Describing Convex Semialgebraic Sets by Linear Matrix Inequalities

A semialgebraic set is a set described by a boolean combination of real polynomial inequalities in several variables. A linear matrix inequality (LMI) is a condition expressing that a symmetric matrix whose entries are affine-linear combinations of variables is positive semidefinite. We call solution sets of LMIs spectrahedra and their linear images semidefinite representable. Every spectrahedron satisfies a condition called rigid convexity, and every semidefinite representable set is convex and semialgebraic. Helton, Vinnikov and Nie recently showed in several seminal papers [21, 4, 3; 2] that the converse statements are true in surprisingly many cases and conjectured that they remain true in general. This shows the need for symbolic algorithms to compute LMI descriptions of convex semialgebraic sets. Once such a description is computed, it makes the corresponding semialgebraic set amenable to efficient numerical computation. Indeed, spectrahedra are the feasible sets in semidefinite programming (SDP), just in the same way as (convex closed) polyhedra are the feasible sets in linear programming (LP). The aim of this tutorial talk is to convince the audience that a symbolic interface between semialgebraic geometry and SDP has to be developed, and to initiate in the basic theory of LMI representations known so far. This theory is based to a large extent on determinantal representations of polynomials and on positivity certificates involving sums of squares.