On Polynomial Optimization Over Non-compact Semi-algebraic Sets

Abstract

The optimal value of a polynomial optimization over a compact semialgebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semialgebraic feasible sets, for which the associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. We show that for such problems, the corresponding hierachy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.

Convergent Semidefinite Programming Relaxations for Global Bilevel Polynomial Optimization Problems

“…The convergence of the proposed semidefinite programming approximation scheme relies on Assumption 2.1 which requires that the feasible set of the bilevel problem is bounded. The proposed scheme can also be extended to cover possible unbounded feasible sets by exploiting coercivity of the objective function of the upper/lower level problem as in our recent papers [21,22,23] where the convergence of the sequence of semidefinite programming relaxations was established for polynomial optimization problems with unbounded feasible sets.…”

Untitled

“…We shall refer to this as the Sum of Squares (SoS) proof system. In fact, provided a certain compactness condition holds, the certificate of infeasibility (i. e., the set {q S : S ⊆ [m]}) can always be assumed to have q S = 0 for |S| > 1; this is due to Putinar [25].…”

“…While the results in [35,56,25] were well-known in the algebraic geometry community and are intimately tied to Hilbert's seventeenth problem [23], the interest in the theoretical computer science community is relatively new. In the late 1980s, Shor [55] introduced the idea of replacing "positivity constraints" by "sum-of-squares constraints" in optimization problems and also noted that the latter type of constraints (for a fixed degree d) can be enforced using semidefinite programming.…”

A bilevel Farkas lemma to characterizing global solutions of a class of bilevel polynomial programs

“…Various sufficient conditions, including easily numerically verifiable conditions, for coercivity of a polynomial have been recently established in [17,13]. In particular, for a convex polynomial f on R n is coercive if there exists…”

“…This technique has been used in [13,17] to obtain convergent sequence of semidefinite programming relaxations for polynomial optimization problems over possibly noncompact feasible sets. (c) A very useful feature of the conclusion of Theorem 2.1 is that condition (ii) can be verified by solving a sequence of semidefinite linear programs.…”