Multiobjective Optimization on Function Spaces: A Kolmogorov Approach

Abstract : This report makes explicit that the Kolmogorov Criterion can specialize with sufficient detail to yield concrete and computationally viable tests that identify solutions to difficult optimization problems. Specifically, the classical equal-ripple characterization of best polynomial approximation is generalized to nonlinear polynomial optimization, and then generalized again to multiobjective polynomial optimization. Thus, results in polynomial optimization stretching over this last century readily fit into a single framework and are illustrated with applications in filter design and control theory. In addition to the finite-dimensional polynomials, the Kolmogorov Criterion also applies to the infinite-dimensional disk algebra. The disk algebra is basic to signal processing and control theory. Many engineering problems in these disciplines are optimization problems on the disk algebra. The Kolmogorov Criterion readily characterizes the minimizers of these nonlinear optimization problems. By making explicit the Kolmogorov Criterion and working specific examples, this report equips researchers with a general approach to optimization on spaces of functions and a collection of accessible research problems.