Chordal and factor-width decomposition methods for semidefinite programming and
polynomial optimization have recently enabled the analysis and control of large-scale linear …

Slater's condition–existence of a “strictly feasible solution”–is a common assumption in conic
optimization. Without strict feasibility, first-order optimality conditions may be meaningless …

We develop a practical semidefinite programming (SDP) facial reduction procedure that
utilizes computationally efficient approximations of the positive semidefinite cone. The …

In the conic optimization problems, it is well-known that a positive duality gap may occur,
and that solving such a problem is numerically difficult or unstable. For such a case, we …

Semidefinite programming (SDP) solvers are increasingly used as primitives in many
program verification tasks to synthesize and verify polynomial invariants for a variety of …

A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern,
called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and …

A closed convex cone K in a finite dimensional Euclidean space is called nice if the set K∗+
F⊥ is closed for all F faces of K, where K∗ is the dual cone of K, and F⊥ is the orthogonal …

When sum-of-squares (SOS) programs are recast as semidefinite programs (SDPs) using
the standard monomial basis, the constraint matrices in the SDP possess a structural …

Despite the vast literature on DRS and ADMM, there has been very little work analyzing their
behavior under pathologies. Most analyses assume a primal solution exists, a dual solution …

We develop a monomial basis selection procedure for sum-of-squares (SOS) programs
based on facial reduction. Using linear programming and polyhedral approximations, the …