mompy is a package for solving polynomial optimization and the Generalized Moment Problem by relaxing to a semidefinite program. These techniques are described by Lasserre et al. and Parrilo el al. in the references.

Various examples are included in the polynomial optimization worksheet and the extra examples worksheet, which includes a mixture of Gaussians example and a maxcut example.

mompy was used in our paper, which also describes the estimating mixture model setup:

Sida I. Wang, Arun Tejasvi Chaganty, Percy Liang. Estimating mixture models via mixtures of polynomials. NIPS 2015

You need sympy and cvxopt in order to use mompy.

The polynomial optimization problem is to

minimize f(x) subject to g_i(x) >= 0, for i=1,...,N.

where x is a real vector and f(x), g_i(x) are polynomials. mompy is an implementation of the Lassarre/SOS relaxations to a semidefinite program. One can easily encode NP-hard problems in this so specialized solvers are probably needed if you want to run on large problems, which is out of scope of mompy. But it is really easy to try this technique on small problems where we want to minimize f subject to constraints g:

x,y = sp.symbols('x,y')
f =  x**2 + y**2
gs = [x+y>=4, x+y<=4]
sol = mp.solvers.solve_GMP(f, gs)

We also support direct moment constraints in GMP, see the worksheets.

J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796–817, 2001

J. B. Lasserre. Moments, Positive Polynomials and Their Applications. Imperial College Press, 2011.

J. B. Lasserre. A semidefinite programming approach to the generalized problem of moments. Mathematical Programming, 112(1):65–92, 2008.

P. A. Parrilo and B. Sturmfels. Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 60:83–99, 2003

P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical programming, 96(2):293–320, 2003.

D. Henrion and J. Lasserre. Detecting global optimality and extracting solutions in GloptiPoly. In Positive polynomials in control, pages 293–310, 2005 pdf

Our paper is here

@inproceedings{wang2015polynomial, 
author = {Sida I. Wang and Arun Chaganty and Percy Liang}, 
booktitle = {Advances in Neural Information Processing Systems (NIPS)}, 
title = {Estimating Mixture Models via Mixture of Polynomials}, 
year = {2015} }