On the Rank of Cutting-Plane Proof Systems

We introduce a natural abstraction of propositional proof systems that are based on cut- ting planes. This leads to a new class of proof systems that includes many well-known meth- ods, such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts, or split cuts. The rank of a proof system corresponds to the number of rounds that … Read more

Principle of optimal accuracy classification for metrological purposes

The conceptions of optimizing a quantitative classification hierarchy and the criterion of informational optimality have been used for modeling universal accuracy classification scale. Proposed model is based on test uncertainty ratios determined in conformity with optimal levels of confidence in evaluating measurement uncertainty, as well as on the optimal classification structure. The author believes that … Read more

A class of derivative-free nonmonotone optimization algorithms employing coordinate rotations and gradient approximations

In this paper we study a class of derivative-free unconstrained minimization algorithms employing nonmonotone inexact linesearch techniques along a set of suitable search directions. In particular, we define globally convergent nonmonotone versions of some well-known derivative-free methods and we propose a new algorithm combining coordinate rotations with approximate simplex gradients. Through extensive numerical experimentation, we … Read more

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization

Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, R\’e and Tropp (`Factoring nonnegative matrices with linear programs’, NIPS 2012) proposed a linear programming … Read more

A New Class of Valid Inequalities for Nonlinear Network Design Problems

We consider a nonlinear nonconvex network design problem that arises in the extension of natural gas transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow … Read more

Analytical formulas for calculating the extremal ranks and inertias of + BXB^{*}$ when $ is a fixed-rank Hermitian matrix

The rank of a matrix and the inertia of a square matrix are two of the most generic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix. Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in … Read more

Lot Sizing with Piecewise Concave Production Costs

We study the lot-sizing problem with piecewise concave production costs and concave holding costs. This problem is a generalization of the lot-sizing problem with quantity discounts, minimum order quantities, capacities, overloading, subcontracting or a combination of these. We develop a dynamic programming (DP) algorithm to solve this problem and answer an open question in the … Read more

Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property

This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a … Read more

Novel update techniques for the revised simplex method

This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming (LP) problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple … Read more

A big bucket time indexed formulation for nonpreemptive single machine scheduling problems

A big bucket time indexed mixed integer linear programming formulation for nonpreemptive single machine scheduling problems is presented in which the length of each period can be as large as the processing time of the shortest job. The model generalises the classical time indexed model to one in which at most two jobs can be … Read more