Read e-book online Adaptive Scalarization Methods in Multiobjective PDF

By Gabriele Eichfelder

ISBN-10: 3540791574

ISBN-13: 9783540791577

ISBN-10: 3540791590

ISBN-13: 9783540791591

This e-book provides adaptive resolution equipment for multiobjective optimization difficulties in line with parameter established scalarization techniques. With assistance from sensitivity effects an adaptive parameter keep watch over is constructed such that top quality approximations of the effective set are generated. those examinations are in response to a unique scalarization technique, however the software of those effects to many different famous scalarization tools is usually provided. Thereby very common 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 equipment is verified with numerous try out difficulties in addition to with a contemporary challenge in intensity-modulated radiotherapy. The ebook concludes with another software: a method for fixing multiobjective bilevel optimization difficulties is given and is utilized to a bicriteria bilevel challenge in scientific engineering.

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Let K ⊂ Rm be a closed pointed convex cone with int(K) = ∅ and let the set f (Ω) + K be closed and convex. If there is a parameter (a, r) ∈ Rm × int(K) such that (SP(a, r)) has no minimal solution then E(f (Ω), K) = ∅. 5 for an arbitrary choice of parameters (a, r) ∈ Rm ×int(K), then we either get a weakly K-minimal solution or we get the information that there are no efficient points of the problem (MOP). This property is not satisfied by all scalarization problems as e. g. not by the ε-constraint method as we will see later in Sect.

9. If the point (t¯, x ¯) is an image-unique minimal solution of the scalar problem (SP(a, r)) w. r. t. f , i. e. there is no other minimal solution (t, x) with f (x) = f (¯ x), then x ¯ is a K-minimal solution of the multiobjective optimization problem (MOP). 7]) derive a criterion for checking whether a point is K-minimal or not. 10. A point x ¯ is a K-minimal solution of the multiobjective optimization problem (MOP) if i) there is some t¯ ∈ R so that (t¯, x ¯) is a minimal solution of (SP(a, r)) for some parameters a ∈ R and r ∈ int (K) and ii) for k := a + t¯r − f (¯ x) it is ((a + t¯r) − ∂K) ∩ (f (¯ x) − ∂K) ∩ f (Ω) = {f (¯ x)}.

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. e. K ∗ = {y ∗ ∈ Rm | (y ∗ ) y ≥ 0 for all y ∈ K}, are solved. Another scalarization especially for calculating EP-minimal points is based on the minimization of only one of the m objectives while all the other objectives are transformed into constraints by introducing upper bounds. 1) fi (x) ≤ εi , i ∈ {1, . . , m} \ {k}, x ∈ Ω.

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Adaptive Scalarization Methods in Multiobjective Optimization (Vector Optimization) by Gabriele Eichfelder

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