Section: New Results

Optimization

Participants : Jean Charles Gilbert, Émilie Joannopoulos, Cédric Josz.

A polynomial optimization problem (POP) consists in minimizing a multivariate real polynomial on a set $K$ defined by polynomial inequalities and equalities. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than a decade ago, J. B. Lasserre proposed to solve a POP by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size and precision. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment expression of the POP) and a dual SDP formulation (a sum-of-squares polynomial relaxation of the POP). In [18] , we show that there is no duality gap between each primal and dual SDP problem in Lasserre's hierarchy, provided one of the constraints in the description of set $K$ is a ball constraint. Our proof uses elementary results on SDP duality and it does not assume that $K$ has a strictly feasible point.

Convex quadratic optimization deals with problems consisting in minimizing a convex quadratic function on a polyhedron. In [3] , we analyzed the behavior of the augmented Lagrangian algorithm when it deals with an infeasible convex quadratic optimization problem; this situation is important to master in order to be able to solve correctly the QPs that are generated by the SQP (or Newton-like) algorithm to solve a nonlinear optimization problem, QPs whose feasibility is not guaranteed. It is shown that the algorithm finds a point that, on the one hand, satisfies the constraints shifted by the smallest possible shift that makes them feasible and, on the other hand, minimizes the objective on the corresponding shifted constrained set. The speed of convergence to such a point is globally linear, with a rate that is inversely proportional to the augmentation parameter. This suggests a rule for determining the augmentation parameter that aims at controlling the speed of convergence of the shifted constraint norm to zero; this rule has the advantage of generating bounded augmentation parameters even when the problem is infeasible. The approach has also been implemented in the pieces of software Oqla and Qpalm during the ADT Minoqs (see section  5.2 and [16] , [14] , [15] ).