Exact Relaxations Research Articles

Robust SOS-convex polynomial optimization problems: exact SDP relaxations

This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained programs, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex programs. They also yield exact second-order cone relaxation for robust quadratically constrained programs.

Exact SDP relaxations for classes of nonlinear semidefinite programming problems

This paper addresses the issue of which nonlinear semidefinite linear programming problems possess exact semidefinite linear programming (SDP) relaxations under a constraint qualification. We establish exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials. These classes include SOS-convex semidefinite programming problems and fractional semidefinite programming problems with SOS-convex polynomials. The class of SOS-convex polynomials contains convex quadratic functions and separable convex polynomials. We also derive numerically checkable conditions, completely characterizing minimizers of these classes of problems.

Exact relaxations of non-convex variational problems

Here, we solve non-convex, variational problems given in the form $$\min_{u} I(u) = \int\limits_{0}^{1} f(u'(x))dx \quad {\rm s.t.} \quad u(0) = 0, u(1) = a, \quad(1)$$ where u ∈ (W1,∞(0, 1))k and \(f : {\mathbb{R}}^k \rightarrow {\mathbb{R}}\) is a non-convex, coercive polynomial. To solve (1) we analyse the convex hull of the integrand at the point a, so that we can find vectors \(a^1,\ldots,a^N \in {\mathbb{R}}^k\) and positive values λ1, . . . , λN satisfying the non-linear equation $$(1, a, f_c(a)) = \sum\limits_{i=1}^{N}\lambda_i(1, a^i, f(a^i)). \quad (2)$$ Thus, we can calculate minimizers of (1) by following a proposal of Dacorogna in (Direct Methods in the Calculus of Variations. Springer, Heidelberg, 1989). Indeed, we can solve (2) by using a semidefinite program based on multidimensional moments.

How to Construct Asymptotically Exact Relaxations in Robust Control?

How to Construct Asymptotically Exact Relaxations in Robust Control?