Polynomial Optimization with Real Varieties

We consider the optimization problem of minimizing a polynomial f(x) subject to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence of sum of squares relaxations for finding the global minimum. Let K be the feasible set. We prove the following results: i) If the real variety V_R(h) is finite, then Lasserre's hierarchy has finite convergence, no matter the complex variety V_C(h) is finite or not. This solves an open question in Laurent's survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite convergence of Lasserre's hierarchy is independent of the choice of defining polynomials for the real variety V_R(h). iii) When K is finite, a refined version of Lasserre's hierarchy (using the preordering of g) has finite convergence.Comment: 12 page

Linear Optimization with Cones of Moments and Nonnegative Polynomials

Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A that admit representing measures supported in K. Our main results are: i) We study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality, memberships), and construct a convergent hierarchy of semidefinite relaxations for each of them. ii) We propose a semidefinite algorithm for solving linear optimization problems with the cones P_A(K) and R_A(K), and prove its asymptotic and finite convergence; a stopping criterion is also given. iii) We show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they do, we show to get get a point in the intersections; if they do not, we prove certificates for the non-intersecting

Generating Polynomials and Symmetric Tensor Decompositions

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.Comment: 35 page

Certifying Convergence of Lasserre's Hierarchy via Flat Truncation

This paper studies how to certify the convergence of Lasserre's hierarchy of semidefinite programming relaxations for solving multivariate polynomial optimization. We propose flat truncation as a general certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: i) Putinar type Lasserre's hierarchy has finite convergence if and only if flat truncation holds, under some general assumptions, and this is also true for the Schmudgen type one; ii) under the archimedean condition, flat truncation is asymptotically satisfied for Putinar type Lasserre's hierarchy, and similar is true for the Schmudgen type one; iii) for the hierarchy of Jacobian SDP relaxations, flat truncation is always satisfied. The case of unconstrained polynomial optimization is also discussed.Comment: 18 page