Solving Vector-valued Approximation Problems by Semi-innnite Optimization : Numerical and Genericity Aspects

Abstract

We consider the approximation of vector-valued functions f 2 C (I; X) by functions u from a given subset U C (I; X) in the norm jjjf ? ujjjX = maxt2I kf(t) ? u(t)kX , where I = 0; 1], X a Banach space with norm k kX. For numerical purposes we write the approximation problem as a semi-innnite optimization problem. We especially discuss the linear approximation problem which can be put in the form of either a convex problem or in the form of a linear (extended) semi-innnite problem. A comparison of the problems in diierent norms k kX leads to numerically relevant implications of strongly and strictly uniqueness. We give a theoretical and numerical comparison of two diierent approaches for solving the vector-valued approximation problems. One approach is based on nonlinear optimization the other on techniques from linear semi-innnite optimization. The last section is devoted to genericity results.

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