- Abstract operator system on cones of positive semidefinite matrices (arXiv)
Author: Martin Berger, Tim Netzer
Summary: There are several important abstract operator systems with convex cones of positive semidefinite matrices at the first level. The well-known are the operator systems for separable matrices, positive semidefinite matrices, and block positive matrices. From a map perspective, these are operator systems for disentangled maps, fully positive maps, and positive linear maps, respectively. However, there are other interesting and less studied operator systems. For example, fully co-positive maps, double fully positive maps, and decomposable maps all play important roles in quantum information theory. We investigate which of these systems are finitely generated and which admit finite-dimensional realizations in the sense of the Choi-Effros theorem. Completely answer this question for all systems described. Our main contribution is that, while decomposable maps form systems that do not admit finite-dimensional realizations, although finite-dimensional realizations are possible, systems of doubly perfect positive maps do not allow finite-dimensional realizations. There is a realization of dimension, but it is not generated finitely. This also means that there cannot be a finite Choi-type characterization of a doubly completely positive map. △ Few
2. First-order and second-order optimality conditions for quadratic conic and semidefinite programming under constant rank conditions (arXiv)
Authors: Roberto Andreani, Gabriel Heiser, Leonardo M. Mito, Hector Ramírez C., Thiago P. Silveira
Summary: Well-known constant rank constraints [Math. Program. Study 21:110–126, 1984] The problem introduced by Janin for nonlinear programming has recently been extended to the conical context by exploiting the eigenvector structure of the problem. In this paper, we propose a more general and geometric approach to define a new extension of this condition to the conical context. The main advantage of our approach is that we can recompute the strong quadratic properties of the constant rank condition in a conical context. In particular, even though our condition is independent of Robinson's condition, it is stronger than the classical optimality condition obtained under Robinson's constraint in the sense that it holds for all Lagrangian multipliers. The required quadratic optimality condition is obtained.
