Optimality conditions for set-valued optimization problems via scalarization function

One of the most important and popular topics in optimization problems is to find its optimal solutions, especially Pareto optimal points, a well-known solution introduced in multi-objective optimization. This topic is one of the oldest challenges in many issues related to science, engineering and other fields. Many important practical-problems in science and engineering can be expressed in terms of multi-objective/ set-valued optimization problems in order to achieve the proper results/ properties. To find the Pareto solutions, a corresponding scalarization problem has been established and studied. The relationships between the primal problem and its scalarization one should be investigated for finding optimal solutions. It can be shown that, under some suitable conditions, the solutions of the corresponding scalarization problem have uniform spread and have a close relationship to Pareto optimal solutions for the primal one. Scalarization has played an essential role in studying not only numerical methods but also duality theory. It can be usefully applied to get relationships/ important results between other fields, for example optimization, convex analysis and functional analysis. In scalarization, we ussually use a kind of scalarized-functions. One of the first and the most popular scalarized-functions used in scalarization method is the Gerstewitz function. In the paper, we mention some problems in set-valued optimization. Then, we propose an application of the Gerstewitz function to these problems. In detail, we establish some optimality conditions for Pareto/ weak solutions of unconstrained/ constrained set-valued optimization problems by using the Gerstewitz function. The study includes the consideration of problems in theoretical approach. Some examples are given to illustrate the obtained results.

Scalarizing functions in solving multi-objective problem-an evolutionary approach

<p><span>Scalarizing functions had long been observed for optimization of multi-objective problems. Scalarizing functions on multi-objective problem along with Differential Evolution (DE) algorithm variants had been used to analyze the effect of scalarizing functions. The main purpose is to find the better scalarizing function which can be applied for optimization. The effective solution of the multi-objective problem depends on the various factors like the DE algorithm and the scalarizing functions used. Multi objective evolutionary algorithm (MOEA) framework in java had been used for performing the analysis. The Obtained results showed that Tchebysheff scalarization function performs better than the other scalarizing functions on various indicator functions used.</span></p>

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.