Download Adaptive Scalarization Methods In Multiobjective by Gabriele Eichfelder PDF

By Gabriele Eichfelder

This publication provides adaptive resolution tools for multiobjective optimization difficulties in accordance with parameter based scalarization methods. With assistance from sensitivity effects an adaptive parameter regulate is constructed such that fine quality approximations of the effective set are generated. those examinations are according to a distinct scalarization method, however the software of those effects to many different recognized scalarization equipment is additionally offered. Thereby very common multiobjective optimization difficulties are thought of with an arbitrary partial ordering outlined through a closed pointed convex cone within the target house. The effectiveness of those new equipment is tested with numerous attempt difficulties in addition to with a up to date challenge in intensity-modulated radiotherapy. The publication concludes with one more program: a technique for fixing multiobjective bilevel optimization difficulties is given and is utilized to a bicriteria bilevel challenge in scientific engineering.

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Extra info for Adaptive Scalarization Methods In Multiobjective Optimization

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2 Properties of the Pascoletti-Serafini Scalarization 25 (compare with the weighted Chebyshev norm). 1,c)) which is closest to the reference point. The Pascoletti-Serafini problem is also related to a scalarization introduced by Gerstewitz in [91] as well as to the problem discussed in [92, 237] by Tammer, Weidner and Winkler. Further, in [74] Engau and Wiecek examine the Pascoletti-Serafini scalarization concerning εefficiency. Pascoletti and Serafini allow for the parameter r only r ∈ L(K) with L(K) the smallest linear subspace in Rm including K.

SternaKarwat discusses in [213] and [214] also this problem. Helbig ([104, 106]) assumes r ∈ rint(K), i. e. he assumes r to be an element of the relative interior of the closed pointed convex cone K. In [106] Helbig varies not only the parameters a and r, but he also varies the cone K. For s ∈ K ∗ he defines parameter dependent cones K(s) with K ⊂ K(s). For these cones he solves the scalar problems (SP(a, r)). The solutions are then weakly K(s)-minimal, yet w. r. t. the cone K they are even minimal.

The formulation of this scalar optimization problem corresponds to the definition of K-minimality. A point x ¯ ∈ Ω with y¯ = f (¯ x) is K-minimal if (¯ y − K) ∩ f (Ω) = {¯ y }, (see Fig. 1 for m = 2 and K = R2+ ). If we rewrite the problem (SP(a, r)) as follows min t subject to the constraints f (x) ∈ a + tr − K, x ∈ Ω, t ∈ R, we see that for solving this problem the ordering cone −K is moved in direction −r on the line a + t r starting in the point a till the set (a + t r − K) ∩ f (Ω) is reduced to the empty set.

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