Abstract

We study weak sharp minima for optimization problems with cone constraints. Some necessary conditions for weak sharp minima of higher order are established by means of upper Studniarski or Dini directional derivatives. In particular, when the objective and constrained functions are strict derivative, a necessary condition is obtained by a normal cone.

1. Introduction

The notion of a weak sharp minimum in general mathematical program problems was first introduced by Ferris in [1]. It is an extension of a sharp (or strongly unique) minimum in [2]. Weak sharp minima play an important role in the sensitivity analysis [3, 4] and convergence analysis of a wide range of optimization algorithms [5–7]. Recently, the study of weak sharp solution set covers real-valued optimization problems [5, 8–10] and multiobjective optimization problems [11–13]. Moreover, it has been extended to convex-composite optimization with inequality constraints [14] and semi-infinite programs [15].

The weak sharp minima defined in [5] specified first-order growth of the objective function away from the set of optimal solutions. Recently, Studniarski [16] considered a special class of nonsmooth functions which are pointwise maxima of finite collections of strictly differentiable functions and presented a characterization of weak sharp local minima of order one. In addition, Studniarski [17] established the Kuhn-Tucker conditions for a nonlinear programming problem with constraints of both inequality and equality types, where the objective and inequality constrained functions are locally Lipschitzian and the equality constraints are differentiable.

Weak sharp minima of higher order are also of interest in sensitivity analysis in parametric optimization. In particular, the presence of weak sharp minima in parametric optimization leads to Hëlder continuity properties of the associated solution mappings in [18]. Bonnans and Ioffe [19] studied sufficient conditions and characterizations for weak sharp minima of order two in the case that the objective function is a pointwise maximum of twice continuously differentiable convex functions. In [20], Ward presented some necessary conditions for weak sharp minima of higher order for optimization problems with a set constraint. In [21], Studniarski and Ward obtained some sufficient conditions and characterizations for weak sharp local minimizer of higher order in terms of the limiting proximal normal cone and a generalization of the contingent cone.

However, to the best of our knowledge, there has no research concerning weak sharp minima for optimization with cone constraints although conic programming is a very hot research topic in optimization. In this paper, we first discuss necessary conditions for weak sharp minima of higher order in terms of the upper Studniarski and Dini directional derivatives and various tangent cones. In particular, by means of a normal cone, we provide a necessary condition for weak sharp minima of order one when the objective and constrained functions are strict derivative.

This paper is organized as follows. In Section 2, we recall the basic definitions. In Section 3, we establish several necessary conditions for a weak sharp minimizer of higher order.

2. Notions and Preliminaries

Consider the following optimization problem with cone constraints where is finite space and is a normed space. is an extended real-valued function defined on . and are nontrivial closed convex cones in , in which defines an order. is a vector-valued mapping. is a closed convex subset of . Let . Denote by the feasible set, that is, .

Definition 2.1 (see [21]). Let be the Euclidean norm on . Suppose that is a constant on the set , and let and . For , let (a)We say that is a weak sharp minimizer of order with module for (2.1), if there exists such that

Let be a vector-valued mapping. The Hadamard and Dini derivatives of at in a direction are, respectively, defined by

Let be finite at and an integer number. The upper Studniarski and Dini derivatives of order at in a direction are, respectively, defined by

If , then, the indicator function of is if and if . The support function for is defined by .

Let be a closed and convex subset of , we define the projection of a point onto the set , denoted by as follows:

Let be a cone in a norm space . Denote by the dual cone of where is the topological dual of . Note that is a -closed convex cone. Let us introduce the following set:

Definition 2.2. Let be a subset of normed vector space and , then(a)(see [22]) the contingent cone to the set is ,(b)(see [23]) the Clarke tangent cone to the set is ,(c)(see [24]) ,(d)(see [20]) .

It is easy to see that .

A set is said to be regular at if . Obviously, every convex set is regular. Moreover, if is a convex set, we call both the contingent cone and Clarke tangent cone to the set as tangent cone to the set .

For a nonempty set , we define the polar of to be the set . The classic normal cone to at is defined dually by the relation .

Definition 2.3 (see [17]). Let and be subset of , and let . The normal cone to at relative to is defined by

Definition 2.4. Let map to another Banach space . We say that admits a strict derivative at , an element denoted , provided that for each the following holds

3. Necessary Conditions

In the section, we provide necessary optimality conditions for the problem (2.1), which are formulated in terms of the upper Studniarski and Dini derivatives of the objective function, respectively. Simultaneously, we also apply the indicator function of a set to state the necessary conditions.

Theorem 3.1. Suppose that is a closed set. Let be a weak sharp minimizer of order with module for the problem (2.1). Suppose that is Hadamard derivative at in all directions . Then, In particular, if is regular at , then

Proof. Let . By the definition of contingent cone, there exist and such that . In addition, , by assumption .
Since is Hadamard derivative at in the direction , we have that, for , Moreover, , then there exists a natural number such that for , which implies that Hence, According to the definition of weak sharp minimizer of order , we get Consequently, it follows from (3.7) that Taking lim sups of both sides in (3.8) as , we have To establish (3.1), we suffice to show that Set If , then (3.10) holds true. Hence, we may assume that . Then, for any , there exists a number , such that, for all , Set Since the sequence is bounded, we may assume, taking a subsequence if necessary, that it converges to some with . For each , we have Hence and (3.10) holds.
Observe that the regularity at implies that . Hence, the inequality (3.2) holds.

Theorem 3.2. Suppose that is a closed set. Let be a weak sharp minimizer of order with module for the problem (2.1). Suppose that is Hadamard derivative at in all directions . Then, In particular, if is regular at , then

Proof. Suppose that is a weak sharp minimizer of order with module for the problem (2.1), then Since is the Hadamard derivative at in the direction , there exist and such that Moreover, , then there exists a natural number such that, for , which implies that Therefore, If , then, by (3.18), On the other hand, if , then and (3.23) still holds true.
Hence, from (3.23), it follows that Taking the lim sups on both sides in (3.24) as , we get The rest of the proof is similar to Theorem 3.1 and hence omitted.

In what follows, we state other necessary conditions for the weak sharp minimizer of order for the problem (2.1) in terms of the cone and . Note that the necessary conditions do not require the cone to have nonempty interior.

Theorem 3.3. Suppose that is a closed set. Let be a weak sharp minimizer of order with module for the problem (2.1). Suppose that is Dini derivative at in all directions . Then, In particular, if is regular at , then

Proof. Let , then there exists a sequence such that for sufficiently large . In addition, which leads to the following relation where as . Since , for any , , which yields that for large enough . Observing that is a closed convex cone, it follows that is weakly closed and Consequently, for sufficiently large . The rest of the proof is analogue to Theorem 3.1 and thus omitted.

By using the method of Theorems 3.2 and 3.3, we easily establish the following results.

Theorem 3.4. Suppose that is a closed set. Let be a weak sharp minimizer of order with module for the problem (2.1). Suppose that is Dini derivative at in all directions . Then, In particular, if is regular at , then

Since , the following result is direct consequence of Theorem 3.3.

Corollary 3.5. Suppose that is a closed set. Let be a weak sharp minimizer of order with module for the problem (2.1). Suppose that is Dini derivative at in all directions . Then,

Suppose that and are finite spaces. In what follow, we apply the normal cone to present a necessary optimality condition for problem (2.1), where the objective and constrained functions are strict derivative.

Theorem 3.6. Suppose that is a closed set. Let be a weak sharp minimizer of order one with module for the problem (2.1). Suppose that and are strict derivatives at . Then, for any with ,

Proof. Assume that for some . Then, by the definition of normal cone, there exist , , , with , and Since and , we have for sufficiently large, and consequently, .
Observe that the condition (3.36) implies that for , Moreover, for , . We can assume that since . Hence, for any , there exist such that, for all , Since is strict derivative at , It follows that from , there exists a natural number such that, for , which implies that Hence, On the other hand, by assumption, for , , Together with relation (3.38), for , we have Taking the limit when and for the arbitrary , we deduce that , which is a contradiction to the fact that .