Gabriele Eichfelder's Adaptive Scalarization Methods in Multiobjective PDF

By Gabriele Eichfelder

ISBN-10: 3540791574

ISBN-13: 9783540791577

ISBN-10: 3540791590

ISBN-13: 9783540791591

This e-book offers adaptive resolution equipment for multiobjective optimization difficulties in line with parameter established scalarization techniques. With assistance from sensitivity effects an adaptive parameter regulate is constructed such that high quality approximations of the effective set are generated. those examinations are in keeping with a unique scalarization process, however the program of those effects to many different recognized scalarization equipment can be provided. Thereby very basic multiobjective optimization difficulties are thought of with an arbitrary partial ordering outlined via a closed pointed convex cone within the aim house. The effectiveness of those new tools is proven with a number of try out difficulties in addition to with a up to date challenge in intensity-modulated radiotherapy. The ebook concludes with a different program: a strategy for fixing multiobjective bilevel optimization difficulties is given and is utilized to a bicriteria bilevel challenge in scientific engineering.

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L2 f (¯ Proof. 6) we already have 1 1 f (¯ x ) ≤ l f (¯ x2 ). We assume now l1 f (¯ x1 ) = l1 f (¯ x2 ). 15 we get l1 f (x) = l1 f (¯ 1 2 x )) = 0. 15 and thus l (f (x) − f (¯ 2 2 it is l f (x) ≥ l f (¯ x2 ) and hence l2 (f (x) − f (¯ x2 )) ≥ 0 for all x ∈ M(f (Ω), K). Summarizing this results in f (x)−f (¯ x2 ) ∈ K. As x is K-minimal we conclude f (x) = f (¯ x2 ) for all x ∈ M(f (Ω), K) and thus E(f (Ω), K) = 2 {f (¯ x )}. x1 ) = l2 f (¯ x2 ) implies E(f (Ω), K) = {f (¯ x1 )}. Analogously l2 f (¯ ✷ l1 We project the points f (¯ x1 ) and f (¯ x2 ) in direction r onto the line 1 H (compare Fig.

Because of 0 0 ¯ ∈ H . 20) a point s¯ ∈ R with m−1 s¯i v i . 22). Thus it is smin,i i i 0 i = 1, . . , m − 1 and it follows a ¯∈H . ✷ Hence we can also restrict the parameter set for the case of more than two objectives and arbitrary ordering cones K. 23) a + t r − f (x) = 0m , t ∈ R, x ∈ Ω. Here the inequality constraint a + t r − f (x) ∈ K is replaced by an equality constraint. For the connection between the problem (SP(a, r)) and the problem (SP(a, r)) the following theorem is important. 21.

A point x ¯ is a minimal solution of (Pk (ε)) with Lagrange multipliers μ ¯i ∈ R+ for i ∈ {1, . . , m} \ {k}, ν¯ ∈ Rp+ , and ξ¯ ∈ Rq , if and only if (fk (¯ x), x ¯) is a minimal solution of (SP(a, r)) with ¯ with μ Lagrange multipliers (¯ μ, ν¯, ξ) ¯k = 1, and ai = εi , ∀i ∈ {1, . . , m} \ {k}, with ek the kth unit vector in Rm . 25) 50 2 Scalarization Approaches Proof. By introducing the additional variable t ∈ R the scalar optimization problem (Pk (ε)) can be formulated as subject εi − fi (x) ≥ t − fk (x) ≥ ≥ gj (x) = hl (x) t ∈ R, x ∈ Rn .

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

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