Perspective Envelopes for Bilinear Functions
Given the bilinear function f(x,y) = xy, and the constraint x
Linear, Cone and Semidefinite Programming
A Framework for Applying Subgradient Methods to Conic Optimization Problems (version 2)
A framework is presented whereby a general convex conic optimization problem is transformed into an equivalent convex optimization problem whose only constraints are linear equations and whose objective function is Lipschitz continuous. Virtually any subgradient method can be applied to solve the equivalent problem. Two methods are analyzed. (In version 2, the development of algorithms … Read more
Parallelizing the dual revised simplex method
This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational … Read more
Extension and Implementation of Homogeneous Self-dual Methods for Symmetric Cones under Uncertainty
Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space has been proposed by Jin et al. in [12]. Alzalg [8], has adopted their work to derive homogeneous self-dual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these two results to derive homogeneous self-dual algorithms for stochastic programs … Read more
On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
We investigate structural properties of the completely positive semidefinite cone, consisting of all the nxn symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set Q of … Read more
Vector Space Decomposition for Linear Programs
This paper describes a vector space decomposition algorithmic framework for linear programming guided by dual feasibility considerations. The resolution process moves from one basic solution to the next according to an exchange mechanism which is defined by a direction and a post-evaluated step size. The core component of this direction is obtained via the smallest … Read more
Six mathematical gems from the history of Distance Geometry
This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron’s formula, Cauchy’s theorem on the rigidity of polyhedra, Cayley’s generalization of Heron’s formula to higher dimensions, Menger’s characterization of abstract semi-metric spaces, a … Read more
A corrected semi-proximal ADMM for multi-block convex optimization and its application to DNN-SDPs
In this paper we propose a corrected semi-proximal ADMM (alternating direction method of multipliers) for the general $p$-block $(p\!\ge 3)$ convex optimization problems with linear constraints, aiming to resolve the dilemma that almost all the existing modified versions of the directly extended ADMM, although with convergent guarantee, often perform substantially worse than the directly extended … Read more
Exact solutions to Super Resolution on semi-algebraic domains in higher dimensions
We investigate the multi-dimensional Super Resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the l1-minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual … Read more
Submodular Minimization in the Context of Modern LP and MILP Methods and Solvers
We consider the application of mixed-integer linear programming (MILP) solvers to the minimization of submodular functions. We evaluate common large-scale linear-programming (LP) techniques (e.g., column generation, row generation, dual stabilization) for solving a LP reformulation of the submodular minimization (SM) problem. We present heuristics based on the LP framework and a MILP solver. We evaluated … Read more