A Strengthened Barvinok-Pataki Bound on SDP Rank

The Barvinok-Pataki bound provides an upper bound on the rank of extreme points of a spectrahedron. This bound depends solely on the number of affine constraints of the problem, i.e., on the algebra of the problem. Specifically, the triangular number of the rank r is upper bounded by the number of affine constraints. We revisit … Read more

Robust Interior Point Method for Quantum Key Distribution Rate Computation

While the security proof method for quantum key distribution, QKD, based on the numerical key rate calculation problem, is powerful in principle, the practicality of the method is limited by computational resources and the efficiency of the underlying algorithm for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model … Read more

Finite convergence of sum-of-squares hierarchies for the stability number of a graph

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp.875–892], who conjectured convergence to $\alpha(G)$ in $r=\alpha(G) -1$ steps. Even the weaker conjecture claiming finite convergence is still open. … Read more

Weak notions of nondegeneracy in nonlinear semidefinite programming

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

A Geometric View of SDP Exactness in QCQPs and its Applications

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

A Reformulation-Linearization Technique for Optimization over Simplices

We study non-convex optimization problems over simplices. We show that for a large class of objective functions, the convex approximation obtained from the Reformulation-Linearization Technique (RLT) admits optimal solutions that exhibit a sparsity pattern. This characteristic of the optimal solutions allows us to conclude that (i) a linear matrix inequality constraint, which is often added … Read more

Graph Recovery From Incomplete Moment Information

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

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

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

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2 = Y$, the matrix analog of binary variables that satisfy $z^2 = z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization … Read more

SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the … Read more