joint work with Lieven Vandenberghe
We will present new primal-dual algorithms for hyperbolic programming problems. Our algorithms have many desired properties including polynomial-time iteration-complexity for computing an approximate solution (for well-posed instances). In the special case of dense SDPs (as well as for symmetric cone programming problems with self-scaled barriers), our algorithms specialize to Nesterov-Todd algorithm. However, for certain sparse SDPs, our algorithms are new and have additional desired properties.
joint work with Cunlu Zhou
We construct a new class of objective functions on a symmetric cone compatible in the sense of Nesterov and Nemirovsky with the standard self-concordant barrier attached to the symmetric cone. The construction involves a matrix monotone scalar function on a positive semi-line. This allows us to develop a long-step path-following algorithm for symmetric programming problems with both equality and inequality constraints and objective functions from the class. In particular, the objective function arising in quantum entanglement problem belongs to this class. Numerical results are presented.
joint work with C.J. Argue
A convex cone K that is a subset of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank 1 matrices. ROG property is closely related to the characterizations of exactness of SDP relaxations, e.g., of nonconvex quadratic programs. We consider the case when K is obtained as the intersection of the PSD cone with finitely many linear (or conic) matrix inequalities, and identify sufficient conditions that guarantee that K is ROG. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient condition.