An exact Jacobian SDP relaxation for polynomial optimization

Abstract

Given polynomials f (x), g i (x), h j (x), we study how to minimize f (x) on the setLet f min be the minimum of f on S. Suppose S is nonsingular and f min is achievable on S, which are true generically. This paper proposes a new type semidefinite programming (SDP) relaxation which is the first one for solving this problem exactly. First, we construct new polynomials ϕ 1 , . . . , ϕ r , by using the Jacobian of f, h i , g j , such that the above problem is equivalent toSecond, we prove that for all N big enough, the standard N -th order Lasserre's SDP relaxation is exact for solving this equivalent problem, that is, its optimal value is equal to f min . Some variations and examples are also shown.

Border basis relaxation for polynomial optimization

“…It has also been shown that the limit can even be reached in a finite number of steps in some cases, see e.g. (Lasserre et al, 2009;Nie et al, 2006;Marshall, 2009;Ha and Pham, 2010;Nie, 2011;Abril Bucero and Mourrain, 2013). In this case, the relaxation is said to be exact.…”

“…Then the relaxation associated to the preordering sequence L ⋆ t,G is exact and yields I min (see (Ha and Pham, 2010;Abril Bucero and Mourrain, 2013) or (Nie, 2011) for Cregularity and constraints G 0 that involve minors of A ν ). If I min is non-empty and finite then the border basis relaxation (2) yields the points V min and the border basis of I min .…”

“…The sequence of minima converges to the actual infimum f * of the function under some hypotheses (Lasserre, 2001). In some cases, the sequence even reaches the infimum in a finite number of steps (Laurent, 2007;Nie et al, 2006;Marshall, 2009;Demmel et al, 2007;Ha and Pham, 2010;Nie, 2011). This approach has proved to be particularly fruitful in many problems (Lasserre, 2009).…”

“…As shown in (Abril Bucero and Mourrain, 2013;Nie et al, 2006;Demmel et al, 2007;Marshall, 2009;Nie, 2011;Ha and Pham, 2010), an exact SDP relaxation can be constructed for "well-posed" optimization problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points, or zero-dimensional G-radicals.…”

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

“…Thus we can WLOG add these as constraints. It was shown in [12] that for general polynomial optimization problems, adding the KKT conditions yields an SDP hierarchy with finite convergence. Moreover, the number of levels necessary for convergence is a function only of the number of variables and the degrees of the objective and constraint polynomials.…”