Polynomial Optimization II

TD-04: Polynomial Optimization II
Stream: Polynomial Optimization
Room: Lagrange
Chair(s): Corbinian Schlosser

Topology optimization of discrete structures by polynomial optimization
Marek Tyburec, Jan Zeman, Martin Kružík, Didier Henrion
Although finding continuous cross-sections of minimum-compliance frame structures is a fundamental problem of structural design, only local optimization techniques are adopted. Here, we develop a semidefinite programming formulation with a non-convex polynomial matrix inequality constraint and solve it by the moment-sum-of-squares hierarchy. Besides lower bounds from the hierarchy, we recover a feasible solution in each relaxation, building a sequence of upper bounds. The gap between the bounds quantifies the solution quality and establishes a simple sufficient condition for global optimality.

The approximation of measures by positive moment sequences
Lorenzo Baldi, Bernard Mourrain
To solve Polynomial Opt. problems Lasserre introduced the Moment Matrix hierarchy, approximating measures with a convex cone of positive linear functionals, and proved the convergence of optimal value to the minimum in the Archimedean case. We analyse the convex cone of truncated positive moment sequences and show that truncated measures with low rank moment matrix are extremal rays. We show the convergence in Hausdorff distance of a convex section of the outer approximation to probability measures, and describe the order of convergence using an effective version of Putinar Positivstellensatz.

RAPOSa: a polynomial optimization solver
Brais González Rodríguez, Julio González-Díaz, Ángel Manuel González Rueda, Beatriz Pateiro-López, Diego Rodriguez Martinez, David R Penas
In this talk we will present RAPOSa, a new solver for polynomial optimization problems based on the reformulation linearization techniques (originally introduced by Sherali and Tuncbilek in 1991). The current implementation incorporates most of the features discussed in past literature and other additional enhancements. The present version of the algorithm has proven to be very efficient, and comparisons with other global optimization solvers will be presented.

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