Convexity in semi-algebraic geometry and polynomial optimization

Details Convexity in semi-algebraic geometry and polynomial optimization Convexity in semi-algebraic geometry and polynomial optimization

2008-06-23

arxiv.org/abs/0806.3784v3

Mathematics

Optimization and Control

21 pages

Scientific paper

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.

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