The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation … Read more
The dimer method is a Hessian-free algorithm for computing saddle points. We augment the method with a linesearch mechanism for automatic step size selection as well as preconditioning capabilities. We prove local linear convergence. A series of numerical tests demonstrate significant performance gains. Citationhttp://arxiv.org/abs/1407.2817ArticleDownload View PDF
We consider an extension of a noncooperative game where players have joint binding constraints. In this model, the constrained equilibrium can not be implemented within the same noncooperative framework and requires some other additional regulation procedures. We consider several approaches to resolution of this problem. In particular, a share allocation method is presented and substantiated. … Read more
This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semi-definite … Read more
It is often the case that the computed optimal solution of an optimization problem cannot be implemented directly, irrespective of data accuracy, due to either (i) technological limitations (such as physical tolerances of machines or processes), (ii) the deliberate simplification of a model to keep it tractable (by ignoring certain types of constraints that pose … Read more
The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this … Read more
This paper follows the recent discussion on the sparse solution recovery with quasi-norms Lq; q\in(0,1) when the sensing matrix possesses a Restricted Isometry Constant \delta_{2k} (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if \delta_{2k}\le 1/2, … Read more
The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$ denote the $k$–th iterate of CG. In this paper we obtain an expression for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use … Read more
Operations Research (OR) is the discipline of applying advanced analytical methods to help make better decisions (Horner (2003)). OR is characterized by its broad applicability and its interdisciplinary nature. Currently, in addition to mathematics, many other undergraduate programs such as management sciences, business, economics, electrical engineering, civil engineering, chemical engineering, and related fields, have incorporated … Read more
Numerical experiments have indicated that the reweighted $\ell_1$-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. Thus it is important to carry out a further investigation of this class of methods. In this paper, we point out that reweighted $\ell_1$-methods are intrinsically associated with the minimization of the so-called merit … Read more