Necessary Conditions for Weak Sharp Minima in Cone-Constrained Optimization Problems

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. constrained


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.

Notions and Preliminaries
Consider the following optimization problem with cone constraints where X is finite space and Y is a normed space.f is an extended real-valued function defined on X. S and K are nontrivial closed convex cones in Y , in which S defines an order.g : Let g : X → Y be a vector-valued mapping.The Hadamard and Dini derivatives of g at x in a direction v ∈ X are, respectively, defined by 2.4 Let f : X → R ∪ { ∞} be finite at x and m ≥ 1 an integer number.The upper Studniarski and Dini derivatives of order m at x in a direction v ∈ X are, respectively, defined by Let A be a closed and convex subset of X, we define the projection of a point x ∈ X onto the set A, denoted by P A, x as follows: Let K be a cone in a norm space Z. Denote by K * the dual cone of K where Z * is the topological dual of Z.Note that K * is a w * -closed convex cone.Let us introduce the following set: Definition 2.2.Let A be a subset of normed vector space Z and x ∈ clA, then a see 22 the contingent cone to the set

It is easy to see that IK A, x ⊂ T A, x ⊂ T A, x .
A set A is said to be regular at x ∈ A if T A, x K A, x .Obviously, every convex set is regular.Moreover, if A is a convex set, we call both the contingent cone and Clarke tangent cone to the set A as tangent cone to the set A.
For a nonempty set A ⊂ X, we define the polar of A to be the set A • {x * ∈ X * : x * , x ≤ 1 ∀x ∈ A}.The classic normal cone to A at x is defined dually by the relation N A, x T A, x with x n ∈ S, w n ∈ P E, x n and y n x n − w n t n , ∀n .

2.9
Definition 2.4.Let f map X to another Banach space Y .We say that f admits a strict derivative at x, an element L X, Y denoted ∇f x , provided that for each the following holds

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 S is a closed set.Let x ∈ S be a weak sharp minimizer of order m with module α > 0 for the problem 2.1 .Suppose that g is Hadamard derivative at x in all directions v ∈ X.Then, In particular, if S is regular at x, then Hence, According to the definition of weak sharp minimizer of order m, we get

3.7
Consequently, it follows from 3.7 that

3.8
Taking lim sups of both sides in 3.8 as n → ∞, we have 3.9 To establish 3.1 , we suffice to show that lim inf

3.11
If L ∞, then 3.10 holds true.Hence, we may assume that L < ∞.Then, for any > 0, there exists a number N, such that, for all k > N,

3.12
Set Since the sequence {z n k } is bounded, we may assume, taking a subsequence if necessary, that it converges to some z ∈ X with z m ≤ L. For each k > N, we have Observe that the regularity at x ∈ S implies that T S, x K S, x .Hence, the inequality 3.2 holds.Theorem 3.2.Suppose that S is a closed set.Let x ∈ S be a weak sharp minimizer of order m with module α > 0 for the problem 2.1 .Suppose that g is Hadamard derivative at x in all directions v ∈ X.Then,

3.16
In particular, if S is regular at x, then 3.17 Proof.Suppose that x is a weak sharp minimizer of order m with module α > 0 for the problem 2.1 , then , ∀x ∈ M.

3.18
Since g is the Hadamard derivative at x in the direction v ∈ X, there exist t n → 0 and Moreover, dg x, v ∈ − int K, then there exists a natural number N 2 such that, for n ≥ N 2 , which implies that Therefore,

3.23
On the other hand, if x t n v n / ∈ Q, then i Q x t n v n ∞ and 3.23 still holds true.Hence, from 3.23 , it follows that

3.24
Taking the lim sups on both sides in 3.24 as n → ∞, we get

3.25
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 m for the problem 2.1 in terms of the cone K 0 and T Q, x .Note that the necessary conditions do not require the cone K to have nonempty interior.Theorem 3.3.Suppose that S is a closed set.Let x ∈ S be a weak sharp minimizer of order m with module α > 0 for the problem 2.1 .Suppose that g is Dini derivative at x in all directions v ∈ X.Then,

3.26
In particular, if S is regular at x, then By using the method of Theorems 3.2 and 3.3, we easily establish the following results.
Theorem 3.4.Suppose that S is a closed set.Let x ∈ S be a weak sharp minimizer of order m with module α > 0 for the problem 2.1 .Suppose that g is Dini derivative at x in all directions v ∈ X.Then,

3.32
In particular, if S is regular at x, then

3.33
Since IK A, x ⊂ T A, x , the following result is direct consequence of Theorem 3.3.
Corollary 3.5.Suppose that S is a closed set.Let x ∈ bdS be a weak sharp minimizer of order m with module α > 0 for the problem 2.1 .Suppose that g is Dini derivative at x in all directions v ∈ X.Then, Suppose that X and Y 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 S is a closed set.Let x ∈ bdS be a weak sharp minimizer of order one with module α > 0 for the problem 2.1 .Suppose that f and g are strict derivatives at x.Then, for any

3.44
Taking the limit when n → ∞ and for the arbitrary > 0, we deduce that lim n → ∞ v n 0, which is a contradiction to the fact that v / 0.
By the definition of contingent cone, there exist t n → 0 and v n → v such that x t n v n ∈ Q.In addition, v / 0, by assumptiond H g x, v ∈ − int K.Since g is Hadamard derivative at x in the direction v ∈ X, we have that, for t n → 0 , ∈ − int K, then there exists a natural number N 1 such that for n ≥ N 1 , then there exists a sequence t n → 0 such that x t n v ∈ Q for sufficiently large n.In addition, ∈ M for sufficiently large n.The rest of the proof is analogue to Theorem 3.1 and thus omitted.
→ v, we have v n / 0 for n sufficiently large, and consequently, x n − w n d x n , S .Observe that the condition 3.36 implies that for n → ∞, → ∞,w n − x ≤ w n − x n x n − x ≤ 2 x n − x → 0. We can assume that lim n → ∞ f w n t n v n − f w n /t n ≤ 0 since ∇f x , v ≤ 0.Hence, for any > 0, there exist N 1 such that, for all n > N 1 , It follows that from ∇g x , v ∈ − int K, there exists a natural number N 2 such that, for n ≥ N 2 , On the other hand, by assumption, for n ≥ max{N 1 , N 2 }, x n w n t n v n ,f w n t n v n − f x f w n t n v n − f w n ≥ αd x n , S α x n − w n .3.43Together with relation 3.38 , for n ≥ max{N 1 , N 2 }, we have x ∩ {u : ∇g x , u ∈ − int K}.Then, by the definition of normal cone, there exist v n → v, x n → x, t n ∈ 0, ∞ , w n ∈ X with x n ∈ Q, w n ∈ P S, x n and