quasidifferentiable with ⭸ K f Ž x . s 04 and ⭸ K f Ž x . s ⭸ K f Ž x .. It is well known that the Clarke derivative is bounded and convex; therefore there exists the Clarke subdifferential ⭸ ⬚ f [ ⭸ T C l f. If f is directionally differentiable Ži.e., Dy f Ž x, ⭈ . s Dq f Ž x, ⭈ . [ f ⬘Ž x, ⭈ .. and it is F-subdifferentiable we say that f is quasidifferentiable in the sense of Pshenichnyi w21x, while if f is F-quasidifferentiable we say that f is quasidifferentiable in the sense of Demyanov and Rubinov w8x. DEFINITION 1.3. Let f : X ª ⺢ and K be a local cone approximation; x g X is said to be a K-inf-stationary point for f if f K Ž x, y . G 0 for each y g ⺢ n. The following result gives the characterization of a K-inf-stationary point for K-quasidifferentiable functions. THEOREM 1.1. Let f : X ª ⺢ and K be a local cone approximation. If f is K-quasidifferentiable, then x g X is a K-inf-stationary point for f if and only if ⭸ K f Ž x . : ⭸ K f Ž x .. Proof. Let x be a K-inf-stationary point and let us suppose by contradiction that there exists x* g ⭸ K f Ž x . such that x* f ⭸ K f Ž x .. Then there exist y g ⺢ n and ) 0 such that ² x*, y : G ² x*, y : q ,
᭙ x* g ⭸ K f Ž x . ;
hence ² x*, y : G
max
x*g ⭸ K f Ž x .
² x*, y : q ,
and then y G
max
x*g ⭸ K f Ž x .
² x*, y : y ² x*, y : G f K Ž x, y .
that contradicts the assumption. The converse implication is immediate. Remark 1.2. In particular, if f is K-subdifferentiable, then x g X is a K-inf-stationary point for f if and only if 0 g ⭸ K f Ž x .. In w3, 4x it was proved that if K is an isotone local approximation Ži.e., K Ž A, x . : K Ž B, x . for each A : B ., then every local minimizer of f over ⺢ n is a K-inf-stationary point for f. Unfortunately, in general it is not possible to deduce that a K-inf-stationary point is a local optimal solution. For this reason, in Section 2, we will introduce the concept of K-invexity.
NONSMOOTH INVEX FUNCTIONS
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2. K-INVEXITY In this section we propose a unifying definition of invexity for nonsmooth functions. DEFINITION 2.1. Let K be a local cone approximation; the function f : X ª ⺢ is said to be K-in¨ ex if there exists a function : X = X ª ⺢ n such that f Ž x 2 . y f Ž x1 . G f K Ž x1 , Ž x1 , x 2 . . ,
᭙ x1 , x 2 g X .
The function is said to be the kernel of the K-invexity. By means of Definition 2.1, we can obtain all the definitions of invexity for nonsmooth functions. For instance if we use Clarke’s tangent cone for locally Lipschitz functions, we recover the concept of invexity introduced by Reiland w22x; if we consider the class of the directionally differentiable functions and we take K s F, we get the d-invexity given by Ye w26x. Moreover if f is also F-subdifferentiable or F-quasidifferentiable, we obtain the P-invexity and the DR-invexity, respectively, studied in w12x. Finally, we observe that if f K 1 Ž x, ⭈ . G f K 2 Ž x, ⭈ . and f is K 1-invex then f is K 2-invex with respect to the same kernel; in particular every locally Lipschitz TC l-invex function w22x is also K-invex for K s F, WF. The following result shows the characterization of K-invexity for K-subŽor quasi-. differentiable functions. THEOREM 2.1. Let f : X ª ⺢ and K be a local cone approximation. If f is K-quasidifferentiable, then f is K-in¨ ex with respect to the kernel if and only if for each x 1 , x 2 g X and for each x* g ⭸ K f Ž x 1 . there exists x*Ž x 1 , x 2 . g ⭸ K f Ž x 1 . such that f Ž x 2 . y f Ž x 1 . G ² x* y x* Ž x 1 , x 2 . , Ž x 1 , x 2 . : . Proof. Let f be K-invex; then for each x 1 , x 2 g X and for each x* g ⭸ K f Ž x 1 . f Ž x 2 . y f Ž x1 . G f K Ž x1 , Ž x1 , x 2 . . G ² x*, Ž x 1 , x 2 . : y
max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . : .
Since ⭸ K f Ž x 1 . is a compact set, there exists x*Ž x 1 , x 2 . g ⭸ K f Ž x 1 . such that max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . : s ² x* Ž x 1 , x 2 . , Ž x 1 , x 2 . :
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and the thesis is achieved. For the converse, by assumption f Ž x 2 . y f Ž x1 . G G
max
² x*, Ž x 1 , x 2 . : y ² x* Ž x 1 , x 2 . , Ž x 1 , x 2 . :
max
² x*, Ž x 1 , x 2 . : y
x*g ⭸ K f Ž x 1 .
x*g ⭸ K f Ž x 1 .
max
x*g ⭸ K f Ž x 1 .
² x*, Ž x 1 , x 2 . :
s f K Ž x1 , Ž x1 , x 2 . . .
Remark 2.1. In particular, if f is K-subdifferentiable, then f is K-invex with respect to the kernel if and only if for each x 1 , x 2 g X f Ž x 2 . y f Ž x 1 . G ² x*, Ž x 1 , x 2 . : ,
᭙ x* g ⭸ K f Ž x 1 . .
To deepen the analysis of the structure of this class of functions, we consider a generalization of the result given by Craven and Glover w7x and Ben-Israel and Mond w1x and proved for differentiable functions and adapted by Giorgi and Guerraggio w12x for Lipschitz nonsmooth functions. THEOREM 2.2. Let f : X ª ⺢ and K be a local cone approximation; f is K-in¨ ex if and only if e¨ ery K-inf-stationary point is a global minimum point. Proof. Consider the following two cases. Ža. If x 1 , x 2 g X are such that f Ž x 2 . G f Ž x 1 . we take Ž x 1 , x 2 . [ 0. Žb. If x 1 , x 2 g X are such that f Ž x 2 . - f Ž x 1 ., then x 1 cannot be a K-inf-stationary point and therefore there exists a direction y g ⺢ n such that f K Ž x 1 , y . - 0. If we consider the function
Ž x1 , x 2 . [
f Ž x 2 . y f Ž x1 . f K Ž x1 , y .
y
then f K Ž x1 , Ž x1 , x 2 . . s
f Ž x 2 . y f Ž x1 . f K Ž x1 , y .
f K Ž x1 , y . s f Ž x 2 . y f Ž x1 . ,
hence f is K-invex with respect to . The converse implication is immediate.
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NONSMOOTH INVEX FUNCTIONS
EXAMPLE 2.1. Given f Ž x . s y5 x 5, where 5 ⭈ 5 is the Euclidean norm in ⺢ n, then
¡ 1 ¢5 y 5 , ¡y 1 ² x, y :, ~ 5 5 f ⬘ Ž x, y . s ¢y5 yx5, ² x, y : , y f⬚ Ž x, y . s ~ 5 x 5
if x / 0, if x s 0, if x / 0, if x s 0.
Since x s 0 is a TC l-inf-stationary point but it is not a global minimum, for Theorem 2.2, f is not TC l-invex; on the contrary f is F ŽWF .-invex with respect to the kernel
¡5 x 5 x , Ž x , x . s~ 5 x 5 ¢x , 2
if x 1 / 0,
1
1
1
2
if x 1 s 0.
2
The function f : ⺢ ª ⺢ defined by f Ž x. s
½
< x <, y< x < ,
if x g ⺡, if x f ⺡
is not F-invex because x s 0 is a F-inf-stationary point Ž Dq f Ž0, y . s < y <. but it is not a global minimum; on the contrary it is WF-invex with respect to the kernel
¡ x Ž < x < y < x <. , Ž x , x . s~ < x < ¢< x < , 1
1
1
2
2
if x 1 / 0,
1
2
if x 1 s 0.
Straightforward extensions of K-invexity can be made as follows. DEFINITION 2.2. Let K be a local cone approximation; the function f : X ª ⺢ is said to be 䢇
K-pseudoin¨ ex if there exists a function : X = X ª ⺢ n such that f K Ž x1 , Ž x1 , x 2 . . G 0 « f Ž x 2 . G f Ž x1 . ,
䢇
᭙ x1 , x 2 g X ;
K-quasi-in¨ ex if there exists a function : X = X ª ⺢ n such that f Ž x 2 . F f Ž x 1 . « f K Ž x 1 , Ž x 1 , x 2 . . F 0,
᭙ x1 , x 2 g X ;
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MARCO CASTELLANI
strictly K-quasi-in¨ ex if there exists a functional : X = X ª ⺢ n such that 䢇
f Ž x 2 . F f Ž x 1 . « f K Ž x 1 , Ž x 1 , x 2 . . - 0,
᭙ x1 / x 2 g X .
We observe that every K-invex function is both K-pseudoinvex and K-quasi-invex with respect to the same kernel. In w19x it was shown the existence of a K-pseudoinvex function with respect to the kernel which is not K-invex with respect to the same . Nevertheless, from the characterization given in Theorem 2.2 for K-invex functions, it is immediate to deduce that the two definitions coincide. In other words a K-pseudoinvex function may be not K-invex for the same but will be K-invex for some . With the next proof we emphasize this fact and we show how the kernel of invexity can change. THEOREM 2.3. Let f : X ª ⺢ and K be a local cone approximation; if f is K-pseudoin¨ ex then f is K-in¨ ex. Proof. Let f be K-pseudoinvex with respect to and, in order to show that f is K-invex, we consider the following two cases. Ža. If x 1 , x 2 g X are such that f Ž x 2 . G f Ž x 1 . it is sufficient to take Ž x 1 , x 2 . [ 0. Žb. If x 1 , x 2 g X are such that f Ž x 2 . - f Ž x 1 ., then f K Ž x 1 , Ž x 1 , x 2 .. - 0. If we consider the function
Ž x1 , x 2 . [
f Ž x 2 . y f Ž x1 . f
K
f
K
Ž x1 , Ž x1 , x 2 . .
Ž x1 , x 2 .
then f K Ž x1 , Ž x1 , x 2 . . s
f Ž x 2 . y f Ž x1 .
Ž x1 , Ž x1 , x 2 . .
f K Ž x1 , Ž x1 , x 2 . .
s f Ž x 2 . y f Ž x1 . , hence f is K-invex with respect to . Remark 2.2. We observe that no assumption is required for proving Theorem 2.3. The same result was established by Giorgi and Guerraggio w12x for the case of locally Lipschitz functions and via Clarke’s tangent cone. Moreover, the same result was obtained by Reiland w22x but under the unnecessary condition that the cone
D Ž ⭸ ⬚ f Ž x 1 . = Ž f Ž x 2 . y f Ž x 1 . . 4 .
G0
be closed.
NONSMOOTH INVEX FUNCTIONS
327
3. SUFFICIENT OPTIMALITY CONDITIONS In this section we study the extremum problem min f 0 Ž x . : f i Ž x . F 0, i g I 4 ,
Ž P.
where X : ⺢ n is an open set, f 0 , f i : X ª ⺢, and I [ 1, . . . , m4 . For every feasible point x we denote I Ž x . [ i g I: f i Ž x . s 04 ; moreover I0 [ I j 04 and I0 Ž x . [ I Ž x . j 04 . In the last two decades many generalizations of the Kuhn᎐Tucker necessary optimality condition for the problem Ž P . have been stated without assuming the differentiability of the functions f i Žsee w5, 16, 23, 24x and references therein .. It has been proved Žsee for instance w3, 4x. that these necessary optimality conditions are equivalent to the impossibility of suitable systems of sublinear functions. DEFINITION 3.1. Let x be a feasible point for Ž P . and K i , with i g I0 Ž x ., be local cone approximations; the point x is said to be: a weakly stationary point for the problem Ž P . with respect to K i if the following system is impossible 䢇
½
f 0K 0 Ž x, y . - 0, f iK i Ž x, y . - 0,
i g IŽ x. ;
Ž S1 .
a strongly stationary point for the problem Ž P . with respect to K i if the following system is impossible 䢇
½
f K 0 Ž x, y . - 0, f iK i Ž x, y . F 0,
i g IŽ x. .
Ž S2 .
In w3, 4, 23, 25x it was shown that it is always possible to choose suitable local cone approximations K i , with i g I0 Ž x ., which do not depend on the functions f i and such that every local optimal solution x is a weakly stationary point with respect to K i . Such local cone approximations are called admissible. For instance, K 0 s WF and K i s F, or, if f i are locally Lipschitz functions, K 0 s K i s T, are admissible. Moreover, if some regularity condition holds, it is possible to prove w2x that every weakly stationary point is a strongly stationary point. Finally, if f iK i Ž x, ⭈ ., with i g I0 Ž x ., are convex Žor difference of convex functions or, more generally, pointwise minimum of sublinear functions., it is possible to prove, through theorems of the alternative w13, 14x, that the impossibility of the systems Ž S1 . and Ž S2 . are equivalent to the generaliza-
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tions of the John and Kuhn᎐Tucker necessary optimality conditions, respectively. In other words if, for instance, f iK i Ž x, ⭈ ., with i g I0 Ž x ., are convex, the impossibility of Ž S1 . is equivalent to the existence of the John multipliers i G 0, with i g I0 Ž x ., not all zero, such that 0g
Ý
i ⭸ K i fi Ž x . ;
Ž J.
igI 0 Ž x .
while the impossibility of Ž S2 ., under a suitable regularity condition, is equivalent to the existence of the Kuhn᎐Tucker multipliers i G 0, with i g I Ž x ., such that 0 g ⭸ K 0 f0 Ž x . q
Ý
i ⭸ K i fi Ž x . .
Ž KT .
igI Ž x .
All the sufficiency results stated for nonsmooth invex functions are deduced from the necessary optimality conditions Ž J . or Ž KT .; hence they require the convexity of the directional derivatives w19, 22, 26x. The results of this section show that, under suitable assumptions of invexity, it is possible to deduce sufficient optimality conditions directly from the impossibility of the systems Ž S1 . and Ž S2 .. THEOREM 3.1. Let x be a strongly stationary point for the problem Ž P . with respect to K i , i g I0 Ž x .. If f 0 is K 0-pseudoin¨ ex and f i are K i-quasi-in¨ ex with respect to the same kernel then x is a global optimal solution for Ž P .. Proof. Let x be any feasible point for Ž P .. Then fi Ž x . F 0 s fi Ž x . ,
᭙i g IŽ x. .
By the K i-quasiinvexity of f i , we have f iK i Ž x, Ž x, x . . F 0,
᭙i g IŽ x. .
Since Ž S2 . is impossible f 0K 0 Ž x, Ž x, x . . G 0, and, by the K 0-pseudoinvexity of f 0 , we conclude f 0 Ž x . G f 0 Ž x .. EXAMPLE 3.1. Given the problem min f 0 Ž x 1 , x 2 . s x 12 y 2 < x 1 < q < x 2 < , f 1 Ž x 1 , x 2 . s x 14 q 5 < x 1 < y 2 < x 2 < F 0,
NONSMOOTH INVEX FUNCTIONS
329
if we consider the admissible pair ŽWF, F ., we have f 0X Ž Ž x 1 , x 2 . , Ž 1 , 2 . . s 2 x 11 y 2 Ž x 1 , 1 . q Ž x 2 , 2 . , f 1X Ž Ž x 1 , x 2 . , Ž 1 , 2 . . s 4 x 131 q 5 Ž x 1 , 1 . y 2 Ž x 2 , 2 . , where : ⺢ 2 ª ⺢ is defined by
¡ ¢
, Ž x, . s < < , y ,
~
if x ) 0, if x s 0, if x - 0.
It is immediate to see that x s Ž0, 0. is a strongly stationary point, that is, the following system is impossible
½
f 0X Ž Ž 0, 0 . , Ž 1 , 2 . . s y2 <1 < q < 2 < - 0, f 1X Ž Ž 0, 0 . , Ž 1 , 2 . . s 5 <1 < y 2 < 2 < F 0.
Since f 0 is WF-pseudoinvex and f 1 is F-quasi-invex with respect to the same kernel
Ž Ž x 1 , x 2 . , Ž y 1 , y 2 . . s Ž yx 1 q Ž x 1 , < y 1 < . , yx 2 q Ž x 2 , < y 2 < . . , the conditions of Theorem 3.1 are satisfied and x s Ž0, 0. is an optimal solution of Ž P .. Some remarks are needed: first of all we observe that f iX ŽŽ0, 0., Ž1 , 2 .. are not sublinear and therefore we cannot consider a necessary optimality condition of Kuhn᎐Tucker-type with subdifferentials w26x. Nevertheless the functions are quasidifferentiable w8x and therefore it is possible to have a necessary optimality condition expressed via quasidifferentials w8x. Moreover, since f i are Lipschitzian, we can exploit the Clarke’s directional derivative. In such a way we get the impossibility of the system
½
< < < < fT 0 Ž Ž 0, 0 . , Ž 1 , 2 . . s 2 1 q 2 - 0, < < < < fT 0 Ž Ž 0, 0 . , Ž 1 , 2 . . s 5 1 q 2 2 F 0.
We observe that x s Ž0, 0. is a TC l-inf-stationary point for f 0 but it is not a global minimum; therefore f 0 is not TC l-pseudoinvex. Hence we cannot apply Theorem 3.1 and related theory w19, 22x. We have noted that the impossibility of Ž S2 . descends from the impossibility of Ž S1 . in the presence of a regularity condition. Nevertheless, even if we do not have regularity but we strengthen the hypothesis of invexity of the constraint functions, the impossibility of the system Ž S1 . implies the optimality of x.
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THEOREM 3.2. Let x be a weakly stationary point for the problem Ž P . with respect to K i , i g I0 Ž x .. If f 0 is K 0-pseudoin¨ ex and the f i are strictly K i-quasi-in¨ ex with respect to the same kernel then x is a global optimal solution for Ž P .. Proof. The proof is the same as that of Theorem 3.1 except that, by the strict K i-quasi-invexity of f i we can deduce f iK i Ž x, Ž x, x .. - 0. 4. DUALITY We conclude giving weak and strong duality results for problems with K i-subdifferentiable functions. Consider the following modified Mond᎐ Weir w20x dual problem:
¡max f Ž u. ~
0 g ⭸ K 0 f0 Ž u. q
Ý i ⭸ K fi Ž u . , i
igI
i f i Ž u . G 0, i G 0.
Ž D.
¢
The following duality results are established for Ž P . and Ž D .. THEOREM 4.1 ŽWeak Duality.. Let x be feasible for Ž P . and Ž u, . be feasible for Ž D .. If f 0 is K 0-pseudoin¨ ex and f i is K i-quasi-in¨ ex with respect to the same kernel , then f 0 Ž x . G f 0 Ž u.. Proof. Since i G 0 and f i Ž x . F 0 we have i f i Ž x . F i f i Ž u.. By K-quasi-invexity of f i we obtain ² uUi , Ž x, u . : F 0,
᭙ uUi g ⭸ K i f i Ž u . .
Moreover, by feasibility of Ž u, ., there exists uUi g ⭸ K i f i Ž u. such that uU0 q
Ý i uUi s 0. igI
Therefore 0G
Ý i ² uUi , Ž x, u . : s ² y uU0 , Ž x, u . : . igI
Since f 0 is K-pseudoinvex we achieve the thesis. THEOREM 4.2 ŽStrong Duality.. Let x be a local minimum of Ž P . and assume that a regularity condition holds at x. Then there exists such that
NONSMOOTH INVEX FUNCTIONS
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Ž x, . is feasible for Ž D .. Moreo¨ er if f 0 is K 0-pseudoin¨ ex and f i is K i-quasi-in¨ ex with respect to the same kernel , then x and Ž x, . are global minima of Ž P . and Ž D ., respecti¨ ely. Proof. By assumption there exists such that Ž x, . is feasible for Ž D .. From Theorem 4.1 we achieve the thesis. When the f i are locally Lipschitz, choosing K i s TC l , we get a result similar to one expressed in w19, 22x.
REFERENCES 1. A. Ben-Israel and B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 Ž1986., 1᎐9. 2. M. Castellani and F. Romeo, On Constraint Qualifications in Nonlinear Programming, Technical Report, University of Pisa, 1998. 3. M. Castellani and M. Pappalardo, First order cone approximations and necessary optimality conditions, Optimization 35 Ž1995., 113᎐126. 4. M. Castellani and M. Pappalardo, Local second-order approximations and applications in optimization, Optimization, 37 Ž1996., 305᎐321. 5. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Wiley, New York, 1984. 6. B. D. Craven, Invex functions and constrained local minima, Bull. Austral. Math. Soc. 24 Ž1981., 357᎐366. 7. B. D. Craven and B. M. Glover, Invex functions and duality, J. Austral. Math. Soc. Ser. A 39 Ž1985., 1᎐20. 8. V. F. Demyanov and M. A. Rubinov, On quasidifferentiable functions, So¨ iet Math. Dokl. 21 Ž1980., 14᎐17. 9. K. H. Elster and J. Thierfelder, On cone approximations and generalized directional derivatives, in ‘‘Nonsmooth Optimization and Related Topics’’ ŽF. H. Clarke, V. F. Demyanov, and F. Giannessi, Eds.., pp. 133᎐159, 1989. 10. K. H. Elster and J. Thierfelder, On cone approximations and generalized directional derivatives, in ‘‘Nonsmooth Optimization and Related Topics’’ ŽF. H. Clarke, V. F. Demyanov, and F. Giannessi, Eds.., pp. 133᎐159, 1989. 11. G. Giorgi, A note on the relationship between convexity and invexity, J. Austral. Math. Soc. Ser. B 32 Ž1990., 97᎐99. 12. G. Giorgi and A. Guerraggio, Various types of nonsmooth invex functions, J. Inform. Optim. Sci. 17 Ž1996., 137᎐150. 13. B. M. Glover, Y. Ishizuka, V. Jeyakumar, and H. D. Tuan, Complete characterizations of global optimality for problems involving the pointwise minimum of sublinear functions, SIAM J. Optim. 6 Ž1996., 362᎐372. 14. B. M. Glover, V. Jeyakumar, and W. Oettli, A Farkas lemma for difference sublinear systems and quasidifferentiable programming, Math. Programming 63 Ž1994., 109᎐125. 15. M. A. Hanson, On sufficiency of the Kuhn᎐Tucker conditions, J. Math. Anal. Appl. 80 Ž1981., 545᎐550. 16. J. B. Hiriart-Urruty, On optimality conditions in nondifferentiable programming, Math. Programming 14 Ž1978., 73᎐86. 17. V. Jeyakumar, On optimality conditions in nonsmooth inequality constrained minimization, Numer. Funct. Anal. Optim. 9 Ž1987., 535᎐546.
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18. R. N. Kaul and S. Kaur, Optimality criteria in nonlinear programming involving nonconvex functions, J. Math. Anal. Appl. 105 Ž1985., 104᎐112. 19. R. N. Kaul, S. K. Suneja, and C. S. Lalitha, Generalized nonsmooth invexity, J. Inform. Optim. Sci. 15 Ž1994., 1᎐17. 20. B. Mond and T. Weir, Generalized convexity and duality, in ‘‘Generalized Concavity in Optimization and Economics’’ ŽS. Schaible and W. T. Ziemba, Eds.., pp. 263᎐280, 1981. 21. B. N. Pshenichnyi, ‘‘Necessary Conditions for an Extremum,’’ Dekker, New York, 1971. 22. T. W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc. 42 Ž1990., 437᎐446. 23. D. E. Ward, Isotone tangent cones and nonsmooth optimization, Optimization 18 Ž1987., 769᎐783. 24. D. E. Ward, Directional derivative calculus and optimality conditions in nonsmooth mathematical programming, J. Inform. Optim. Sci. 10 Ž1989., 81᎐96. 25. D. E. Ward, Convex directional derivatives in optimization, in Lecture Notes in Econom. and Math. Systems, Vol. 345, pp. 36᎐51, Springer-Verlag, New YorkrBerlin, 1990. 26. Y. L. Ye, d-invexity and optimality conditions, J. Math. Anal. Appl. 162 Ž1991., 242᎐249.