Adaptive Scalarization Methods in Multiobjective by Gabriele Eichfelder

By Gabriele Eichfelder

This booklet offers adaptive answer tools for multiobjective optimization difficulties in response to parameter established scalarization methods. Readers will enjoy the new adaptive tools and concepts for fixing multiobjective optimization.

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R. t. the cone K: (SP(a,r)) min t subject to the constraints a + t r − f (x) ∈ K, g(x) ∈ C, h(x) = 0q , t ∈ R, x ∈ S. This problem has the parameter dependent constraint set Σ(a, r) := {(t, x) ∈ Rn+1 | a + t r − f (x) ∈ K, x ∈ Ω}. We assume that the cone K is a nonempty closed pointed convex cone. 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+ ).

A + (t¯ − t + tˆ) r − f (ˆ ˆ) is feasible for (SP(a, r)) with t¯ − t + tˆ < t¯ in Hence (t¯ − t + tˆ, x contradiction to the minimality of (t¯, x ¯) for (SP(a, r)). 22. Thus for a minimal solution (t¯, x ¯) of the scalar optimization problem (SP(a, r)) with ¯ a + t¯r − f (¯ x) = k, k¯ = 0m , ¯) is a minimal there is a parameter a ∈ H and some t ∈ R so that (t , x solution of (SP(a , r)) with x) = 0 m a + t r − f (¯ (see Fig. 7) and hence (t , x ¯) is also a minimal solution of (SP(a , r)). 46 2 Scalarization Approaches Fig.

7. If the objective value of the optimization problem (SP(a, r)) for a ∈ Rm , r ∈ int(Rm + ) is not bounded from below then ) = ∅, i. e. there exists no EP-minimal point of the related M(f (Ω), Rm + multiobjective optimization problem. 1,a)). 3]). 8. If the point (t¯, x ¯) is a minimal solution of (SP(a, r)) ¯ ¯ with k := a + t r − f (¯ x) and if there is a point y = f (x) ∈ f (Ω) dominating the point f (¯ x) w. r. t. the cone K, then the point (t¯, x) is also a minimal solution of (SP(a, r)) and there exists a k ∈ ∂K, k = 0m , with a + t¯r − f (x) = k¯ + k.

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