Semidefinite programs (SDPs) are some of the most popular and broadly applicable optimization problems to emerge in the last thirty years. A curious pathology of SDPs, illustrated by a classical example of Khachiyan, is that their solutions may need exponential space to even write down. Exponential size solutions are the main obstacle to solve a … Read more
The moment sum of squares (moment-SOS) hierarchy produces sequences of upper and lower bounds on functionals of the exit time solution of a polynomial stochastic differential equation with polynomial constraints, at the price of solving semidefinite optimization problems of increasing size. In this note we use standard results from elliptic partial differential equation analysis to … Read more
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, $\ell$-dimensional kernel of the constraint matrix, by the linear independence of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this … Read more
Let S denote a subset of Rn defined by quadratic equality and inequality constraints and let S denote its projected semidefinite program (SDP) relaxation. For example, take S and S to be the epigraph of a quadratically constrained quadratic program (QCQP) and the projected epigraph of its SDP relaxation respectively. In this paper, we suggest … Read more
We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as for instance in some problems of optimal transport or density estimation. We show that the sole knowledge of first degree moments of the function, namely linear measurements, is sufficient to … Read more
A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has nearly feasible solutions that approximate the constraint set to arbitrary precision. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to “bad” linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe … Read more
A common approach to solve or find bounds of polynomial optimization problems like Max-Cut is to use the first level of the Lasserre hierarchy. Higher levels of the Lasserre hierarchy provide tighter bounds, but solving these relaxations is usually computationally intractable. We propose to strengthen the first level relaxation for sparse Max-Cut problems using constraints … Read more
We study an extended trust region subproblem minimizing a nonconvex function over the hollow ball $r \le \|x\| \le R$ intersected with a full-dimensional second order cone (SOC) constraint of the form $\|x – c\| \le b^T x – a$. In particular, we present a class of valid cuts that improve existing semidefinite programming (SDP) … Read more
The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is … Read more
We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite … Read more