## Modeling probability densities with sums of exponentials via polynomial approximation

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**Modeling probability densities with sums of exponentials via polynomial approximation.** / Dumitrescu, Bogdan; Şicleru, Bogdan C.; Avram, Florin.

Research output: Contribution to journal › Article › Scientific › peer-review

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*Journal of Computational and Applied Mathematics*, vol. 292, pp. 513–525. https://doi.org/10.1016/j.cam.2015.07.032

### APA

*Journal of Computational and Applied Mathematics*,

*292*, 513–525. https://doi.org/10.1016/j.cam.2015.07.032

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TY - JOUR

T1 - Modeling probability densities with sums of exponentials via polynomial approximation

AU - Dumitrescu, Bogdan

AU - Şicleru, Bogdan C.

AU - Avram, Florin

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Density fitting

KW - Optimization

KW - Polynomial approximation

KW - Semi-infinite programming

KW - Sum of exponentials

U2 - 10.1016/j.cam.2015.07.032

DO - 10.1016/j.cam.2015.07.032

M3 - Article

VL - 292

SP - 513

EP - 525

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -