Johannes Jahn

Introduction to the Theory of Nonlinear Optimization

This book serves as an introductory text to optimization theory in normed spaces and covers all areas of nonlinear optimization. It presents fundamentals with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory. The reader is expected to have a basic knowledge of linear functional analysis.

Contingent epiderivatives and set-valued optimization

In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable assumptions.

Optimality conditions for set-valued optimization problems

Abstract. The generalized contingent epiderivative of set-valued maps is introduced in this paper and its relationship to the contingent epiderivative is investigated. A unified necessary and sufficient optimality condition is derived in terms of the generalized contingent epiderivative. The existence of weak subgradients of set-valued maps is proved, and a sufficient optimality condition of set-valued optimization problems is obtained in terms of weak subgradients.

Scalarization in vector optimization

In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.

Duality in vector optimization

In this paper the problem dual to a convex vector optimization problem is defined. Under suitable assumptions, a weak, strong and strict converse duality theorem are proved. In the case of linear mappings the formulation of the dual is refined such that well-known dual problems of Gale, Kuhn and Tucker [8] and Isermann [12] are generalized by this approach.

The Lagrange Multiplier Rule in Set-Valued Optimization

The known Lagrange multiplier rule is extended to set-valued constrained optimization problems using the contingent epiderivative as differentiability notion. A necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions.

GENERALIZED CONTINGENT EPIDERIVATIVES IN SET-VALUED OPTIMIZATION: OPTIMALITY CONDITIONS

ABSTRACT Necessary and sufficient optimality conditions are given for various optimality notions in set-valued optimization. These optimality conditions are given by employing the generalized contingent epiderivative and the weak contingent epiderivative of the objective set-valued map and the set-valued map defining the constraints. The known Lagrange multiplier rule and the so-called Zowe-Kurcyusz-Robinson (cf. Robinson, S.M. Stability Theory for Systems of Inequalities. II, Differentiable Nonlinear Systems. SIAM J. Numer. Anal. 1976, 13, 497–513. Zowe, J.; Kurcyusz, S. Regularity and Stability for the Mathematical Programming Problem in Banach spaces, Appl. Math. Optim 1979, 5, 49–62.) regularity condition are extended using these differentiability notions.

A generalization of a theorem of Arrow, Barankin, and Blackwell

This paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell (“Admissible points of convex sets,” in Contributions to the Theory of Games, H. W. Kuhn and A. W. Tucker, eds., Princeton University Press, Princeton, NJ, 1953). This result is shown to hold even in a real normed space partially ordered by a Bishop-Phelps cone.

Reference point approximation method for the solution of bicriterial nonlinear optimization problems

This paper presents a reference point approximation algorithm which can be used for the interactive solution of bicriterial nonlinear optimization problems with inequality and equality constraints. The advantage of this method is that the decision maker may choose arbitrary reference points in the criteria space. Moreover, a special tunneling technique is given for the computation of global solutions of certain subproblems. Finally, the proposed method is applied to a mathematical example and a problem in mechanical engineering.

A characterization of properly minimal elements of a set

In this paper properly minimal elements of a set are characterized as minimal solutions of appropriate approximation problems without any convexity assumptions.