By Gabriele Eichfelder

ISBN-10: 3540791574

ISBN-13: 9783540791577

This booklet provides adaptive resolution tools for multiobjective optimization difficulties in line with parameter based scalarization techniques. With assistance from sensitivity effects an adaptive parameter keep watch over is built such that top quality approximations of the effective set are generated. those examinations are in response to a different scalarization method, however the program of those effects to many different famous scalarization equipment can also be provided. Thereby very normal multiobjective optimization difficulties are thought of with an arbitrary partial ordering outlined by means of a closed pointed convex cone within the aim house. The effectiveness of those new tools is proven with numerous try out difficulties in addition to with a contemporary 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 clinical engineering.

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4) As the cone K is pointed and because r = 0m it is k + (t¯ − t ) r = 0m >0 and thus f (¯ x) = f (x ). 4) we conclude f (¯ x) ∈ f (x ) + K \ {0m } ¯ K-minimal. ✷ for x ∈ Ω in contradiction to x In the following we give a stricter restriction of the set from which we have to choose the parameter a such that we are still able to ﬁnd all K-minimal points. We ﬁrst consider the bicriteria case before we come to the more general case of an arbitrary multiobjective optimization problem. 1 Bicriteria Case In this section we only consider biobjective problems, i.

We start by showing that it is suﬃcient to vary the parameter a on a hyperplane H = {y ∈ Rm | b y = β} with b ∈ Rm \ {0m }, β ∈ R. 11. Let x ¯ be K-minimal for (MOP) and deﬁne a hyperplane H = {y ∈ Rm | b y = β} with b ∈ Rm \ {0m } and β ∈ R. Let r ∈ K with b r = 0 be arbitrarily given. Then there is a parameter a ∈ H and some t¯ ∈ R so that (t¯, x ¯) is a minimal solution of (SP(a, r)). This holds for instance for x) − β b f (¯ t¯ = b r and a = f (¯ x) − t¯r. Proof. For x) − β b f (¯ and a = f (¯ x) − t¯r t¯ = b r we have a ∈ H and the point (t¯, x ¯) is feasible for (SP(a, r)).

The connections will be studied in this section. The relations are important for applying the results about an adaptive parameter control gained in the following chapters for the general scalarization to the special problems, too. 1 ε-Constraint Problem We start with a common method called ε-constraint method ([54, 98, 60, 159, 165]). It is a very wide spread method especially in engineering design for ﬁnding EP-minimal points, because the method is very intuitive and the parameters are easy to interpret as upper bounds.

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