European Journal of Operational Research 212 (2011) 235–241

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Continuous Optimization

Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials Phan Quoc Khanh a, Ho Thuc Quyen b, Jen-Chih Yao c,⇑ a

Department of Mathematics, International University of Hochiminh City, Khu Pho 6, Linh Trung, Thu Duc, Hochiminh City, Vietnam Department of Mathematics, Ho Chi Minh City, University of Architecture, 196 Pasteur St., D.3, HoChiMinh City, Vietnam c Center for General Education, Kaohsiung Medical University, Kaohsiung 80702, Taiwan b

a r t i c l e

i n f o

Article history: Received 3 October 2008 Accepted 15 January 2011 Available online 4 February 2011 Keywords: Optimality conditions Convex sublevel sets Normal cones Star subdifferentials Adjusted subdifferentials

a b s t r a c t A set-constrained optimization problem and a mathematical programming problem are considered. We assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex. Using the two star subdifferentials and the adjusted subdifferential, we establish optimality conditions for usual minima and strict minima. Our results contain and improve some recent ones in the literature. Examples are provided to explain the advantages of each of our results. 2011 Elsevier B.V. All rights reserved.

1. Introduction Optimality conditions for nonconvex-nonsmooth problems have been intensively studied for a long time (see e.g. important books [5,22–24,32] and some papers of our group [11–20] among numerous works of other authors), since convexity and/or differentiability conditions are often not satisﬁed for optimization-related problems in practice. A large number of classes of such problems have been proposed and investigated due to demands of practical applications and also to motivations for mathematical researchers. Qua-siconvex functions constitute an important class of nonconvex functions as is evident from their application in optimization and economic modelling, see e.g. [1], and their structures are convenient for employing mathematical tools, including convex analysis. The reader can refer to [2–4,7,21,26–29] and references therein for recent developments in quasiconvex optimization. A function f from a normed space X to R :¼ R [ fþ1g is called quasi-convex if its sublevel set Lf(x) :¼ {u 2 X: f(u) 6 f(x)} at x is convex for all x 2 X or, equivalently, if for each r 2 R the strict sublevel set {u 2 X:f(u) < r} is convex. Hence, f is quasiconvex if and only if the strict sublevel set L< f ðxÞ :¼ fu 2 X : f ðuÞ < f ðxÞg is convex for all x 2 X. Another equivalent statement, which is often met in the literature, is that f is quasiconvex if for all x, y 2 dom f :¼ {x 2 X: f(x) < +1}, all t 2 [0, 1], f((1 t)xj + ty) 6 max{f(x),f(y)}. An optimization problem is quasiconvex if the objective is quasiconvex and ⇑ Corresponding author. Tel.: +886 7 5253816; fax: +886 7 5253809. E-mail address: [email protected] (J.-C. Yao). 0377-2217/$ - see front matter 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.01.024

the constraint set is convex. The generalized subdifferentials used in [21,27] were the lower subdifferen-tial (known also as Plastria subdifferential) [30], and the infradifferential (or Gutiérrez subdifferential) [9]. These subdifferentials enjoy many helpful properties and hence are convenient to be applied. However, they may be empty in a number of cases. Many simple cases are given in examples of Sections 3 and 4; even constantwise functions may have both Plastria and Gutiérrez subdifferentials being empty, cf. Example 3.3. There exist also differentiable quasiconvex functions with empty Plastria (Gutiérrez) subdifferentials at each point. Reference [21] even required f to be a Plastria or Gutiérrez function (see the deﬁnitions below). In [26,28] the Greenberg–Pierskalla subdifferential [8], a kind of normal-cone subdifferentials, i.e. those that are conic-valued, was additionally used. In [4,29] the two star subdifferentials, which are similar to the Greenberg–Pierskalla subdifferential, were introduced. They are nonempty under weak conditions. Moreover, they are closed convex cones, and quite different from the classical Fenchel subdifferential, which is very often bounded. However, these two star subdifferentials are in general neither quasimonotone nor cone-upper semicontinuous [2,4]. (Roughly speaking, a cone-value mapping is cone-upper semicontinuous, cone-u.s.c. in short, if the correspondence forms via its images with some hyperplane is itself Berge u.s.c..) Hence, they are not suitable for relating minimization problems to variational inequalities. Motivated by this, the authors of [2] introduced the adjusted sublevel set Laf ðxÞ, which is between L< f ðxÞ and Lf(x), and the corresponding adjusted subdifferential, deﬁned similarly as the star subdifferentials, and used them to deal with the solution

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existence of a quasiconvex set-constraint minimization problem. In [3] the adjusted subdifferential together with the limiting (or Mordukhovich) subdifferential [23] were employed to study the existence of solutions and optimality conditions for quasiconvex problems with a locally star-shaped constraint set. They also applied these results to mathematical programming with equilibrium constraints, where the equality constraint is quasiconvex and the equality and equilibrium constraints are quasiafﬁne. There have been also a number of papers using other kinds of generalized derivatives to deal with quasiconvex optimization problems. For instance, in recent reference [7], the Dini directional derivatives are employed. But in this paper we are concerned only with the afore-mentioned subdifferentials. We observe that the changes of the properties of quasiconvex functions, when we assume that the sublevel sets are convex only at a point x under consideration, not at each point, can be controlled when considering optimality conditions. Then the assumptions become remarkably less restrictive. These observations have motivated us to study optimality conditions involving a class of functions that need only admit a convex sublevel set at the optimal point. We do not impose any differentiability assumption and use the star and adjusted sub differentials. The organization of this paper is as follows. Section 2 contains deﬁnitions and preliminaries needed in the sequel. Section 3 is devoted to optimality conditions for a minimization problem with a convex constraint set. In Section 4 optimality conditions are established for the mathematical programming problem.

n o o< f ðxÞ :¼ x 2 X : 8x 2 L < 2; f ðxÞ ¼ 1 > > 2; > : x; L< f ðxÞ

0 small enough such that v þ th 2 C n f xg as this set is open. Therefore:

thu ; hi ¼ hu ;

v x þ thi hu ; v xi P 0;

which implies that u⁄ = 0, again a contradiction.

h

The following example illustrates advantages of Theorem 3.4. Example 3.5. Let C ¼ fðx1 ; x2 Þ : x2 < 0g [ fð0; 0Þg; x ¼ ð0; 0Þ and f : R2 ! R be given by

f ðx1 ; x2 Þ ¼

1; if ðx1 ; x2 Þ–ð0; 0Þ and x2 P 0; 0; if ðx1 ; x2 Þ ¼ ð0; 0Þ or x2 < 0:

Then C n fxg is open. Since o f ð xÞ ¼ o< f ðxÞ ¼ ;, Propositions 5 of [21], Proposition 2.1 of [26] and the results of [27,28] using these two subdifferentials cannot be in use. But the assumptions of Theorem 3.4 (ii) are fulﬁlled, since ð0; 1Þ 2 o~ f ðxÞ \ NðC; xÞ. It is easy to see directly that x is a minimizer of f on C. Now we prove a necessary condition for strict solutions to (3). Theorem 3.6. Let x be a strict solution to (3) and an extreme point of C. Assume that Lf ð xÞ is convex, C is not reduced to f xg and either of the conditions: (i)–(iii) of Theorem 3.1 holds. Then

ov f ðxÞ \ ðNðC; xÞÞ–f0g:

ð6Þ

Proof. Since x is an extreme point of C and C–f xg, the set C n f xg is convex and nonempty. As x is a strict solution to (3), C n f xg and Lf ð xÞ are disjoint. For (i), by the Hahn–Banach separation theorem, there exists some c 2 R and 0 – u⁄ 2 X⁄ such that the following inequalities hold, for all w 2 Lf ð xÞ and x 2 C n f xg:

hu ; x xi P c P hu ; w xi:

This result can be derived directly from Theorem 3.6. Indeed, there exists u⁄ – 0 such that u 2 NðC; xÞ and u 2 ov f ð xÞ ¼ Rþ o6 f ðxÞ, since f is a Gutiérrez function at x. So, one can ﬁnd s > 0 and x 2 o6 f ðxÞ such that u⁄ = sx⁄ and – x 2 NðC; xÞ. However, being a Gutiérrez function may be a severe restriction as shown by the following example. Example 3.8. Let C ¼ ½0; 1; x ¼ 0 and f : R ! R be given by

f ðxÞ ¼

1; for x < 0; x;

for x P 0:

Then Lf ðxÞ ¼ ð1; 0 and o6 f ðxÞ ¼ ;. So Proposition 6 of [21] cannot be applied. But ov f ðxÞ ¼ Rþ and NðC; xÞ ¼ Rþ and hence the conclusion of Theorem 3.6 is true. The assumptions of Theorem 3.6 can also be easily veriﬁed. Furthermore, o< f ðxÞ is also empty and then the results involving these two subdifferentials of [21,26–28] are not applicable. Passing to sufﬁcient conditions for strict solutions, we need the following strict convexity, used in [21] among others. Let X be a normed space. A # X is called strictly convex at x if hx ; x xi < 0 for every x 2 A n f xg and x 2 NðA; xÞ n f0g. Suggested by a referee, we discuss relations j between this strict convexity with some close known notions. Denote A(x) :¼ cone (A x). x 2 A is called an extreme point of A if Að xÞ \ ðAð xÞÞ ¼ f0g, and a strictly extreme point of A if clAð xÞ \ ðclAð xÞÞ ¼ f0g. x 2 A is said to be an exposed point of A if there exists x⁄ 2 X⁄ such that hx ; xi < hx ; xi for all x 2 A n f xg, and a strictly exposed point of A if this strict inequality holds for all x 2 clA n f xg. Moreover, x 2 A is called strongly strictly exposed point of A provided that x⁄ satisﬁes additionally that if hx ; xn i ! hx ; xi for xn 2 clA then xn ! x. In [31] the following relations were established. (i) Strong strict exposedness ) strict exposedness ) exposedness ) extremeness. (ii) A strictly exposed point is always a strictly extreme point. The converse is true if X is separable and A is convex. A strictly extreme point is incomparable with an exposed point in general. Example 6 of [31] gave a strictly extreme point which is not an exposed point. Conversely, boundary points of a disk are exposed but not strictly extreme. (iii) If X is ﬁnite dimensional and A is convex, the three notions of strong strict exposedness, strict exposedness and strict extremeness are equivalent. Now we compare with strict convexity. It is clear that if A is strictly convex at x, then this point is an extreme point. Unfortunately, strict convexity is incomparable with other abovementioned properties. We give illustrative examples.

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(i) The set A = {(x,y):y > 0} [ {(0,0)} in R2 is strictly convex at (0, 0), but this point is not strictly extreme. While (0, 0) is a strictly extreme point of R2þ but this set is not strictly convex at (0, 0). (ii) The set A in (i) is strictly convex at (0, 0), but this point is not strictly exposed. On the other hand, (0, 0) is strongly strictly exposed point of R2þ but this set is not strictly convex at (0, 0).

Then, oa f ðxÞ ¼ ½0; þ1Þ and NðC; xÞ ¼ ½0; þ1Þ and the mentioned modiﬁed conditions are fulﬁlled but x is not a strict solution. This fact agrees with Theorem 3.9, since ov f ð xÞ ¼ f0g. But x is clearly a solution, which can also be obtained from Theorem 3.4.

We have the following simple sufﬁcient condition for strict solutions.

Let us consider now the case in which the constraint set C is deﬁned by a ﬁnite family of inequalities, so that problem (3) turns into the mathematical programming problem:

Theorem 3.9. x is a strict solution of (3) if (6) is satisﬁed and either of the following conditions holds:

minimize f ðxÞ subject to g 1 ðxÞ 6 0; . . . ; g n ðxÞ 6 0:

(i) either C is strictly convex at x or C n f xg is open. (ii) Lf ð xÞ n f xg is open.

8w 2 Lf ðxÞ;

x 2 C;

then hu ; v xi ¼ 0 for any v 2 ðLf ðxÞ \ CÞ n f xg. For (i), observe ﬁrst that C n fxg is open implies that C is strictly convex at x. Indeed, if C n f xg is open, it is equal to int C. Hence hx ; x xi < 0 for every x 2 C n f xg and x 2 NðC; xÞ n f0g, i.e. C is strictly convex at x. Now we have to consider only the case where C is strictly convex at x. Then, by the strict convexity, hu ; x xi < 0 for all x 2 C n f xg, a contradiction. For (ii), let h 2 X be nonzero and arbitrary. As Lf ð xÞ n fxg is open, there exists t > 0 small enough such that v þ th 2 Lf ðxÞ. Then

g ¼ maxi6j6n g i ; C ¼ g ð1; 0; I ¼ fi : g i ðxÞ ¼ 0g

ð7Þ and

1 (i) L< f ðxÞ [ fxg and g i ð1; 0 are convex for i = 1, . . . , n. (ii) gi are u.s.c. at x for i = 1,. . .,n. (iii) either of the following regularity conditions holds: (a) there exists k 2 I such that Lgk ð xÞ \ fx 2 X : x 2 L< gi ðxÞ; 8i 2 I n fkgg–; (Slater condition). (b) X is complete, Lgi ð xÞ is closed for each i 2 I and Rþ ðD Pi2I Lgi ð xÞÞ is a closed subspace, where D = {(xi)i 2 I: "j, k 2 I;xj = xk} is the diagonal of XI. (iv) either X is ﬁnite dimensional or f is u.s.c. at some point of L< f ðxÞ.If x is a solution but not a local minimizer of f on X, then:

! X v o f ðxÞ \ o g I ðxÞ –f0g: ~

t hu ; hi ¼ hu ; v x þ thi hu ; v xi 6 0:

We denote h ¼ maxi2I g i .

1

Theorem 4.1. Assume for problem (7) that:

Proof. Suppose, ad absurdum, ðLf ð xÞ \ CÞ n f xg–;. The relation (6) implies that there exists 0 – u⁄ 2 X⁄ such that:

hu ; x xi P 0 P hu ; w xi;

4. Optimality conditions for the mathematical programming problem

ð8Þ

i2I ⁄

Hence u = 0, again a contradiction. h Note that, like Theorem 3.4, in this sufﬁcient condition no convexity condition is imposed on f. Note further that Theorem 3.9 (i) sharpens the sufﬁcient condition of Proposition 6 in [21], where the Gutiérrez subdifferential is employed. However, Gutiérrez subdifferential may be empty in many cases as shown in Example 3.8, where ov f ð xÞ \ NðC; xÞ ¼ Rþ , and so Theorem 3.9 is directly veriﬁed (and can be applied). The condition that Lf ð xÞ n f xg is open looks restrictive. So we illustrate it in the following example. Example 3.10. Let f : R2 ! R be given by

C ¼ f0g ð1; 0; ð x1 ; x2 Þ ¼ ð0; 0Þ

and

8 > < 1; if x1 –0 and x2 > 0; f ðx1 ; x2 Þ ¼ x22 ; if x2 < 0 or x1 ¼ 0; > : 2 x1 ; if x2 ¼ 0: Then Lf ð xÞ ¼ fðx1 ; x2 Þ : x2 > 0; x1 –0g [ fxg. C is not strictly convex at x since h(1, 0), xi = 0 for all x 2 C. So Proposition 6 of [21] does not work. But Lf ð xÞ n f xg is open. ov f ð xÞ ¼ f0g ð1; 0 and v ð0; 1Þ 2 o f ðxÞ \ ðNðC; xÞÞ and hence Theorem 3.9 concludes that x is a strict solution. A natural question is whether we can replace (6) by the following weaker relation:

oa f ðxÞ \ ðNðC; xÞÞ–f0g; in Theorem 3.9. The following example yields a negative answer. þ

Example 3.11. Let X ¼ R; C ¼ R ; x ¼ 0 and

f ðxÞ ¼

0;

if x P 0;

1; if x < 0:

This implies the following usual form, for some k1 ; . . . ; kn 2 Rþ , not all zero, such that:

0 2 o~ f ðxÞ þ

n X

kj ov g j ðxÞ;

ð9Þ

j¼0

kj g j ðxÞ ¼ 0; j ¼ 1; . . . ; n:

ð10Þ ~

If additionally, f is l.s.c. at x then, in (8) and (9), o f ðxÞ can be replaced by oa f ðxÞ.

Proof. Observe that C is convex and contained in Lh ð xÞ. So NðLh ð xÞ; xÞ # NðC; xÞ. To prove the reverse inclusion we show that TðLh ð xÞ; xÞ # TðC; xÞ. By the assumed convexity we have TðLh ð xÞ; xÞ ¼ clconeðLh ð xÞ xÞ, i.e. any v 2 TðLh ð xÞ; xÞ is of the form lim t k ðxk xÞ, where tk > 0 and xk 2 Lh ð xÞ. On the other hand, let x 2 Lh ð xÞ be arbitrary. If i 2 I and xt :¼ x þ tðx xÞ, then gi(xt) 6 0 for t 2 [0, 1] by the convexity. For i R I; x 2 int g 1 i ð1; 0 by the assumed upper semicontinuity. So, for t > 0 small enough, gi(xt) 6 0. Hence xt 2 C. Therefore, tðx xÞ 2 coneðC xÞ for any x 2 Lh ð xÞ and any t > 0. It follows that the above-mentioned lim t k ðxk xÞ belongs to clcone ðC xÞ ¼ TðC; xÞ. Thus, TðLh ð xÞ; xÞ # TðC; xÞ, and then NðC; xÞ # NðLh ð xÞ; xÞ. Thus we have equality. In case (a) with the Slater condition, one has Lgk ð xÞ\ ð\i2Infkg intLgi ð xÞÞ–; (by the assumed upper semicontinuity, cf. the proof of Theorem 3.1) and hence, by the Moreau–Rockafellar theorem:

NðC; xÞ ¼ NðLh ðxÞ; xÞ ¼

X

NðLgi ðxÞ; xÞ:

i2I

In case (b), by Theorem 4.3 of [25], we also have this relation. By (iv) we can apply Theorem 3.1 to get (8). Taking ki 2 Rþ n f0g arbitrarily for i 2 I and ki = 0 for i 2 I we obtain (9) and (10).

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, applying the last assertion of Theorem 3.1, the If f is l.s.c. at x counterpart of (8)–(10) in this theorem, involving oaf, is proved. h

! X v o f ðxÞ \ o g i ðxÞ –f0g: v

ð12Þ

i2I

For the special case, where f is a Plastria function at x and gi are Gutierrez functions at x for all i 2 I (severe conditions), Theorem 4.1 with f being u.s.c. in (iv) collapses to Theorem 10 of [21] and relation (9) takes the form:

0 2 o< f ðxÞ þ

n X

kj o6 g i ðxÞ:

ð11Þ

Hence there are k1 ; . . . ; kn 2 Rþ , not all zero, such that:

0 2 ov f ðxÞ þ

n X

kj ov g j ðxÞ;

ð13Þ

j ¼ 1; . . . ; n:

ð14Þ

j¼0

kj g j ðxÞ ¼ 0;

i¼0

If, more speciﬁcally, f and gi,i = 1, . . . , n are Gateaux differentiable, then (9) and (10) become the classical Kuhn–Tucker multiplier rule. It should be noted here that (9) and (10) together are still weaker than (8), but they look more similar to the mentioned classical rule. Theorem 4.1 with dimX being ﬁnite in (iv) is new and indicated in the following example to be conveniently applied in some cases where Theorem 10 of [21] fails in use. Example 4.2. Let X ¼ R; x ¼ 0; g 1 ðxÞ ¼ x and f : R ! R be given by

f ðxÞ ¼

x

Proof. By applying Theorem 3.6 and the same arguments as in the proof of Theorem 4.1, we get the result. h Remark 4.6. The condition that x is an extreme point of C can be guaranteed by a stronger but easily checked one that it is an extreme point of Lgi ð xÞ, for some i⁄ 2 I. This condition in turn is weaker than the strictly quasiconvexity of g i . If f and gi are Gutiérrez functions at x, for all i 2 I, relation (13) takes the form:

0 2 o6 f ðxÞ þ

if x 6 0;

X

yi o6 g i ðxÞ:

ð15Þ

i2I

x þ 1 if x > 0:

L< f ðxÞ

Then, ¼ ð1; 0Þ and Lg ðxÞ ¼ ½0; þ1Þ. We can compute directly the subdifferentials o< f ð xÞ ¼ ½l; þ1Þ; o~ f ðxÞ ¼ ½0; þ1Þ; o6 g 1 ðxÞ ¼ v ð1; 1 and o g 1 ðxÞ ¼ ð1; 0. Consequently, f is a Plastria function at x and g1 is a Gutiérrez functions at x. The Slater condition is satisﬁed at x. x is not a local minimizer of f on R but is a solution of problem (7). Therefore, we can employ Theorem 4.1. In fact, taking k1 = 1 we get (11) and k1 g 1 ð xÞ ¼ 0. Note that Theorem 10 of [21] cannot be used as f is not u.s.c. at x ¼ 0. The following theorem gives a simple sufﬁcient condition for problem (7). Theorem 4.3. Let f be u.s.c. and x be a feasible solution of problem (7). Then, relation (8) implies that x is a solution. Proof. Let D = h1(1, 0]. Then, C # D. Observe that x is a solution to problem (7) if and only if L< ð x Þ \ C ¼ ;. We shall prove a stronf < ger conclusion that L< f ðxÞ \ D ¼ ;. Since f is u.s.c., Lf ðxÞ is open. Note that we always have:

NðD; xÞ

X X v N Lgi ðxÞ; x ¼ o g i ðxÞ: i2I

i2I

Now applying Theorem 3.4, we see that x is a solution to the following set-constrained problem:

minimize f ðxÞ subject to x 2 D: This completes the proof. h The following example yields a case where Theorem 4.3 is more advantageous than Theorem 12 of [21]. Example 4.4. Let x ¼ 0; f : R ! R be given by f(x) = 1 for x < 0, f(x) = x for x P 0, and g 1 : R ! R be given by g1(x) = x. Then f is u.s.c. but o

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Continuous Optimization

Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials Phan Quoc Khanh a, Ho Thuc Quyen b, Jen-Chih Yao c,⇑ a

Department of Mathematics, International University of Hochiminh City, Khu Pho 6, Linh Trung, Thu Duc, Hochiminh City, Vietnam Department of Mathematics, Ho Chi Minh City, University of Architecture, 196 Pasteur St., D.3, HoChiMinh City, Vietnam c Center for General Education, Kaohsiung Medical University, Kaohsiung 80702, Taiwan b

a r t i c l e

i n f o

Article history: Received 3 October 2008 Accepted 15 January 2011 Available online 4 February 2011 Keywords: Optimality conditions Convex sublevel sets Normal cones Star subdifferentials Adjusted subdifferentials

a b s t r a c t A set-constrained optimization problem and a mathematical programming problem are considered. We assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex. Using the two star subdifferentials and the adjusted subdifferential, we establish optimality conditions for usual minima and strict minima. Our results contain and improve some recent ones in the literature. Examples are provided to explain the advantages of each of our results. 2011 Elsevier B.V. All rights reserved.

1. Introduction Optimality conditions for nonconvex-nonsmooth problems have been intensively studied for a long time (see e.g. important books [5,22–24,32] and some papers of our group [11–20] among numerous works of other authors), since convexity and/or differentiability conditions are often not satisﬁed for optimization-related problems in practice. A large number of classes of such problems have been proposed and investigated due to demands of practical applications and also to motivations for mathematical researchers. Qua-siconvex functions constitute an important class of nonconvex functions as is evident from their application in optimization and economic modelling, see e.g. [1], and their structures are convenient for employing mathematical tools, including convex analysis. The reader can refer to [2–4,7,21,26–29] and references therein for recent developments in quasiconvex optimization. A function f from a normed space X to R :¼ R [ fþ1g is called quasi-convex if its sublevel set Lf(x) :¼ {u 2 X: f(u) 6 f(x)} at x is convex for all x 2 X or, equivalently, if for each r 2 R the strict sublevel set {u 2 X:f(u) < r} is convex. Hence, f is quasiconvex if and only if the strict sublevel set L< f ðxÞ :¼ fu 2 X : f ðuÞ < f ðxÞg is convex for all x 2 X. Another equivalent statement, which is often met in the literature, is that f is quasiconvex if for all x, y 2 dom f :¼ {x 2 X: f(x) < +1}, all t 2 [0, 1], f((1 t)xj + ty) 6 max{f(x),f(y)}. An optimization problem is quasiconvex if the objective is quasiconvex and ⇑ Corresponding author. Tel.: +886 7 5253816; fax: +886 7 5253809. E-mail address: [email protected] (J.-C. Yao). 0377-2217/$ - see front matter 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.01.024

the constraint set is convex. The generalized subdifferentials used in [21,27] were the lower subdifferen-tial (known also as Plastria subdifferential) [30], and the infradifferential (or Gutiérrez subdifferential) [9]. These subdifferentials enjoy many helpful properties and hence are convenient to be applied. However, they may be empty in a number of cases. Many simple cases are given in examples of Sections 3 and 4; even constantwise functions may have both Plastria and Gutiérrez subdifferentials being empty, cf. Example 3.3. There exist also differentiable quasiconvex functions with empty Plastria (Gutiérrez) subdifferentials at each point. Reference [21] even required f to be a Plastria or Gutiérrez function (see the deﬁnitions below). In [26,28] the Greenberg–Pierskalla subdifferential [8], a kind of normal-cone subdifferentials, i.e. those that are conic-valued, was additionally used. In [4,29] the two star subdifferentials, which are similar to the Greenberg–Pierskalla subdifferential, were introduced. They are nonempty under weak conditions. Moreover, they are closed convex cones, and quite different from the classical Fenchel subdifferential, which is very often bounded. However, these two star subdifferentials are in general neither quasimonotone nor cone-upper semicontinuous [2,4]. (Roughly speaking, a cone-value mapping is cone-upper semicontinuous, cone-u.s.c. in short, if the correspondence forms via its images with some hyperplane is itself Berge u.s.c..) Hence, they are not suitable for relating minimization problems to variational inequalities. Motivated by this, the authors of [2] introduced the adjusted sublevel set Laf ðxÞ, which is between L< f ðxÞ and Lf(x), and the corresponding adjusted subdifferential, deﬁned similarly as the star subdifferentials, and used them to deal with the solution

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existence of a quasiconvex set-constraint minimization problem. In [3] the adjusted subdifferential together with the limiting (or Mordukhovich) subdifferential [23] were employed to study the existence of solutions and optimality conditions for quasiconvex problems with a locally star-shaped constraint set. They also applied these results to mathematical programming with equilibrium constraints, where the equality constraint is quasiconvex and the equality and equilibrium constraints are quasiafﬁne. There have been also a number of papers using other kinds of generalized derivatives to deal with quasiconvex optimization problems. For instance, in recent reference [7], the Dini directional derivatives are employed. But in this paper we are concerned only with the afore-mentioned subdifferentials. We observe that the changes of the properties of quasiconvex functions, when we assume that the sublevel sets are convex only at a point x under consideration, not at each point, can be controlled when considering optimality conditions. Then the assumptions become remarkably less restrictive. These observations have motivated us to study optimality conditions involving a class of functions that need only admit a convex sublevel set at the optimal point. We do not impose any differentiability assumption and use the star and adjusted sub differentials. The organization of this paper is as follows. Section 2 contains deﬁnitions and preliminaries needed in the sequel. Section 3 is devoted to optimality conditions for a minimization problem with a convex constraint set. In Section 4 optimality conditions are established for the mathematical programming problem.

n o o< f ðxÞ :¼ x 2 X : 8x 2 L < 2; f ðxÞ ¼ 1 > > 2; > : x; L< f ðxÞ

0 small enough such that v þ th 2 C n f xg as this set is open. Therefore:

thu ; hi ¼ hu ;

v x þ thi hu ; v xi P 0;

which implies that u⁄ = 0, again a contradiction.

h

The following example illustrates advantages of Theorem 3.4. Example 3.5. Let C ¼ fðx1 ; x2 Þ : x2 < 0g [ fð0; 0Þg; x ¼ ð0; 0Þ and f : R2 ! R be given by

f ðx1 ; x2 Þ ¼

1; if ðx1 ; x2 Þ–ð0; 0Þ and x2 P 0; 0; if ðx1 ; x2 Þ ¼ ð0; 0Þ or x2 < 0:

Then C n fxg is open. Since o f ð xÞ ¼ o< f ðxÞ ¼ ;, Propositions 5 of [21], Proposition 2.1 of [26] and the results of [27,28] using these two subdifferentials cannot be in use. But the assumptions of Theorem 3.4 (ii) are fulﬁlled, since ð0; 1Þ 2 o~ f ðxÞ \ NðC; xÞ. It is easy to see directly that x is a minimizer of f on C. Now we prove a necessary condition for strict solutions to (3). Theorem 3.6. Let x be a strict solution to (3) and an extreme point of C. Assume that Lf ð xÞ is convex, C is not reduced to f xg and either of the conditions: (i)–(iii) of Theorem 3.1 holds. Then

ov f ðxÞ \ ðNðC; xÞÞ–f0g:

ð6Þ

Proof. Since x is an extreme point of C and C–f xg, the set C n f xg is convex and nonempty. As x is a strict solution to (3), C n f xg and Lf ð xÞ are disjoint. For (i), by the Hahn–Banach separation theorem, there exists some c 2 R and 0 – u⁄ 2 X⁄ such that the following inequalities hold, for all w 2 Lf ð xÞ and x 2 C n f xg:

hu ; x xi P c P hu ; w xi:

This result can be derived directly from Theorem 3.6. Indeed, there exists u⁄ – 0 such that u 2 NðC; xÞ and u 2 ov f ð xÞ ¼ Rþ o6 f ðxÞ, since f is a Gutiérrez function at x. So, one can ﬁnd s > 0 and x 2 o6 f ðxÞ such that u⁄ = sx⁄ and – x 2 NðC; xÞ. However, being a Gutiérrez function may be a severe restriction as shown by the following example. Example 3.8. Let C ¼ ½0; 1; x ¼ 0 and f : R ! R be given by

f ðxÞ ¼

1; for x < 0; x;

for x P 0:

Then Lf ðxÞ ¼ ð1; 0 and o6 f ðxÞ ¼ ;. So Proposition 6 of [21] cannot be applied. But ov f ðxÞ ¼ Rþ and NðC; xÞ ¼ Rþ and hence the conclusion of Theorem 3.6 is true. The assumptions of Theorem 3.6 can also be easily veriﬁed. Furthermore, o< f ðxÞ is also empty and then the results involving these two subdifferentials of [21,26–28] are not applicable. Passing to sufﬁcient conditions for strict solutions, we need the following strict convexity, used in [21] among others. Let X be a normed space. A # X is called strictly convex at x if hx ; x xi < 0 for every x 2 A n f xg and x 2 NðA; xÞ n f0g. Suggested by a referee, we discuss relations j between this strict convexity with some close known notions. Denote A(x) :¼ cone (A x). x 2 A is called an extreme point of A if Að xÞ \ ðAð xÞÞ ¼ f0g, and a strictly extreme point of A if clAð xÞ \ ðclAð xÞÞ ¼ f0g. x 2 A is said to be an exposed point of A if there exists x⁄ 2 X⁄ such that hx ; xi < hx ; xi for all x 2 A n f xg, and a strictly exposed point of A if this strict inequality holds for all x 2 clA n f xg. Moreover, x 2 A is called strongly strictly exposed point of A provided that x⁄ satisﬁes additionally that if hx ; xn i ! hx ; xi for xn 2 clA then xn ! x. In [31] the following relations were established. (i) Strong strict exposedness ) strict exposedness ) exposedness ) extremeness. (ii) A strictly exposed point is always a strictly extreme point. The converse is true if X is separable and A is convex. A strictly extreme point is incomparable with an exposed point in general. Example 6 of [31] gave a strictly extreme point which is not an exposed point. Conversely, boundary points of a disk are exposed but not strictly extreme. (iii) If X is ﬁnite dimensional and A is convex, the three notions of strong strict exposedness, strict exposedness and strict extremeness are equivalent. Now we compare with strict convexity. It is clear that if A is strictly convex at x, then this point is an extreme point. Unfortunately, strict convexity is incomparable with other abovementioned properties. We give illustrative examples.

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(i) The set A = {(x,y):y > 0} [ {(0,0)} in R2 is strictly convex at (0, 0), but this point is not strictly extreme. While (0, 0) is a strictly extreme point of R2þ but this set is not strictly convex at (0, 0). (ii) The set A in (i) is strictly convex at (0, 0), but this point is not strictly exposed. On the other hand, (0, 0) is strongly strictly exposed point of R2þ but this set is not strictly convex at (0, 0).

Then, oa f ðxÞ ¼ ½0; þ1Þ and NðC; xÞ ¼ ½0; þ1Þ and the mentioned modiﬁed conditions are fulﬁlled but x is not a strict solution. This fact agrees with Theorem 3.9, since ov f ð xÞ ¼ f0g. But x is clearly a solution, which can also be obtained from Theorem 3.4.

We have the following simple sufﬁcient condition for strict solutions.

Let us consider now the case in which the constraint set C is deﬁned by a ﬁnite family of inequalities, so that problem (3) turns into the mathematical programming problem:

Theorem 3.9. x is a strict solution of (3) if (6) is satisﬁed and either of the following conditions holds:

minimize f ðxÞ subject to g 1 ðxÞ 6 0; . . . ; g n ðxÞ 6 0:

(i) either C is strictly convex at x or C n f xg is open. (ii) Lf ð xÞ n f xg is open.

8w 2 Lf ðxÞ;

x 2 C;

then hu ; v xi ¼ 0 for any v 2 ðLf ðxÞ \ CÞ n f xg. For (i), observe ﬁrst that C n fxg is open implies that C is strictly convex at x. Indeed, if C n f xg is open, it is equal to int C. Hence hx ; x xi < 0 for every x 2 C n f xg and x 2 NðC; xÞ n f0g, i.e. C is strictly convex at x. Now we have to consider only the case where C is strictly convex at x. Then, by the strict convexity, hu ; x xi < 0 for all x 2 C n f xg, a contradiction. For (ii), let h 2 X be nonzero and arbitrary. As Lf ð xÞ n fxg is open, there exists t > 0 small enough such that v þ th 2 Lf ðxÞ. Then

g ¼ maxi6j6n g i ; C ¼ g ð1; 0; I ¼ fi : g i ðxÞ ¼ 0g

ð7Þ and

1 (i) L< f ðxÞ [ fxg and g i ð1; 0 are convex for i = 1, . . . , n. (ii) gi are u.s.c. at x for i = 1,. . .,n. (iii) either of the following regularity conditions holds: (a) there exists k 2 I such that Lgk ð xÞ \ fx 2 X : x 2 L< gi ðxÞ; 8i 2 I n fkgg–; (Slater condition). (b) X is complete, Lgi ð xÞ is closed for each i 2 I and Rþ ðD Pi2I Lgi ð xÞÞ is a closed subspace, where D = {(xi)i 2 I: "j, k 2 I;xj = xk} is the diagonal of XI. (iv) either X is ﬁnite dimensional or f is u.s.c. at some point of L< f ðxÞ.If x is a solution but not a local minimizer of f on X, then:

! X v o f ðxÞ \ o g I ðxÞ –f0g: ~

t hu ; hi ¼ hu ; v x þ thi hu ; v xi 6 0:

We denote h ¼ maxi2I g i .

1

Theorem 4.1. Assume for problem (7) that:

Proof. Suppose, ad absurdum, ðLf ð xÞ \ CÞ n f xg–;. The relation (6) implies that there exists 0 – u⁄ 2 X⁄ such that:

hu ; x xi P 0 P hu ; w xi;

4. Optimality conditions for the mathematical programming problem

ð8Þ

i2I ⁄

Hence u = 0, again a contradiction. h Note that, like Theorem 3.4, in this sufﬁcient condition no convexity condition is imposed on f. Note further that Theorem 3.9 (i) sharpens the sufﬁcient condition of Proposition 6 in [21], where the Gutiérrez subdifferential is employed. However, Gutiérrez subdifferential may be empty in many cases as shown in Example 3.8, where ov f ð xÞ \ NðC; xÞ ¼ Rþ , and so Theorem 3.9 is directly veriﬁed (and can be applied). The condition that Lf ð xÞ n f xg is open looks restrictive. So we illustrate it in the following example. Example 3.10. Let f : R2 ! R be given by

C ¼ f0g ð1; 0; ð x1 ; x2 Þ ¼ ð0; 0Þ

and

8 > < 1; if x1 –0 and x2 > 0; f ðx1 ; x2 Þ ¼ x22 ; if x2 < 0 or x1 ¼ 0; > : 2 x1 ; if x2 ¼ 0: Then Lf ð xÞ ¼ fðx1 ; x2 Þ : x2 > 0; x1 –0g [ fxg. C is not strictly convex at x since h(1, 0), xi = 0 for all x 2 C. So Proposition 6 of [21] does not work. But Lf ð xÞ n f xg is open. ov f ð xÞ ¼ f0g ð1; 0 and v ð0; 1Þ 2 o f ðxÞ \ ðNðC; xÞÞ and hence Theorem 3.9 concludes that x is a strict solution. A natural question is whether we can replace (6) by the following weaker relation:

oa f ðxÞ \ ðNðC; xÞÞ–f0g; in Theorem 3.9. The following example yields a negative answer. þ

Example 3.11. Let X ¼ R; C ¼ R ; x ¼ 0 and

f ðxÞ ¼

0;

if x P 0;

1; if x < 0:

This implies the following usual form, for some k1 ; . . . ; kn 2 Rþ , not all zero, such that:

0 2 o~ f ðxÞ þ

n X

kj ov g j ðxÞ;

ð9Þ

j¼0

kj g j ðxÞ ¼ 0; j ¼ 1; . . . ; n:

ð10Þ ~

If additionally, f is l.s.c. at x then, in (8) and (9), o f ðxÞ can be replaced by oa f ðxÞ.

Proof. Observe that C is convex and contained in Lh ð xÞ. So NðLh ð xÞ; xÞ # NðC; xÞ. To prove the reverse inclusion we show that TðLh ð xÞ; xÞ # TðC; xÞ. By the assumed convexity we have TðLh ð xÞ; xÞ ¼ clconeðLh ð xÞ xÞ, i.e. any v 2 TðLh ð xÞ; xÞ is of the form lim t k ðxk xÞ, where tk > 0 and xk 2 Lh ð xÞ. On the other hand, let x 2 Lh ð xÞ be arbitrary. If i 2 I and xt :¼ x þ tðx xÞ, then gi(xt) 6 0 for t 2 [0, 1] by the convexity. For i R I; x 2 int g 1 i ð1; 0 by the assumed upper semicontinuity. So, for t > 0 small enough, gi(xt) 6 0. Hence xt 2 C. Therefore, tðx xÞ 2 coneðC xÞ for any x 2 Lh ð xÞ and any t > 0. It follows that the above-mentioned lim t k ðxk xÞ belongs to clcone ðC xÞ ¼ TðC; xÞ. Thus, TðLh ð xÞ; xÞ # TðC; xÞ, and then NðC; xÞ # NðLh ð xÞ; xÞ. Thus we have equality. In case (a) with the Slater condition, one has Lgk ð xÞ\ ð\i2Infkg intLgi ð xÞÞ–; (by the assumed upper semicontinuity, cf. the proof of Theorem 3.1) and hence, by the Moreau–Rockafellar theorem:

NðC; xÞ ¼ NðLh ðxÞ; xÞ ¼

X

NðLgi ðxÞ; xÞ:

i2I

In case (b), by Theorem 4.3 of [25], we also have this relation. By (iv) we can apply Theorem 3.1 to get (8). Taking ki 2 Rþ n f0g arbitrarily for i 2 I and ki = 0 for i 2 I we obtain (9) and (10).

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, applying the last assertion of Theorem 3.1, the If f is l.s.c. at x counterpart of (8)–(10) in this theorem, involving oaf, is proved. h

! X v o f ðxÞ \ o g i ðxÞ –f0g: v

ð12Þ

i2I

For the special case, where f is a Plastria function at x and gi are Gutierrez functions at x for all i 2 I (severe conditions), Theorem 4.1 with f being u.s.c. in (iv) collapses to Theorem 10 of [21] and relation (9) takes the form:

0 2 o< f ðxÞ þ

n X

kj o6 g i ðxÞ:

ð11Þ

Hence there are k1 ; . . . ; kn 2 Rþ , not all zero, such that:

0 2 ov f ðxÞ þ

n X

kj ov g j ðxÞ;

ð13Þ

j ¼ 1; . . . ; n:

ð14Þ

j¼0

kj g j ðxÞ ¼ 0;

i¼0

If, more speciﬁcally, f and gi,i = 1, . . . , n are Gateaux differentiable, then (9) and (10) become the classical Kuhn–Tucker multiplier rule. It should be noted here that (9) and (10) together are still weaker than (8), but they look more similar to the mentioned classical rule. Theorem 4.1 with dimX being ﬁnite in (iv) is new and indicated in the following example to be conveniently applied in some cases where Theorem 10 of [21] fails in use. Example 4.2. Let X ¼ R; x ¼ 0; g 1 ðxÞ ¼ x and f : R ! R be given by

f ðxÞ ¼

x

Proof. By applying Theorem 3.6 and the same arguments as in the proof of Theorem 4.1, we get the result. h Remark 4.6. The condition that x is an extreme point of C can be guaranteed by a stronger but easily checked one that it is an extreme point of Lgi ð xÞ, for some i⁄ 2 I. This condition in turn is weaker than the strictly quasiconvexity of g i . If f and gi are Gutiérrez functions at x, for all i 2 I, relation (13) takes the form:

0 2 o6 f ðxÞ þ

if x 6 0;

X

yi o6 g i ðxÞ:

ð15Þ

i2I

x þ 1 if x > 0:

L< f ðxÞ

Then, ¼ ð1; 0Þ and Lg ðxÞ ¼ ½0; þ1Þ. We can compute directly the subdifferentials o< f ð xÞ ¼ ½l; þ1Þ; o~ f ðxÞ ¼ ½0; þ1Þ; o6 g 1 ðxÞ ¼ v ð1; 1 and o g 1 ðxÞ ¼ ð1; 0. Consequently, f is a Plastria function at x and g1 is a Gutiérrez functions at x. The Slater condition is satisﬁed at x. x is not a local minimizer of f on R but is a solution of problem (7). Therefore, we can employ Theorem 4.1. In fact, taking k1 = 1 we get (11) and k1 g 1 ð xÞ ¼ 0. Note that Theorem 10 of [21] cannot be used as f is not u.s.c. at x ¼ 0. The following theorem gives a simple sufﬁcient condition for problem (7). Theorem 4.3. Let f be u.s.c. and x be a feasible solution of problem (7). Then, relation (8) implies that x is a solution. Proof. Let D = h1(1, 0]. Then, C # D. Observe that x is a solution to problem (7) if and only if L< ð x Þ \ C ¼ ;. We shall prove a stronf < ger conclusion that L< f ðxÞ \ D ¼ ;. Since f is u.s.c., Lf ðxÞ is open. Note that we always have:

NðD; xÞ

X X v N Lgi ðxÞ; x ¼ o g i ðxÞ: i2I

i2I

Now applying Theorem 3.4, we see that x is a solution to the following set-constrained problem:

minimize f ðxÞ subject to x 2 D: This completes the proof. h The following example yields a case where Theorem 4.3 is more advantageous than Theorem 12 of [21]. Example 4.4. Let x ¼ 0; f : R ! R be given by f(x) = 1 for x < 0, f(x) = x for x P 0, and g 1 : R ! R be given by g1(x) = x. Then f is u.s.c. but o