JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
202, 900]919 Ž1996.
0353
Inequality Systems and Global OptimizationU V. Jeyakumar School of Mathematics, Uni¨ ersity of New South Wales, Sydney 2052, Australia
A. M. Rubinov School of Information Technology and Mathematical Sciences, Uni¨ ersity of Ballarat, Victoria, Australia
B. M. Glover School of Information Technology and Mathematical Sciences, Uni¨ ersity of Ballarat, Victoria, Australia
and Y. Ishizuka Department of Mechanical Engineering, Sophia Uni¨ ersity, Tokyo, Japan Submitted by George Leitmann Received November 13, 1995
Solvability results for infinite inequality systems involving convex and difference of convex ŽDC. functions are given. Generalizations of Farkas’ lemma are obtained. These results are presented in terms of epigraphs of conjugate functions. Applications are given for characterizing global e-optimality Žand optimality. of difference of convex ŽDC. optimization problems with convex inequality constraints. Solvability results for cone convex inequality systems are also given. Q 1996 Academic Press, Inc.
1. INTRODUCTION Conditions which characterize the solvability of inequality systems have played an important role in the development of optimization. Applications range from classical optimization theory to modern areas of optimization such as nonsmooth optimization Žsee w2, 5, 20x., nonlinear programming U
This research was partially supported by an Australian Research Council Grant. 900
0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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Žsee w3x. and global nonconvex optimization Žsee w13, 27, 17, 19x.. Solvability results for linear inequality systems have been used to develop Lagrange multiplier theory, duality results, and minimax theories in classical optimization and linear programming Žsee, for instance, w2, 25x.. Corresponding results for nonlinear inequality systems have been applied to derive necessary optimality conditions for nonsmooth optimization problems and for certain nonlinear programming problems Žsee w16x.. Dual conditions which characterize solvability of infinite convex inequality systems have recently been employed for developing necessary and sufficient optimality conditions for certain constrained global optimization problems Žsee w17, 4x.. Solvability theorems for inequality systems have recently been exploited in the study of constrained global optimization by providing a mechanism for characterizing optimality for a range of problems including Difference of Convex ŽDC.-optimization, convex maximization, and fractional programming problems. This is not a surprising development given that the solvability of inequality systems can often be viewed as the solvability of appropriate global constrained optimization problems Žsee w14x.. This approach has been used to obtain dual conditions characterizing global optimality of constrained difference sublinear optimization problems w3x and convex maximization problems w18x. Very recently, dual characterizations of solvability of convex inequality systems were obtained in w17, 4x and as an application necessary and sufficient conditions for global optimality of multi-objective convex optimization problems and certain DCoptimization problems were presented. In this paper, we present dual conditions characterizing solvability for infinite inequality systems involving difference of convex functions. The dual conditions are given in terms of epigraphs of conjugate functions. We also show how a generalization of the Motzkin Theorem of the alternative can be obtained to cone convex inequality systems, which is known to provide applications in multi-objective optimization. We then apply the solvability results to characterizing global e-optimality for DC-optimization problems with convex inequality constraints. We further show how optimality conditions for DC optimization problems involving DC constraints can be established. The optimality conditions are given in terms of e-subdifferentials. These results now provide general global dual optimality conditions in constrained DC optimization Žcf. w17, 27x..
2. EPIGRAPHS OF CONJUGATE FUNCTIONS Let X be a locally convex Hausdorff topological vector space Žl.c.H.t.v.s.. with conjugate space X X . We will consider a pair h s Ž ¨ , c . g X X = R as
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JEYAKUMAR ET AL.
an affine function on the space X, i.e., h Ž x . s ¨ Ž x . y c. Now let g: X ª Rq` s R j q`4 be a lower semicontinuous Žl.s.c.. convex function. We denote by g U the Fenchel]Moreau conjugate function
Ž;¨ g XX .
g U Ž ¨ . s sup Ž ¨ Ž x . y g Ž x . . xgX
Then g s g UU , that is, g Ž x . s sup Ž ¨ Ž x . y g U Ž ¨ . . ¨ gX
s s
X
sup
¨ gdom g U
Ž ¨ Ž x . y gU Ž ¨ . .
sup
hs Ž ¨ , g U Ž ¨ ..gG Ž g U .
hŽ x . .
Here dom g U s ¨ : g U Ž ¨ . - q`4 and G Ž g U . s Ž ¨ , g U Ž ¨ ..: ¨ g dom g U 4 is the graph of the function g U . Let us denote by D Ž g . the set of all x g X such that subdifferential g Ž x . is not empty. Clearly, int dom g ; D Ž g . and if x g D Ž g . then g Ž x . q gU Ž ¨ . s ¨ Ž x . for ¨ g g Ž x .. So for x g D Ž g . the following formula holds: g Ž x . s maxU h Ž x . . hgG Ž g .
Recall some results of convex analysis which will be used in the paper. Let us consider the set sŽ f . of all affine functions hŽ x . s ¨ Ž x . y c which are majorized by function f : s Ž f . s Ž ¨ , c . g X X = R: ¨ Ž x . y c F f Ž x . ; x g X 4 . It follows directly from the definition of the conjugate function that sŽ f . s epi f U for a l.s.c. convex function f. It is well known w11x that f Ž x . s sup hŽ x .: h g sŽ f .4 . Using separation arguments, we can show that s sup f j s cl co Ž D j g J S Ž f j . .
ž
jgJ
/
for each arbitrary family Ž f j . j g J of l.s.c. convex functions. Therefore U
epi Ž sup f j . s cl co D j g J epi f jU .
Ž 1.
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Note that for a set A the convex hull of A is denoted by coŽ A. and the closure of A is denoted by clŽ A.. Applying the well-known Moreau]Rockafellar theorem we have for l.s.c. convex functions f and g U
epi Ž f q g . s epi Ž cl Ž f U [ g U . . . Recall that for a function f defined on X, closure of f, cl f, is given by the formula epiŽcl f . s clŽepi f . and that f U [ g U is the inf-convolution. Since epi cl Ž f U [ g U . s cl Ž epi f U q epi g U . , we have U
epi Ž f q g . s cl Ž epi f U q epi g U . .
Ž 2.
This equation leads us to examine conditions which guarantee that the sum of two closed convex sets is closed and which are easily verifiable for epigraphs of l.s.c. convex functions. We will denote the recession cone of a convex set Z by rc Z. Recall that by definition rc Z s u: Ž; y g Z . y q a u g Z ;a ) 04 . We need the following lemmas for which we provide proofs for the sake of completion. The related results can be found in, for instance, w23x. LEMMA 2.1. Let Z be a closed con¨ ex set in X. If the sequences u n g Z and a n g R, a n ª q`, Ž u nra n . ª u, then u g rc Z. Proof. Take y g Z and a ) 0. We have
ž
1y
a an
/
yq
a an
un g Z
;a n ) a .
Hence, y q a u g Z ; y g Z and ;a ) 0. LEMMA 2.2. Let Z1 and Z2 be v U-closed con¨ ex subsets of the conjugate space X X of X. Assume that Ža. Žrc Z1 . l rcŽyZ2 . s 04 Žb. if u n g Z1 and u n is unbounded then there exist a subsequence Ž un . k vU y1 Ž . of the sequence u n and a number sequence a n k such that a n k u n k ª u / 0. Then the set Z1 q Z2 is v U-closed. vU
Proof. Let u n g Z1 , v n g Z2 Ž n s 1, 2, . . . . and u n q v n ª ¨ . We consider two cases
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Ži. Suppose that u n is bounded. Then we can assume by taking a subsequence if necessary that there exists u g Z1 such that v U-lim u n s u. Clearly v n converges too and so ¨ g Z1 q Z2 . Žii. Suppose now that u n is unbounded. Applying Žb. we can find a subsequence Ž u n k . and a real sequence a n k such that ay1 n k ? u n k ª u, u / 0. Clearly a n k ª q`. Since u n k q v n k ª ¨ we have ay1 n k v n k ª yu. Then, Lemma 2.1 shows that u g rc Z1 and u g yrc Z2 s rcŽyZ2 .. This contradicts Ža.. Now we see that when the set Z1 s epi f U in Lemma 2.2, the condition Žb. in this lemma is easily satisfied. Assume that X is a Banach space. If f is a l.s.c. con¨ ex
PROPOSITION 2.1. function satisfying
sup f Ž x . F r
5 x 5Fc
for some real numbers c ) 0 and r, then assumption Žb. of Lemma 2.2 holds for the set Z1 s epi f U . Proof. We have for ¨ g X X , f U Ž ¨ . s sup Ž ¨ Ž x . y f Ž x . . G sup ¨ Ž x . y r G c 5 ¨ 5 y r . 5 x 5Fc
x
If Ž ¨ , a . g epi f U then
a 5¨ 5
G
fUŽ¨.
Gcy
5¨ 5
r 5¨ 5
s
c5¨ 5 y r 5¨ 5
.
Take an unbounded sequence Ž ¨ n , a n . g epi f U . Without loss of generality we can assume that 5Ž ¨ n , a n .5 ª q`. If the sequence Ž ¨ n . is bounded then a n ª q` and we have
an
s
¨n
ž / an
,1
vU
Ž 0, 1 . / 0.
6
Ž ¨n , an .
Now let the sequence Ž ¨ n . be unbounded. Assume 5 ¨ n 5 ª q`. If c 5 ¨ n 5 ) r then 5¨n 5
an
F
5¨n 5 c5¨n 5 y r
.
So the sequence 5 ¨ n 5ra n is bounded and we can find subsequence Ž ¨ n , a n . such that limŽŽ ¨ n , a n .ra n . s Ž ¨ , 1. / 0. i i i i i
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Remark 2.1. It is known that under mild conditions on f we could find real numbers c ) 0 and r g R such that sup 5 x 5 F c f Ž x . F r which is used in Proposition 2.1. For instance, this condition holds if 0 g intŽdom f .. We include the following result for the sake of completeness Žsee w11x for the finite dimensional version.. We provide a simple proof. LEMMA 2.3. If f is a l.s.c. con¨ ex function with dom f s X then f U is a coerci¨ e function, that is, fUŽ¨.
as 5 ¨ 5 ª q`.
ª q`
5¨ 5 Proof. For l ) 0, define
r Ž l . s sup m : m B ; Sl Ž f . 4 , where B s x g X: 5 x 5 F 14 is the unit ball and SlŽ f . s x g X: f Ž x . F l4 is the level set of the function f. Clearly r Ž l. is increasing. Since dom f s X it follows that r Ž l. ª q` as l ª q`. Hence we have fUŽ¨. 5¨ 5
s G G
1 5¨ 5
sup Ž ¨ Ž x . y f Ž x . . x
1 5¨ 5
sup f Ž x .F 5 ¨ 5
sup 5 x 5Fr Ž 5 ¨ 5.
ž
Ž ¨ Ž x . y 5 ¨ 5.
¨ Ž x.
5¨ 5
y 1 s r Ž 5 ¨ 5 . y 1.
/
As 5 ¨ 5 ª q`, r Ž5 ¨ 5. y 1 ª q` and so the conclusion follows. Let us now describe the recession cone rcŽepi f U . for a function f with dom f s X. PROPOSITION 2.2. Let f be a l.s.c. con¨ ex function defined on a Banach space X with dom f s X. Then rc Ž epi f U . s Ž 0, a . : a G 0 4 . Proof. Let Ž u, a . g rcŽepi f U . and Ž ¨ , g . g epi f U . Then we have for all m ) 0,
Ž ¨ , g . q m Ž u, a . s Ž ¨ q m u, g q ma . g epi f U ; that is, g q ma G f U Ž ¨ q m u.. Now, Lemma 2.3 shows that
grm q a 5 ¨ rm q u 5
s
g q ma 5 ¨ q m u5
G
f U Ž ¨ q m u. 5 ¨ q m u5
ª q`
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JEYAKUMAR ET AL.
as m ª `. It is only possible for u s 0, a ) 0. On the other hand,
Ž ¨ , g . q b Ž 0, a . g epi f U
; Ž ¨ , g . g epi f U ;b ) 0.
Hence the conclusion follows.
3. INEQUALITY SYSTEMS INVOLVING CONVEX AND DC-FUNCTIONS In this section we develop a dual description of the following implication which would then allow us to obtain dual conditions characterizing global optimality of certain DC-optimization problems. Ži. Ž; i g I . pi Ž x . F 0 « g Ž x . y f Ž x . F 0, where I is an arbitrary nonempty index set, pi , i g I, f and g are l.s.c. convex functions defined on a l.c.H.t.v.s. X. THEOREM 3.1. The following statements are equi¨ alent:
Ž i.
Ž ; i g I . pi Ž x . F 0 « g Ž x . y f Ž x . F 0
Ž ii .
epi g U ; cl epi f U q cl cone co Ž D i epi pUi . .
Ž 3.
Proof. Let p Ž x . s sup pi Ž x . ;
C p s x : p Ž x . F 04 .
i
Clearly the statement Ži. holds if and only if Cp ; D ,
Ž 4.
where D s x g X: g Ž x . y f Ž x . F 04 . Clearly the function p is a l.s.c. and convex function as it is the pointwise supremum of l.s.c. convex functions pi . Therefore C p is a closed convex set. We will denote the indicator function d of the set C p by
d Ž x. s
½
0,
x g Cp
q`,
x f Cp .
Ž 5.
It is easy to check that inclusion Ž4. holds if and only if the inequality gFfqd
Ž 6.
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holds. At the same time Ž6. is equivalent to the inequality g U G Ž f q d .U which can be rewritten in the form U
epi g U ; epi Ž f q d . .
Ž 7.
Applying Ž2. we get U
epi Ž f q d . s cl Ž epi f U q epi d U . .
Ž 8.
Now let us calculate epi d U . It is easy to see that d Ž x . s supl) 0 l pŽ x .. So
d Ž x . s sup sup l pi Ž x . s l)0 igI
sup l )0, igI
l pi Ž x . .
Applying Ž1. we can deduce that epi d U s cl co cone D i g I epi pUi
Ž 9.
and that Ž7. is equivalent to Ž3.; hence the conclusion holds. Remark 3.1. Since clŽ A q cl B . s clŽ A q B . for arbitrary sets A and B in a topological vector space we can substitute Ž3. for epi g U ; cl Ž epi f U q cone co D i epi pUi . .
Ž 10 .
Now we give conditions which guarantee that the set on the right-hand side in formula Ž3. is closed. THEOREM 3.2.
Let f, g, pi Ž i g I . be functions as abo¨ e.
Ži. Assume that sup f Ž x . F r
5 x 5Fc
for some numbers c ) 0 and r ) 0 and rc Ž epi f U . l Ž ycl cone co D i g I epi pUi . s 0 4 .
Ž 11 .
Then assertion Ži. of Theorem 3.1 is equi¨ alent to the inclusion epi g U ; epi f U q cl Ž cone co D i g I epi pUi . .
Ž 12 .
Žii. If dom f s X then assertion Ži. of Theorem 3.1 is equi¨ alent to inclusion Ž12.. Proof. Ži. Proposition 2.1 shows that assumption Žb. from Lemma 2.2 is fulfilled. The result follows from this lemma because Ž11. shows that assumption Ža. is also fulfilled.
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JEYAKUMAR ET AL.
Žii. Applying Ž9. we have cl cone co D i g I epi pUi s epi d U . If the set C p s x g X: pŽ x . F 04 is empty then d U s y` and the set epi f U q epi d U s X X = R is closed. Otherwise Ž0, y1. f epi d U because d U Ž0. s sup x g C p 0Ž x . s 0. Since epi d U is a closed convex cone we have
Ž 0, 1 . f y Ž rc Ž epi d U . . s yepi d U . On the other hand Proposition 2.1 shows that rcŽepi f U . s lŽ0, 1..: l G 04 . So Žepi f U . l yŽrcŽepi d U .. s 04 and assumption Ža. of Lemma 2.2 is fulfilled. Clearly assumption Žb. of Lemma 2.2 is also fulfilled.
4. SOLVABILITY OF DC-INEQUALITY SYSTEMS In this section, we extend the dual descriptions to the following implication involving DC-functions:
Ž ; i g I . f i Ž x . y g i Ž x . F 0 « g 0 Ž x . y f 0 Ž x . F 0. Here I is an arbitrary nonempty index set, 0 f I, f i , g i Ž i g I j 04. are l.s.c. convex functions mapping l.c.H.t.v.s. X into Rq` . Let C s x : fi Ž x . y g i Ž x . F 0 ; i g I 4 , and D0 s x : g 0 Ž x . y f 0 Ž x . F 0 4 .
Ž 13 .
Assume that C is nonempty. Then clearly statement Ži. holds if and only if C ; D0 . In general C and D0 are nonconvex sets. We will show that under appropriate assumptions it is possible to derive a family of convex systems which is equivalent to the given nonconvex system. We will apply for this purpose the linearization of function g i Ž i g I . Žsee Section 2.: If x g D Ž g i . then g i Ž x . s max h i Ž x . : h i g G Ž g Ui . 4 ,
Ž 14 .
where G Ž g Ui . s Ž ¨ , g Ui Ž ¨ ..: ¨ g X X 4 is the graph of the function g Ui . For an affine function h and i g I, define Ci Ž h . s x : h Ž x . G f i Ž x . 4 ,
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and Gs
Ł G Ž g Ui . s h s Ž h i . igI : h i g G Ž g Ui . ; i g I 4 . igI
PROPOSITION 4.1.
If
C s x g X : fi Ž x . y g i Ž x . F 0 ; i g I 4 then D h g G F h i g h Ci Ž h i . ; C. If C ; D Ž g i . ; i g I then C s D h g G F h i g h Ci Ž h i . . Proof. Let h s Ž h i . i g I g G and x g F h i g h Ci Ž h i .. We have fi Ž x . F h i Ž x . F g i Ž x .
; i g I.
Hence x g C. Now assume C ; D Ž g i . ; i g I, and x g C. Applying Ž14. we can find h i g G Ž g i . such that h i Ž x . s g i Ž x . ; i g I. Let h s Ž h i . g G. Clearly x g Ci Ž h i . ; i g I. So x g F h i g h Ch i ; D h g G F h i g h Ch i . Thus the result is established. This proposition shows that inclusion C ; D0 implies the following system of inclusions F h i g h Ch i ; D0
;h g G.
Ž 15 .
If we assume that C ; D Ž gi .
; i g I,
Ž 16 .
then the inclusion C ; D0 is equivalent to system Ž15.. Clearly, F h i g h Ch i s x : f i Ž x . y h i Ž x . F 0 ; i g I 4 . Let us denote f i y h i s pi . Since pi is a l.s.c. convex function we can apply Theorems 3.1 and 3.2 for studying assertion Žii.. Let us note that epi pUi s epi f iU y h i . Proposition 3.3, Theorem 3.1, and Theorem 3.2 show that the following results hold. THEOREM 4.1. Let the functions f i , g i Ž i g I j 04. be as abo¨ e; let C / B. Consider the following statements Ži. Ž; i g I . f i Ž x . y g i Ž x . F 0 « g 0 Ž x . y f 0 Ž x . F 0.
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Žii. For e¨ ery h s Ž h i . i g I g G, epi g U0 ; cl Ž epi f 0U . q cone co D i g I Ž epi f iU y h i . . Then Ži. « Žii.. Furthermore Žii. « Ži. is true pro¨ ided that assumption Ž16. holds. THEOREM 4.2. Let X be a Banach space, let f i , g i Ž i g I j 04. be functions as abo¨ e, and C / B. Ž1. Assume that sup f 0 Ž x . F r ,
5 x 5-c
for some c and r, and for all h s Ž h i . i g I g G rc Ž epi f 0U . l Ž ycl cone co D i epi f iU y h i . s 0 4 . Then Ži. « Žiii., where Žiii. for all h s Ž h i . i g I g G epi g U0 ; epi f 0U q cl cone co Ž D i epi f iU y h i . . Also Ži. is equi¨ alent to Žiii. pro¨ ided that Ž16. holds. Ž2. Assume that dom f 0 s X. Then Ži. « Žiii. and Ži. is equi¨ alent to Žiii. if the assumptions Ž16. hold.
5. GLOBAL DC-OPTIMIZATION In this section we apply the results of Sections 3 and 4 to establish dual characterizations of e-global optimality of difference of convex maximization problems. The dual conditions are given in terms of approximate subdifferentials of the functions involved. The dual optimality conditions extend corresponding results in w17x. We have shown in the proof of Theorem 3.1 that the implication ; i g I,
pi Ž x . F 0 « g Ž x . y f Ž x . F 0
is equivalent to the inclusion U
epi g U ; epi Ž f q d . ,
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where pi , i g I, f and g are l.s.c. convex functions, and d is defined by the formula Ž5.
d Ž x. s
½
0
if x g C p
q`
if x f C p ,
where C p s x g X: pŽ x . F 04 and pŽ x . s sup i g I pi Ž x .. Now, consider the DC-maximization problem ŽDC. maximize g Ž x . y f Ž x . subject to pi Ž x . F 0, i g I, where I is an arbitrary index set and g, f, pi : X ª R are l.s.c. convex functions. The problem ŽDC. can be rewritten as maximize g Ž x . y f Ž x . subject to p Ž x . F 0, where p Ž x . s sup i g I pi Ž x . . Recall that the feasible point x 0 g X of ŽDC. is an e-maximizer of the problem ŽDC. if for each x g C p , Ž g y f .Ž x . F Ž g y f .Ž x 0 . q e . So, the point x 0 g X is an e-maximizer of the problem ŽDC. if and only if the following implication holds: p Ž x . F 0 « Ž g Ž x . y g Ž x 0 . . y Ž f Ž x . y f Ž x 0 . . y e F 0.
Ž 17 .
Let f˜Ž x . s f Ž x . y f Ž x 0 . and ˜ g Ž x . s g Ž x . y Ž g Ž x 0 . q e .. Now, the implication Ž17. is equivalent to the inclusion U
epi ˜ g U : epi Ž f˜q d . .
Ž 18 .
Let us give a description of e-optimality in terms of approximate subdifferentials. PROPOSITION 5.1. The feasible point x 0 g X is an e-maximizer for the problem ŽDC. if and only if ;h G 0,
h ˜ g Ž x 0 . ; eq h Ž f˜q d . Ž x 0 . .
Proof. Clearly, 0 epi ˜ g U s epi g U q . g Ž x0 . q e
ž
/
Ž 19 .
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JEYAKUMAR ET AL.
So, epi ˜ gU s Dh G 0 q
ž
l g X X = R: l g h g Ž x 0 . , l s h q l Ž x 0 . y g Ž x 0 . l
½ž /
5
0 . g Ž x0 . q e
/
Thus, it can be rewritten as epi ˜ gU s Dh G 0
l g X X = R: l g h g Ž x 0 . , l
½ž /
l s h q l Ž x0 . q e .
5
Since x 0 g C p , d Ž x 0 . s 0 and so, Ž f˜q d .Ž x 0 . s f˜Ž x 0 . s 0. Hence, we get U
epi Ž f˜q d . s D h G 0
l g X X = R: l g h Ž f˜q d . Ž x 0 . , l
½ž /
l s h q l Ž x0 . .
5
The conclusion now follows by noting that for each h G 0, l g h g Ž x 0 . and l s h q l Ž x 0 . q e , from Ž18., Ž l, l. g epiŽ f˜q d .U . Thus, there exists hX G 0 such that l g hX Ž f˜q d . Ž x 0 .
l s hX q l Ž x 0 . .
and
Hence, hX s h q e and l g hq e Ž f˜q d .Ž x 0 .. Recall that the e-normal set of C p at x 0 g X is given by Ne Ž x 0 , C p . s l g X X : l Ž x y x 0 . F e , ; x g C p 4 . In w17x, it has been shown that l g Ne Ž x 0 , C p . if and only if
ž
l g cl co cone D l Ž x0 . q e
/
igI G0
li : l i g s pi Ž x 0 . , li
½ž /
5
l i s s q l i Ž x 0 . y pi Ž x 0 . .
Ž 20 .
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THEOREM 5.1. For the problem ŽDC., assume that pi , i g I, f, and g are l.s.c. con¨ ex functions with f continuous at the point x 0 g X. Then the point x 0 is an e-maximizer of ŽDC. if and only if for each h G 0,
h g Ž x 0 . ;
D
eG0, e 2G0 e 1q e 2s eq h
e 1 f Ž x 0 . q Ne 2Ž x 0 , C p . .
Ž 21 .
Proof. The conclusion easily follows from Proposition 5.1 Žsee HiriartUrruty w9x. that
e Ž f q d . Ž x 0 . s
D
e 1G0, e 2G0 e 1q e 2s e
« 1 f Ž x 0 . q « 2 d Ž x 0 . ,
by noting that
e f˜Ž x 0 . s e f Ž x 0 .
and
e d Ž x 0 . s Ne Ž x 0 , c p . .
It is worthwhile observing that if the functions pi , i g I are continuous then the condition Ž21. can be given in terms of approximate subdifferentials of the individual functions involved. Now consider the following more general difference of convex problem ŽCDC. maximize g Ž x . y f Ž x . subject to f i Ž x . y g i Ž x . F 0, i g I, where f i and g i , i g I j 04 are l.s.c. convex functions. Using the approach presented in Section 4, it is easy to see that x 0 is an e-maximizer of the problem ŽCDC. if and only if for each h s Ž h i . g G:' Ł i g I G Ž g Ui ., where G Ž g Ui . is the graph of the function g Ui , p h Ž x . F 0 « Ž g Ž x . y g Ž x 0 . . y Ž f Ž x . y f Ž x 0 . . q e F 0, where p h Ž x . s sup i g I pih Ž x ., pih Ž x . s f i Ž x . y h i Ž x .. Since pih is a l.s.c. convex function, it follows that the above implication coincides with the implication Ž17. with p h s p. Hence, dual conditions characterizing e-maximality of ŽCDC. can be expressed in terms of approximate subdifferentials of the functions f, g, and f i y h i , where h i g G Ž g Ui ., i g I. 6. CONE INEQUALITY SYSTEMS Various generalizations of the Motzkin Theorem of the Alternative have been given in the literature for cone inequality systems involving linear, sublinear, and difference sublinear mappings Žsee w2x.. In these generalizations the positively homogeneous property of the functions involved played
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a crucial role. In the following we establish a generalization of the Motzkin Alternative Theorem for cone convex inequality systems without the positively homogeneous condition. Let X, Y be Banach spaces and S : Y a closed convex cone. Let g: X ª Y be a continuous S-convex function; that is, for each x 1 , x 2 g X and for each a g Ž0, 1.,
a g Ž x 1 . q Ž 1 y a . g Ž x 2 . y g Ž a x 1 q Ž 1 y a . x 2 . g S. LEMMA 6.1. set
Let g: X ª Y be a continuous S-con¨ ex function. Then the A s D lg SU epi Ž l g .
U
is a con¨ ex cone. Proof. Let Ž u, a . g A and g ) 0. Then, there is a l g SU with U
Ž l g . Ž u. F a m sup u Ž x . y l g Ž x . F a xgX
m g sup u Ž x . y l g Ž x . F ga xgX
m sup Ž g u . Ž x . y Ž gl . g Ž x . F ga . xgX
Hence, Žg u, ga . g epiŽgl g .U ; A, so A is a cone. Let Ž u1 , a 1 ., Ž u 2 , a 2 . g A, g g Ž0, 1.. Then there are l1 , l2 g SU with U
Ž l1 g . Ž u1 . F a 1 U
Ž l2 g . Ž u 2 . F a 2 . Consider ¨ s g u1 q Ž1 y g . u 2 . Then, U
Ž gl1 g q Ž 1 y g . l2 g . Ž ¨ . U U s Ž gl1 g . [ Ž Ž 1 y g . l2 g . Ž ¨ . s
U
U
inf
¨ 1q¨ 2s¨
Ž gl1 g . Ž ¨ 1 . q Ž Ž 1 y g . l2 g . Ž ¨ 2 .
U
U
F Ž gl1 g . Ž g u1 . q Ž Ž 1 y g . l2 g .
Ž Ž 1 y g . u2 .
F ga 1 q Ž 1 y g . a 2 . Thus Žg u1 q Ž1 y g . u 2 , ga 1 q Ž1 y g . a 2 . g A and so A is convex.
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THEOREM 6.1. Let f : X ª Z, g: X ª Y, S : Y a closed con¨ ex cone and T : Z a closed con¨ ex cone. Assume f is continuous T-con¨ ex and g is continuous S-con¨ ex. Then exactly one of the following has a solution: Ži. Ž' x g X . y f Ž x . g T, yg Ž x . g int S Žii. Ž'0 / l g SU .0 g epiŽ l g .U q cl D t g T U epiŽ tf .U . Proof. Let C s x: yf Ž x . g T 4 . Then C is a closed convex set. Furthermore Not Ž i .
m~ x g C, yg Ž x . g int S m '0 / l g SU , ; x g C, l g Ž x . G 0
Ž follows by the Basic Alternative Theorem of Craven w 2 x . . m '0 / l g SU x g C « lgŽ x. G 0 m '0 / l g SU yf Ž x . g T « l g Ž x . G 0 m '0 / l g SU
Ž ; t g T U . tf Ž x . F 0 « l g Ž x . G 0 m '0 / l g SU U
0 g epi Ž l g . q cl D t g T U epi Ž tf .
U
Ž by Theorem 3.2 and Lemma 6.1. . We now see that if int S / B and
Ž ' x 0 g X . g Ž x 0 . g yint S then A is weakU closed. To establish this, we require the following technical result. LEMMA 6.2. Let Ž g i . be a sequence of continuous con¨ ex functions such that g i ª g pointwise. Furthermore let u i ª u Ž weakU . in X X . Then g is con¨ ex and lim inf g Ui Ž u i . G g U Ž u . . iª`
Proof. Consider, for any i g I, g Ui Ž u i . s sup u i Ž x . y g i Ž x . . x
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Thus for any x g X g Ui Ž u i . G u i Ž x . y g i Ž x . therefore lim inf g Ui Ž u i . G lim inf u i Ž x . y g i Ž x . iª`
iª`
s uŽ x . y g Ž x . . Since x was arbitrary sup u Ž x . y g Ž x . F lim inf g Ui Ž u i . iª`
x
and the result follows. PROPOSITION 6.1. int S / B and
Let g: X ª Y be a continuous S-con¨ ex function. If
Ž ' x 0 g X . g Ž x 0 . g yint S then A s D lg SU epi Ž l g .
U
is weakU closed. Proof. Let
Ž u n , a n . ª Ž u, a . g cl A with, for some l n g SU , Ž l n g .U Ž u n . F a n . Since int S / B there is a weakU compact convex base B ; SU with 0 f B and SU s cone B. Thus, for gn G 0, l n s gn bn with bn g B. Without loss of generality we can assume gn ) 0 for all n and Žby compactness. bn ª bŽg B ; SU . Žif gn s 0 for infinitely many n then we can assume Ž u n , a n . g epi 0U so that u n s 0 and a n G 0 in which case u s 0 and a G 0 and so Ž u, a . g epi 0U ; A.. We consider the following cases: Ži. gn ª g ) 0. Then U
Ž ln g . Ž u n . F a n U
m Ž gn bn g . Ž u n . F a n U
m gn Ž bn g . Ž u nrgn . F a n U
m Ž bn g . Ž u nrgn . F a nrgn .
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Since bn g ª bg, u nrgn ª urg , a nrgn ª arg , we have U
U
Ž bg . Ž urg . F lim inf Ž bn g . Ž u nrgn . F arg . n
U
Thus Ž urg , arg . g epiŽ bg . and Ž u, a . g epiŽg bg .U : A. Žii. gn ª q`. Then u nrgn ª 0, a nrgn ª 0. Thus U
U
Ž bg . Ž 0 . F lim inf Ž bn g . Ž u nrgn . n
F0 therefore U
Ž bg . Ž 0 . F 0 m sup ybg Ž x . F 0 xgX
m inf bg Ž x . G 0 x
m bg Ž x . G 0,
for all x.
However g Ž x 0 . g yint S and b / 0 so that bg Ž x 0 . - 0 and we have a contradiction. Žiii. gn ª 0. So l n ª 0 Žas Ž bn . is bounded. and l n g ª 0. Thus U
0U Ž u . F lim inf Ž l n g . Ž u n . F a n
therefore 0U Ž u . F a . Now, 0U Ž ¨ . s
½
0 q`
if ¨ s 0 otherwise
so u s 0 and a G 0. Thus U
Ž 0, a . g epi Ž 0 g . ; A. Thus A is weakU closed as required.
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