GloptiPoly 3: moments, optimization and semidefinite programming

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. 1 What is GloptiPoly ? Gloptipoly 3 is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinite-dimensional optimization problem which can be viewed as an extension of the classical problem of moments [8]. From a theoretical viewpoint, the GPM has developments and impact in various areas of mathematics such as algebra, Fourier analysis, functional analysis, operator theory, probability and statistics, to cite a few. In addition, and despite a rather simple and short formulation, the GPM has a large number of important applications in various fields such as optimization, probability, finance, control, signal processing, chemistry, cristallography, tomography, etc. For an account of various methodologies as well as some of potential applications, the interested reader is referred to [1, 2] and the nice collection of papers [5]. The present version of GloptiPoly 3 can handle moment problems with polynomial data. Many important applications in e.g. optimization, probability, financial economics and LAAS-CNRS, University of Toulouse, France Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic Institute of Mathematics, University of Toulouse, France Department of Electrical Engineering, Linköping University, Sweden 1 optimal control, can be viewed as particular instances of the GPM, and (possibly after some transformation) of the GPM with polynomial data. The approach is similar to that used in the former version 2 of GloptiPoly [3]. The software allows to build up a hierarchy of semidefinite programming (SDP), or linear matrix inequality (LMI) relaxations of the GPM, whose associated monotone sequence of optimal values converges to the global optimum. For more details on the approach, the interested reader is referred to [8].