We investigate the completely positive semidefinite cone CS+, a new matrix cone consisting of all n×n matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular, we study relationships between this cone and the completely positive and the doubly nonnegative cone, and between its dual cone and trace positive non-commutative polynomials. We use this n...

These notes give an introduction to duality theory in the context of linear and positive semidefinite programming. These notes are based on material from Convex Analysis and Nonlinear Optimization by Borwein and Lewis and Numerical Optimization by Nocedal and Wright. Two examples are given to show how duality can be used. The first optimization application is to find the matrix in an affine fam...

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semid...

We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is NP -hard for any fixed integer k ≥ 2. Equivalently, for k ≥ 2, it is NP -hard to test membership in the rank constrained elliptope Ek(G), i.e., the set of all partial matrices with of...

A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solving SDP’s require O(n) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os sol...

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative as the inner product two positive (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is equivalent factorization. Applications include combinatorial optimization, quantum-based statistical models, and recommender systems, among others. However, despite increasing interest...