Gabriele Eichfelder

Adaptive scalarization methods in multiobjective optimization

Theory.- Theoretical Basics of Multiobjective Optimization.- Scalarization Approaches.- Sensitivity Results for the Scalarizations.- Numerical Methods and Results.- Adaptive Parameter Control.- Numerical Results.- Application to Intensity Modulated Radiotherapy.- Multiobjective Bilevel Optimization.- Application to Multiobjective Bilevel Optimization.

Local specific absorption rate control for parallel transmission by virtual observation points

The supervision of local specific absorption rate (SAR) in parallel transmission applications in MRI is crucial. One existing approach is to use electromagnetic simulations including human anatomical models and to precalculate the electric field distributions of each individual channel. These can be superposed later with respect to certain combined excitations under investigation, and the local SAR distribution can be evaluated. Local SAR maxima can be obtained by exhaustive search over all investigated subvolumes of the body model. Practical challenges arise for the adequate handling and comparing of precalculated field distributions as long as the expected combined radiofrequency excitations are still undetermined. Worst‐case approximations for local SAR lead to significant radiofrequency pulse performance limitations. Optimizing local SAR in radiofrequency pulse design using constraints for each subvolume is impractical. A method is proposed to significantly reduce the complexity without restriction to particular radiofrequency excitations. By constructing several matrices, it becomes sufficient to consider only these so‐called Virtual Observation Points for an adequate, conservative estimation of the maximum local SAR. The applied techniques involve concepts of vector optimization as well as semidefinite programming. Magn Reson Med, 2011.

Multiobjective bilevel optimization

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.

An Adaptive Scalarization Method in Multiobjective Optimization

This paper presents a new method for the numerical solution of nonlinear multiobjective optimization problems with an arbitrary partial ordering in the objective space induced by a closed pointed convex cone. This algorithm is based on the well-known scalarization approach by Pascoletti and Serafini and adaptively controls the scalarization parameters using new sensitivity results. The computed image points give a nearly equidistant approximation of the whole Pareto surface. The effectiveness of this new method is demonstrated with various test problems and an applied problem from medicine.

Optimality conditions for vector optimization problems with variable ordering structures

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319–377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop–Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.

Scalarizations for adaptively solving multi-objective optimization problems

Abstract In this paper several parameter dependent scalarization approaches for solving nonlinear multi-objective optimization problems are discussed. It is shown that they can be considered as special cases of a scalarization problem by Pascoletti and Serafini (or a modification of this problem). Based on these connections theoretical results as well as a new algorithm for adaptively controlling the choice of the parameters for generating almost equidistant approximations of the efficient set, lately developed for the Pascoletti-Serafini scalarization, can be applied to these problems. For instance for such well-known scalarizations as the ε-constraint or the normal boundary intersection problem algorithms for adaptively generating high quality approximations are derived.

Vector Optimization Problems and Their Solution Concepts

In vector optimization one investigates optimal elements of a set in a pre-ordered space. The problem of determining these optimal elements, if they exist at all, is called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineering and economics. There, these problems are also called multiobjective (or multi criteria or Pareto) optimization problems or one speaks of multi criteria decision making. Vector optimization problems arise, for example, in functional analysis (the Hahn–Banach theorem, the lemma of Bishop–Phelps, Ekeland’s variational principle), multiobjective programming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decades vector optimization has been extended to problems with set-valued maps. This field, called set optimization, has important applications to variational inequalities and optimization problems with multivalued data.

Numerical Procedures in Multiobjective Optimization with Variable Ordering Structures

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space ℝm, a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations are in general not transitive.In this paper, we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals, which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn–Graef–Younes method, known from multiobjective optimization with a partial ordering, is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.

On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

In the paper we prove that any nonconvex quadratic problem over some set K µR n with additional linear and binary constraints can be rewritten as linear problem over the cone, dual to the cone of K-semidefinite matrices.

Properly optimal elements in vector optimization with variable ordering structures

In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions.