Abstract:
The first extension of the moment-sos approach for polynomial optimization, is concerned with semi-algebraic optimization problems, i.e., optimization problems P: min_x {f(x) : x in K}, where the objective function "f" and the defining functions of the feasible set K are semi-algebraic functions (and no only polynomials). Then problem P is equivalent to a polynomial optimization problem in a "lifted" space, and so under appropriate conditions, the standard hierarchy of semidefinite relaxations can be applied with guaranteed convergence to the global optimum.
In a second part, we provide a new characterization of nonnegativity for continuous functions on closed sets. Next, using this characterization when the feasible set K is simple enough (like e.g. R^n or R^n_+, a box, a simplex, an ellipsoid), we can provide a monotone nonincreasing sequence of upper bounds converging to the global minimum. Each upper bound is the optimal value of a semidefinite program with a single variable! (a generalized eigenvalue problem).