Adaptive Scalarization Methods In Multiobjective by Gabriele Eichfelder

By Gabriele Eichfelder

This booklet offers adaptive answer tools for multiobjective optimization difficulties in line with parameter established scalarization ways. With the aid of sensitivity effects an adaptive parameter keep watch over is built such that fine quality approximations of the effective set are generated. those examinations are in keeping with a unique scalarization technique, however the program of those effects to many different recognized scalarization equipment can be awarded. Thereby very basic multiobjective optimization difficulties are thought of with an arbitrary partial ordering outlined by way of a closed pointed convex cone within the target house. The effectiveness of those new tools is verified with numerous try out difficulties in addition to with a contemporary challenge in intensity-modulated radiotherapy. The booklet concludes with yet another program: a strategy for fixing multiobjective bilevel optimization difficulties is given and is utilized to a bicriteria bilevel challenge in clinical engineering.

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18 as it is shown in [72, Theorem 2]. 2 Scalarization Approaches For determining solutions of the multiobjective optimization problem (MOP) (MOP) min f (x) subject to the constraints g(x) ∈ C, h(x) = 0q , x∈S with the constraint set Ω = {x ∈ S | g(x) ∈ C, h(x) = 0q } a widespread approach is the transformation of this problem to a scalar-valued parameter dependent optimization problem. This is done for instance in the weighted sum method ([245]). There the scalar problems m wi fi (x) min x∈Ω i=1 with weights w ∈ K ∗ \ {0m } and K ∗ the dual cone to the cone K, i.

We consider the scalar optimization problem (SP(a, r)) to the multiobjective optimization problem (MOP) with a ∈ Rm , r ∈ L(K). Let (t¯, x ¯) be a minimal solution, then x ¯ is minimal w. r. t. icr(K) ∪ {0m }. For the proof of this theorem we refer to [181]. 2 Properties of the Pascoletti-Serafini Scalarization 29 has no minimal solution, we can conclude under some additional assumptions that the related multiobjective optimization problem has no K-minimal solution at all. 2]. 4. Let K ⊂ Rm be a closed pointed convex cone with int(K) = ∅ and let the set f (Ω) + K be closed and convex.

By solving the scalar problems for a variety of parameters for instance for different weights, several solutions of the multiobjective optimization problem are generated. In the last decades the main focus was on finding one minimal solution e. g. by interactive methods ([166, 165]) whereas objective numerical calculations alternate with subjective decisions done by the decision maker. Based on much better computer performances it is now possible to represent the whole efficient set. Having the whole solution set available the decision maker gets a useful insight in the problem structure.

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