We propose a general method for optimization with semi-infinite constraintsthat involve a linear combination of functions, focusing on the case of theexponential function. Each function is lower and upper bounded on sub-intervalsby low-degree polynomials. Thus, the constraints can be approximated withpolynomial inequalities that can be implemented with linear matrixinequalities. Convexity is preserved, but the problem has now a finite numberof constraints. We show how to take advantage of the properties of theexponential function in order to build quickly accurate approximations. Theproblem used for illustration is the least-squares fitting of a positive sum ofexponentials to an empirical density. When the exponents are given, the problemis convex, but we also give a procedure for optimizing the exponents. Severalexamples show that the method is flexible, accurate and gives better resultsthan other methods for the investigated problems.
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