Well, someone can decide by themselves what they want to do and need to do but sometimes, that kind of person will need some theory of vector optimization references. People with open minded will always try to seek for the new things and information from many sources. On the contrary, people with closed mind will always think that they can do it by their principals. So, what kind of person are you?

Theory Of Vector Optimization

Optimization theory: the finite dimensional case, by Magnus R. Hestenes. Pp xiii, 447. £14. 1976. SBN 0 471 37471 7 (Wiley)

In this paper, by virtue of the image space analysis, we investigate general scalar robust optimization problems with uncertainties both in the objective and constraints. Under mild assumptions, we characterize various robust solutions for different kinds of robustness concepts, by introducing suitable images of the original uncertain problem, or the images of its counterpart problems appropriately, which provide a unified approach to tackling with robustness for uncertain optimization problems. Several examples are employed to show the effectiveness of the results derived in this paper.

A Unified Approach Through Image Space Analysis to Robustness in Uncertain Optimization Problems

In this paper, we characterize robust solutions for uncertain multiobjective optimization problems on the basis of vectorization models by virtue of image space analysis. By introducing cor...

Characterizations of multiobjective robustness on vectorization counterparts

In this paper, the connectedness and path-connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems with respect to addition-invariant set are studied. A class of weak generalized symmetric Ky Fan inequality problems via addition-invariant set is proposed. By using a nonconvex separation theorem, the equivalence between the solutions set for the symmetric Ky Fan inequality problem and the union of solution sets for scalarized problems is obtained. Then, we establish the upper and lower semicontinuity of solution mappings for scalarized problem. Finally, the connectedness and path-connectedness of solution sets for symmetric Ky Fan inequality problems are obtained. Our results are new and extend the corresponding ones in the studies.

Connectedness of Solution Sets for Weak Generalized Symmetric Ky Fan Inequality Problems via Addition-Invariant Sets

In the light, as said in Part I, of showing the main feature of image space analysis—to unify and generalize the several topics of optimization—we continue, in Part II, considering duality and penalization. In the literature, they are distinct sectors of optimization and, as far as we know, they have nothing in common. Here it is shown that they can be derived by the same “root.”

Constrained Extremum Problems and Image Space Analysis—Part II: Duality and Penalization

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.

Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area.

Constrained Extremum Problems and Image Space Analysis—Part III: Generalized Systems

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016. https://doi.org/10.1007/s10957-016-1032-9
 ). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.

Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

This paper mainly focuses on optimality conditions for efficient solutions of a nonconvex constrained multiobjective optimization problem via image space analysis. By virtue of the Gerstewitz funct...

Yangdong Xu

Papers

Constrained Extremum Problems and Image Space Analysis—Part II: Duality and Penalization

Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

Constrained Extremum Problems and Image Space Analysis—Part III: Generalized Systems

Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

Optimality conditions for efficient solutions of nonconvex constrained multiobjective optimization problems via image space analysis