A “joint+marginal” algorithm for polynomial optimization

We present a new algorithm for solving a polynomial program P based on the recent “joint + marginal” approach of the first author for parametric polynomial optimization. The idea is to first consider the variable x₁ as a parameter and solve the associated (n - 1)-variable (x₂, ..., x_n) problem P(x₁) where the parameter x₁ is fixed and takes values in some interval Y₁ ⊂ ℝ, with some probability φ₁ uniformly distributed on Y₁. Then one considers the hierarchy of what we call “joint+marginal” semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations x₁ → p_k(x₁) that converges to the optimal value function x₁ → J(x₁) of problem P(x₁), as k increases. Then with k fixed à priori, one computes x̃₁* ∈ Y₁ which minimizes the univariate polynomial p_k(x₁) on the interval Y₁, a convex optimization problem that can be solved via a single semidefinite program. The quality of the approximation depends on how large k can be chosen (in general for significant size problems k = 1 is the only choice). One iterates the procedure with now an (n - 2)-variable problem P(x₂) with parameter x₂ in some new interval Y₂ ⊂ ℝ, etc. so as to finally obtain a vector x̃ ∈ ℝⁿ. Preliminary numerical results are provided.