Semi-infinite optimization with sums of exponentials via polynomial approximation

TL;DR

This paper introduces a polynomial approximation-based method for semi-infinite optimization problems involving sums of exponentials, enabling efficient and accurate solutions through polynomial inequalities and linear matrix inequalities.

Contribution

The paper presents a novel approach that combines polynomial approximation with semi-infinite optimization, specifically for exponential functions, improving accuracy and computational efficiency.

Findings

Method outperforms existing approaches in accuracy

Approximations are quickly computed and highly precise

Applicable to fitting sums of exponentials in empirical data

Abstract

We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by low-degree polynomials. Thus, the constraints can be approximated with polynomial inequalities that can be implemented with linear matrix inequalities. Convexity is preserved, but the problem has now a finite number of constraints. We show how to take advantage of the properties of the exponential function in order to build quickly accurate approximations. The problem used for illustration is the least-squares fitting of a positive sum of exponentials to an empirical density. When the exponents are given, the problem is convex, but we also give a procedure for optimizing the exponents. Several examples show that the method is flexible, accurate and gives better results than other methods for the investigated problems.