Scalarization and well-posedness for set optimization using coradiant sets

The aim of this paper is to study scalarization and well-posedness for a set-valued optimization problem with order relations induced by a coradiant set. We introduce the notions of the set criterion solution for this problem and obtain some characterizations for these solutions by means of nonlinear scalarization. The scalarization function is a generalization of the scalarization function introduced by Khoshkhabar-amiranloo and Khorram. Moveover, we define the pointwise notions of LP well-posedness, strong DH-well-posedness and strongly B-well-posedness for the set optimization problem and characterize these properties through some scalar optimization problem based on the generalized nonlinear scalarization function respectively.

The GAP function of a parametric mixed strong vector quasivariational inequality problem

The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequality problems, fixed point problems, coincidence point problems, complementary problems etc. There are many authors who have been studied the gap functions for vector variational inequality problem. This problem plays an important role in many fields of applied mathematics, especially theory of optimization. In this paper, we study a parametric gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) in Hausdorff topological vector spaces. (SQVIP) Find x ̅ ∈ K(x ̅ ,γ) and z ̅ ∈ T(x ̅ ,γ) such that < z ̅ , y-x ̅ >+ f(y, x ̅ ,γ) ∈ Rn+ ∀ y ∈ K(x ̅ ,γ), where we denote the nonnegative of Rn by Rn+= {t=(t1 ,t2,…,tn )T ∈ Rn |ti >0, i = 1,2, ...,n}. Moreover, we also discuss the lower semicontinuity, upper semicontinuity and the continuity for the parametric gap function for this problem. To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces. Hence the results presented in this paper (Theorem 1.3 and Theorem 1.4) are new and different in comparison with some main results in the literature.

Generalized Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

We introduce and study well-posedness in connection with the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-Polyak well-posedness. Using the nonlinear scalarization function, we give some sufficient conditions to guarantee the existence of well-posedness for the symmetric vector quasi-equilibrium problem.

Constrained Weak Nash-Type Equilibrium Problems

A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.

A Characterization ofE-Benson Proper Efficiency via Nonlinear Scalarization in Vector Optimization

A class of vector optimization problems is considered and a characterization ofE-Benson proper efficiency is obtained by using a nonlinear scalarization function proposed by Göpfert et al. Some examples are given to illustrate the main results.

LP Well-Posedness for Bilevel Vector Equilibrium and Optimization Problems with Equilibrium Constraints

The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints.

Weak Sharp Minima in Set-Valued Optimization Problems

We introduce the notion of a weakψ-sharp minimizer for set-valued optimization problems. We present some sufficient and necessary conditions that a pair point is a weakψ-sharp minimizer through the outer limit of set-valued map and develop the characterization of the weakψ-sharp minimizer in terms of a generalized nonlinear scalarization function. These results extend the corresponding ones in Studniarski (2007).