We describe simple and exact duals, and certificates of infeasibility and weak infeasibility in conic linear programming which do not rely on any constraint qualification, and retain most of the simplicity of the Lagrange dual. In particular, some of our infeasibility certificates generalize the row echelon form of a linear system of equations, and the … Read more
We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either … Read more
Recently the alternating direction method of multipliers (ADMM) has been widely used for various applications arising in scientific computing areas. Most of these application models are, or can be easily reformulated as, linearly constrained convex minimization models with separable nonlinear objective functions. In this note we show that ADMM can also be easily used for … Read more
We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification … Read more
Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an $\ell_0$-norm penalty. Exactly enforcing the $\ell_0$-norm penalty is computationally intractable for larger scale problems, so different sparsity-inducing penalty … Read more
We develop a new constraint-reduced infeasible predictor-corrector interior point method for semidefinite programming, and we prove that it has polynomial global convergence and superlinear local convergence. While the new algorithm uses HKM direction in predictor step, it adopts AHO direction in corrector step to obtain faster approach to the central path. In contrast to the … Read more
We consider the linear or quadratic 0/1 program \[P:\quad f^*=\min\{ c^Tx+x^TFx : \:A\,x =\b;\:x\in\{0,1\}^n\},\] for some vectors $c\in R^n$, $b\in Z^m$, some matrix $A\in Z^{m\times n}$ and some real symmetric matrix $F\in R^{n\times n}$. We show that $P$ can be formulated as a MAX-CUT problem whose quadratic form criterion is explicit from the data of … Read more
Constraint reduction is an essential method because the computational cost of the interior point methods can be effectively saved. Park and O’Leary proposed a constraint-reduced predictor-corrector algorithm for semidefinite programming with polynomial global convergence, but they did not show its superlinear convergence. We first develop a constraint-reduced algorithm for semidefinite programming having both polynomial global … Read more
By using an axiomatic approach we formalize the concept of size index for closed convex cones in the Euclidean space $\mathbb{R}^n$. We review a dozen of size indices disseminated through the literature, commenting on the advantages and disadvantages of each choice. CitationTo appear in Journal of Convex Analysis (2015) ArticleDownload View PDF
The dual face algorithm uses Cholesky factorization, as would be not very suitable for sparse computations. The purpose of this paper is to present a dual face algorithm using Gauss-Jordan elimination for solving bounded-variable LP problems. ArticleDownload View PDF