From functional optimization to nonlinear programming by the extended Ritz method

This chapter describes the approximate solution of infinite-dimensional optimization problems by the “Extended Ritz Method” (ERIM). The ERIM consists in substituting the admissible functions with fixed-structure parametrized (FSP) functions containing vectors of “free” parameters. The larger the dimensions, the more accurate the approximations of the optimal solutions of the original functional optimization problems. This requires solving easier nonlinear programming problems. In the area of function approximation, we review the definition of approximating sequences of sets, which enjoy the property of density in the sets of functions one wants to approximate. Then, we provide the definition of polynomially complex approximating sequences of sets, which are able to approximate functions provided with suitable regularity properties by using, for a desired arbitrary accuracy, a number of “free” parameters increasing at most polynomially when the number of function arguments grows. In the less studied area of approximate solution of infinite-dimensional optimization problems, the optimizing sequences and the polynomially complex optimizing sequences of FSP functions are defined. Results are presented that allow to conclude that, if appropriate hypotheses occur, polynomially complex approximating sequences of sets give rise to polynomially complex optimizing sequences of FSP functions, possibly mitigating the curse of dimensionality.