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

and Applied Analysis 3 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 dHg x, v lim t→ 0 , u→v g x tu − g x t , dDg x, v lim t→ 0 g x tv − g x t . 2.4 Let f : X → R ∪ { ∞} be finite at x and m ≥ 1 an integer number. The upper Studniarski and Dini derivatives of orderm at x in a direction v ∈ X are, respectively, defined by d m S f x, v lim sup t→ 0 , u→v f x tu − f x tm , d m Df x, v lim sup t→ 0 f x tv − f x tm . 2.5 If A ⊂ X, then, the indicator function of A is iA x 0 if x ∈ A and iA x ∞ if x / ∈ A. The support function for A is defined by ψ∗ A x : sup{〈x∗, x〉 : x∗ ∈ A}. 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: P A,x { y ∈ A : ∥∥x − y∥X min u∈A ‖x − u‖X } . 2.6 Let K be a cone in a norm space Z. Denote by K∗ the dual cone of K K∗ {z∗ : 〈z∗, z〉 ≥ 0, ∀z ∈ K}, 2.7 where Z∗ is the topological dual of Z. Note that K∗ is a w∗-closed convex cone. Let us introduce the following set: K0 {z ∈ K : 〈z∗, z〉 > 0, ∀z∗ ∈ K∗ \ {0}}. 2.8 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 A is T A,x {v ∈ R : ∃tn → 0 , vn → v, with x tnvn ∈ A}, b see 23 the Clarke tangent cone to the set A is K A,x {v ∈ R : ∀A xn → x, ∀tn → 0 , ∃vn → v, with xn tnvn ∈ A}, c see 24 T̃ A,x {v ∈ X : ∃tn → 0 such that x tnv ∈ A, ∀n large enough}, d see 20 IK A,x {v ∈ X : ∃tn → 0 such that ∀vn → v, x tnvn ∈ A, ∀n large enough}. It is easy to see that IK A,x ⊂ T̃ A,x ⊂ T A,x . 4 Abstract and Applied Analysis A setA is said to be regular at x ∈ A if T A,x K A,x . Obviously, every convex set is regular. Moreover, ifA 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 ◦. Definition 2.3 see 17 . Let E and S be subset of R, and let x ∈ clE. The normal cone to E at x relative to S is defined by NS E, x : { y ∈ R : ∃yn −→ y, xn −→ x, tn ∈ 0, ∞ , wn ∈ R with xn ∈ S, wn ∈ P E, xn and yn xn −wn tn , ∀n } . 2.9 Definition 2.4. Let f mapX 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 lim x′ →x,y→v,t→ 0 f ( x′ ty ) − f x′ t 〈∇f x , v〉. 2.10 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 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, d m S f x, v ≥ αdist ( v, T ( S, x ))m , ∀v ∈ T Q,x ∩ {u : dHg x, u ∈ − intK } . 3.1 In particular, if S is regular at x, then d m S f x, v ≥ αdist ( v,K ( S, x )) , ∀v ∈ T Q,x ∩ {u : dHg x, u ∈ − intK } . 3.2 Proof. Let v ∈ T Q,x ∩ {u : dHg x, u ∈ − intK}. By the definition of contingent cone, there exist tn → 0 and vn → v such that x tnvn ∈ Q. In addition, v / 0, by assumption dHg x, v ∈ − intK. Since g is Hadamard derivative at x in the direction v ∈ X, we have that, for tn → 0 , lim n→∞ g x tnvn − g x tn dHg x, v . 3.3 Abstract and Applied Analysis 5 Moreover, dHg x, v ∈ − intK, then there exists a natural number N1 such that for n ≥ N1, g x tnvn − g x tn ∈ −K, 3.4and Applied Analysis 5 Moreover, dHg x, v ∈ − intK, then there exists a natural number N1 such that for n ≥ N1, g x tnvn − g x tn ∈ −K, 3.4 which implies that g x tnvn ∈ g x −K ⊂ −K. 3.5 Hence, x tnvn ∈ M Q ∩G. 3.6 According to the definition of weak sharp minimizer of order m, we get f x tnvn − f x ≥ αdist ( x tnvn, S )m . 3.7 Consequently, it follows from 3.7 that f x tnvn − f x tn ≥ α dist ( x tnvn, S )m tn . 3.8 Taking lim sups of both sides in 3.8 as n → ∞, we have d m S f x, v ≥ lim sup n→∞ f x tnvn − f x tn ≥ α lim sup n→∞ dist ( x tnvn, S )m tn ≥ α lim inf n→∞ dist ( x tnvn, S )m tn . 3.9 To establish 3.1 , we suffice to show that lim inf n→∞ dist ( x tnvn, S )m tn ≥ dist ( v, T ( S, x ))m . 3.10