## Abstract

Abstract We propose a method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on shape-constrained optimization with exponential functions. 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 probability density function. 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.

Original language | English |
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Pages (from-to) | 513–525 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 292 |

Early online date | 30 Jul 2015 |

DOIs | |

Publication status | Published - 2016 |

Publication type | A1 Journal article-refereed |

## Keywords

- Density fitting
- Optimization
- Polynomial approximation
- Semi-infinite programming
- Sum of exponentials

## Publication forum classification

- Publication forum level 2

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics