Continuous approximate solutions in parametric optimization

The question of the existence of approximate solutions in parametric optimization is considered. Most results show that (under hypotheses) if a certain optimization problem has an approximate solution x ₀ for a value p ₀ of a parameter, then an approximate solution x=b(p) can be found for p in P, with b continuous, b(p ₀)=x₀, and any two such bs are homotopic. Some topological methods (use of fibrations) are used to weaken the usual convex hypotheses of such results. An equisemicontinuity condition (relative to a constraint) is introduced to allow some noncompactness. The results are applied to get approximate Nash equilibrium results for games with some nonconvexity in the strategy sets.     

Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle

This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles.     

On approximate solutions in set-valued optimization problems

In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets.     

Bilevel optimization with a multiobjective problem in the lower level

Numerical Algorithms - Bilevel problems model instances with a hierarchical structure. Aiming at an efficient solution of a constrained multiobjective problem according with some pre-defined...     

On comparison of approximate solutions in vector optimization problems

The problem of comparison of approximations (approximate solutions to a vector optimization problem) obtained using different numerical methods is considered. In the absence of a priori information about the set of weakly efficient vectors, a scalar function is introduced that enables pair-wise comparison of approximations and establishes a binary preference relation according to which the approximations close (in the sense of the Hausdorff distance) to the set containing all possible efficient vectors are preferable to other approximations.     

向量优化问题的近似解研究

向量优化是数学规划领域中十分重要的研究方向之一,其相关基础理论与基本方法的研究具有非常重要的理论意义与应用价值.近年来,关于近似解的定义及其性质研究已成为向量优化理论与方法研究的热点.现主要介绍国内学者,特别是我们团队在向量优化问题的各类近似解和统一解概念及其发展和各类近似解与统一解的性质研究方面取得的一些重要进展.最后,提出了与向量优化问题的近似解与统一解相关的一些公开问题.     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Optimality conditions and duality on approximate solutions in vector optimization with arcwise connectivity

In this paper a new class of generalized vector-valued arcwise connected functions, termed sub-arcwise connected functions, is introduced. The properties of sub-arcwise connected functions are derived. The approximate quasi efficient solutions of vector optimization problems are studied, and the necessary and sufficient optimality conditions are obtained under the assumption of arcwise connectivity. An approximate Mond-Weir type dual problem is formulated and the duality theorems are established.     

On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions (\(\epsilon \)-solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for \(\epsilon \)-solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.     

Some properties of approximate solutions for vector optimization problem with set-valued functions

In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions and weak efficient solutions are discussed. In particular, we prove the connectedness of the set of approximate solutions under the condition that the objective functions are quasiconvex set-valued functions.     

Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems

The purpose of this paper is to consider the set-valued optimization problem in Asplund spaces without convexity assumption. By a scalarization function introduced by Tammer and Weidner (J Optim Theory Appl 67:297–320, 1990), we obtain the Lagrangian condition for approximate solutions on set-valued optimization problems in terms of the Mordukhovich coderivative.     

Two-level dynamic optimization methods

The search for the optimum control of complex static or dynamic processes according toconventional optimization methods involves many problems due, in particular, to the volume and complexity of calculations. And, as the dimensions of the systems studied increases steadily, even if a theoretical solution is still possible, an acceptable solution from the economic standpoint can no longer be hoped for. To overcome these difficulties, an effective means is certainly the introduction of multilevel optimization methods. The use of such a technique for a static optimization of complex systems has already been discussed, and the problems of dynamic optimization are dealt with here. As for static systems, three decomposition methods can be applied. The different tasks to which they lead are examined for each of the levels, and the stability of coordinators is studied.     

多面体集下多目标优化问题近似解的若干性质

研究了多目标优化问题的近似解. 首先证明了多面体集是 co-radiant集,并证明了一些性质. 随后研究了多面体集下多目标优化问题近似解的特殊性质.     

拟凸多目标优化问题近似解的最优性条件

研究了拟凸多目标优化问题近似弱有效解、近似有效解的最优性条件.首先,在已有拟凸函数次微分的基础上引进4种近似次微分的概念,并给出它们之间的关系.然后,将4种近似次微分的概念应用到拟凸多目标优化问题中,给出了拟凸多目标优化问题近似弱有效解和近似有效解的充分条件和必要条件,并给出实例加以说明.     

A sequence of lower bound approximate solutions for the problem of secondary creep in pressure vessels

A previously derived [2] monotone non-increasing sequence of approximate solutions to the pressure vessel problem is modified to produce new sharper upper bounds. In addition, by means of a secant method, a monotone non-decreasing sequence of lower bound approximations is obtained. It is proved that this sequence is uniformly convergent and that it furnishes new lower bounds to the exact solution. These results are then used to obtain estimates for the time of initiation of brittle damage.

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Zusammenfassung Eine früher abgeleitete [2] monotone nichtwachsende Folge von Näherungslösungen für das Druckgefässproblem wird modifiziert, zur Gewinnung neuer schärferer oberer Schranken. Zusätzlich wird, mittels einer Sekantenmethode, eine nicht abnehmende Folge von Approximationen der unteren Schranke erhalten. Es wird bewiesen, dass diese Folge gleichmässig konvergiert und dass sie neue untere Schranken für die exakte Lösung liefert. Diese Ergebnisse werdenn dann verwendet, um Abschätzungen für den Zeitpunkt zu finden, in dem der Bruchschaden beginnt.

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This research was supported by the National Science Foundation under Grant MPS-75-07450. This paper was submitted by the first author in partial fulfillment of requirements for a Ph.D. degree at Illinois Institute of Technology, Chicago, Ill. 60616.     

On an algorithm solving two-level programming problems with nonunique lower level solutions

In the paper, an algorithm is presented for solving two-level programming problems. This algorithm combines a direction finding problem with a regularization of the lower level problem. The upper level objective function is included in the regularzation to yield uniqueness of the follower's solution set. This is possible if the problem functions are convex and the upper level objective function has a positive definite Hessian. The computation of a direction of descent and of the step size is discussed in more detail. Afterwards the convergence proof is given.Last but not least some remarks and examples describing the difficulty of the inclusion of upper-level constraints also depending on the variables of the lower level are added.     

An exact penalty on bilevel programs with linear vector optimization lower level

We are interested in a class of linear bilevel programs where the upper level is a linear scalar optimization problem and the lower level is a linear multi-objective optimization problem. We approach this problem via an exact penalty method. Then, we propose an algorithm illustrated by numerical examples.     

Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems

In this paper, we characterize approximate Benson-proper solutions of a constrained vector optimization problem with generalized cone convexity assumptions through approximate solutions of associated scalar optimization problems and also via approximate proper saddle point theorems. These results are based on an approximate version of the well known nearly subconvexlikeness notion and also on a new set-valued Lagrangian and a new concept of approximate proper saddle point.