This is a survey of a recent methodology to solve systems of polynomial equations and inequalities for problems arising in combinatorial optimization. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions.

In this paper we study bipartite quantum correlations using techniques from tracial polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a qua...

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x∗Cx | x∗Akx ≥ 1, k = 0, 1, ...,m, x ∈ Fn} and (2) max{x∗Cx | x∗Akx ≤ 1, k = 0, 1, ..., m, x ∈ Fn}, where F is either the real field R or the complex field C, and Ak, C are ...

This paper aims to find efficient solutions a multi-objective optimization problem (MP) with convex polynomial data. To this end, hybrid method, which allows us transform into scalar \(({\mathrm{P}}_{z})\) and does not destroy the properties of convexity, is considered. First, we show an existence result for under some mild assumption. Then, \((P_{z})\), establish two kinds representations non-...