Algebraic Degree of Polynomial Optimization

TL;DR

This paper investigates the algebraic properties of solutions to polynomial optimization problems, establishing conditions under which optimal solutions are algebraic functions of input coefficients and deriving formulas for their algebraic degrees.

Contribution

It provides a general formula for the algebraic degree of optimal solutions in polynomial optimization, extending to special cases like QCQP, SOCP, and pOCP.

Findings

Optimality conditions hold on solutions under genericity assumptions.

Coordinates of optimizers are algebraic functions of polynomial coefficients.

Derived algebraic degrees for QCQP, SOCP, and pOCP.

Abstract

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the algebraic degree of semidefinite programming.