Abstract
We present a sufficient and necessary condition for weak ψ-sharp minima in infinite-dimensional spaces. Moreover, we develop the characterization of weak ψ-sharp minima by virtue of a nonlinear scalarization function.
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 sharp minimum in 2 . Weak sharp minima play important roles in the sensitivity analysis 3, 4 and convergence analysis of a wide range of optimization algorithms 5 . Recently, the study of weak sharp solution set covers real-valued optimization problems 5-8 and piecewise linear multiobjective optimization problems 9-11 . Most recently, Bednarczuk 12 defined weak sharp minima of order m for vectorvalued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper Hölderness and Hölder calmness of the solution set-valued mappings for a parametric vector optimization problem. In 13 , Bednarczuk discussed the weak sharp solution set to vector optimization problems and presented some properties in terms of well-posedness of vector optimization problems. In 14 , Studniarski gave the definition of weak ψ-sharp local Pareto minimum in vector optimization problems under the assumption that the order cone is convex and presented necessary and sufficient conditions under a variety of conditions. Though the notions in 12, 14 are different for vector optimization problems, they are equivalent for scalar optimization problems. They are a generalization of the weak sharp local minimum of order m.
Introduction
In this paper, motivated by the work in 14, 15 , we present a sufficient and necessary condition of which a point is a weak ψ-sharp minimum for a vector-valued mapping in the 2 Fixed Point Theory and Applications infinite-dimensional spaces. In addition, we develop the characterization of weak ψ-sharp minima in terms of a nonlinear scalarization function. This paper is organized as follows. In Section 2, we recall the definitions of the local Pareto minimizer and weak ψ-sharp local minimizer for vector-valued optimization problems. In Section 3, we present a sufficient and necessary condition for weak ψ-sharp local minimizer of vector-valued optimization problems. We also give an example to illustrate the optimality condition.
Preliminary Results
Throughout the paper, X and Y are normed spaces. B x, δ denotes the open ball with center x ∈ X and radius δ > 0. N x is the family of all neighborhoods of x, and dist x, W is the distance from a point x to a set W ⊂ X. The symbols S c , int S and bds denote, respectively, the complement, interior and boundary of S.
Preliminary Results
Let D ⊂ Y be a convex cone containing 0 . The cone defines an order structure on Y , that is, a relation "
Preliminary Results
Let Ω be an open subset of X, S ⊂ Ω. Given a vector-valued map f : Ω → Y , the following abstract optimization is considered:
Preliminary Results
2.1
Preliminary Results
In the sequel, we always assume that D is a proper closed and convex cone.
Preliminary Results
Definition 2.1. One says that x 0 is a local Pareto minimizer for 2.1 , denoted by x 0 ∈ L Min f, S , if there exists U ∈ N x for which there is no x ∈ S ∩ U such that
Preliminary Results
If one can choose U X, one will say that x 0 is a Pareto minimizer for 2.1 , denoted by
Preliminary Results
Note that 2.2 may be replaced by the simple condition f x − f x 0 ∈ −D \ {0} if we assume that the cone D is pointed. 
Preliminary Results
where 
Preliminary Results
Remark 2.4. In the Definition 2.2, if Y R, D 0, ∞ , and ψ ψ m , then the relation 2.6 becomes the following form:
Preliminary Results
which is the well-known definition of a weak sharp minimizer of order m for 2.1 ; see 16 .
Main Results
In this section, we first generalize the result of 
Main Results
The set Λ is called a base for the dual cone D * . Recall that a point λ is an extremal point of a set Λ if there exist no different points λ 1 , λ 2 ∈ Λ and t ∈ 0, 1 such that λ tλ 1 1 − t λ 2 . 
, S if and only if there exists a covering {S
Fixed Point Theory and Applications
, S if and only if there exists a covering {S
Proof. i Part "only if": by assumption, there exist β > 0 and U ∈ N x 0 such that
, S if and only if there exists a covering {S
Let e ∈ int D be a fixed point. Set β 0 inf λ∈Λ λ, e . Since Λ is w * -compact, the infimum is attained at a point of Q. Namely, β 0 min λ∈Q λ, e . Clearly, λ, e > 0 for any λ ∈ Λ. Hence, β 0 > 0.
, S if and only if there exists a covering {S
For each λ ∈ Q, we define
, S if and only if there exists a covering {S
We will show that
, S if and only if there exists a covering {S
This relation, together with statement λ, e ≥ β 0 yields
, S if and only if there exists a covering {S
Obviously, for any λ ∈ D * , the above relation becomes the following form:
, S if and only if there exists a covering {S
Consequently, by the bipolar theorem, one has
3.10
Therefore,
3.10
and f x − f x 0 d ∈ B 0, βψ dist x, W , which is a contradiction to 3.4 . We have thus proved that S λ covers S ∩ U.
Fixed Point Theory and Applications 5
Now, let x ∈ S λ ∩ U \ W and λ ∈ Q. From the procedure of the above proof, we see that S ∩ U \ W ⊂ ∪ λ∈Q S λ . Hence, by 3.5 , set α ββ 0 / 4 e , inequality 3.2 is true.
Fixed Point Theory and Applications 5
Part "if": we define β 1 sup λ∈Λ λ, e . The supremum is attained at an extremal point because of the w * -compactness of Λ. So β 1 max λ∈Q λ, e > 0 and β −1
Fixed Point Theory and Applications 5
1 λ, e ≤ 1 for any λ ∈ Q. Hence, by assumption, we have
Fixed Point Theory and Applications 5
for x ∈ S λ ∩ U \ W and λ ∈ Q. Now, suppose that for all β > 0, 3.4 is false, then there exist
Fixed Point Theory and Applications 5
Let e ∈ int D be a fixed point, and since D is a cone, there is k > 0 such that B 0,
Fixed Point Theory and Applications 5
3.14 Therefore,
3.15
There is d ∈ D from 3.15 such that
3.17
By choosing β β
3.17
, we obtain a contradiction to 3.12 . ii Part "only if": for each λ ∈ Q, we define,
3.17
Now, we will check that 3.6 holds true. Pick any x ∈ S ∩ U. Suppose that x / ∈ S λ for any λ ∈ Q, then
3.17
Fixed Point Theory and Applications
3.17
Hence, for any λ ∈ cl cone co Q D * , λ, f x − f x 0 ≤ 0. By applying the bipolar theorem, we have
3.17
Combing it with the assumption, we have
3.17
which is a contradiction to 3.19 . So 3.6 holds and 3.3 is satisfied by the definition of S λ . Part "if": suppose that x 0 / ∈ L Min f, S , then there exists x ∈ S ∩ U such that
3.17
Indeed, x ∈ S ∩ U can be replace by x ∈ S ∩ U \ W, because x ∈ W, f x − f x 0 0, which is contradiction to 3.22 . Hence, for
3.17
3.23
3.17
It follows from the assumption that
3.17
Therefore, by 3.3 , we obtain Finally, we apply the nonlinear scalarization function to discuss the weak ψ-sharp minimizer in vector optimization problems.
3.17
Let D ⊂ Y be a closed and convex cone with nonempty interior int D. Given a fixed point e ∈ int D and y ∈ Y , the nonlinear scalarization function ξ : Y → R is defined by ξ y inf t : te ∈ y D .
3.26
This function plays an important role in the context of nonconvex vector optimization problems and has excellent properties such as continuousness, convexity, and strict monotonicity on Y . More results about the function can be found in 17 .
Fixed Point Theory and Applications 7
In what follows, we present several properties about the nonlinear scalarization function. iii ξ y r ⇔ re ∈ y bdD.
Given a vector-valued map
Next, we consider weak ψ-sharp local minimizer for a vector-valued map f through a weak sharp local minimizer of a scalar function ξ • f : X → R.
Given a vector-valued map
3.28
Given a vector-valued map
Proof. Part "only if": let us assume that x 0 ∈ WSL ψ, f, S . Thus, there exist α > 0 and U ∈ N x 0 such that
Given a vector-valued map
Note that, when W is a closed set,
Given a vector-valued map
Therefore,
Given a vector-valued map
By using Lemma 3.3 ii , one has Hence, it follows from the relation that B 0, βψ dist x, W ⊂ βψ dist x, W e − D, ∀x ∈ S ∩ U \ W.
3.42
Combing it with relation 3.40 , we deduce that

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