Modeling probability densities with sums of exponentials via polynomial approximation

• Bogdan Dumitrescu ^[2] ; Bogdan C. Şicleru ^[2] ; Florin Avram ^[1]
  1. [1] University of Pau and Pays de l'Adour

     University of Pau and Pays de l'Adour

     Arrondissement de Pau, Francia

     
  2. [2] University Politehnica of Bucharest, Romania
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 292, Nº 1 (15 January 2016), 2016, págs. 513-525
• Idioma: inglés
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• Resumen

  • 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.