Convexity in SemiAlgebraic Geometry and Polynomial Optimization

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 two algebraic certificate of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied, it 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 [6]. 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.