Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone

Abstract

In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As an application, we consider a class of bounds on the stability number of a graph obtained from the polyhedral approximation hierarchy, and we construct a primal optimal solution with its tensor lifting for each of such linear programs.

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

“…Problem (18) is a natural CP tensor relaxation of problem (17) and by relaxing the equality constaints Z 1,c+n+1 − Z a+1,b+1 = 0, ∀(a, b, c) ∈ S into inequality constraints, we have the following CP tensor relaxation, Proof It is clear that if the coefficients of objective function q 0 (z) are nonnegative, at optimality of problem (19), Z 1,k+n+1 = Z i+1,j+1 holds. And the same objective function values are obtained for problems (18) and (19).…”

Copositive optimization – Recent developments and applications

“…Copositive approximation hierarchies [108,86,18,109,67,126,53] start with the zero-order approximation K (0) = P + N whose dual cone is the above discussed P ∩ N , and consist of an increasing sequence K (r) of cones satisfying cl ∪ r≥0 K (r) = C * . For instance, a higher-order approximation due to [108] uses squaring the variables to get rid of sign constraints: S ∈ C * if and only if y ⊤ Sy ≥ 0 for all y such that y i = x 2 i for some x ∈ R n , and this is guaranteed if the n-variable polynomial of degree 2(r + 2) in x,…”

Copositivity and constrained fractional quadratic problems

“…We will mostly deal with the cases K = C n and K = D * n = P n + N n , but any choice K = K r n for usual SDP-or LP-based approximation hierarchies (K r n ) r∈N would do, where K r n is in some sense close to C n for large r; see [38,11,39,28,50,23], who all more or less follow the ideas first put forward in [37,33]. Recall that checking membership of K r n in any such hierarchy usually involves psd matrices of order n r+1 , rendering these approximations computationally intractable for large r and n.…”

“…Hence ψ cop is a lower bound for (23). In analogy to (22), a stronger lower bound can be found by solving…”

“…Note that A replaces C in the general StFQP formulation (23), and that A plays a different role in the general CFQP formulation. We use (37) and introduce some results that, under a sufficient condition easily verifiable, allows to drop constraint z 1 > 0 in favor of the more manageable constraint z 1 ≥ 0.…”