Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactly and bounds are derived on the number of elementary arithmetic operations necessary, or the cost of all arithmetic operations … Read more
In this paper, we solve a class of two-stage distributionally robust optimization problems which have the property of supermodularity. We exploit the explicit upper bounds on the expectation of supermodular functions and derive the worst-case distribution for the robust counterpart. This enables us to develop an efficient method to derive an exact optimal solution of … Read more
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables (P) when its set of solutions has positive volume. However, when (P) is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two … Read more
Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, such that … Read more
We consider a class of linear-programming based estimators in reconstructing a sparse signal from linear measurements. Specific formulations of the reconstruction problem considered here include Dantzig selector, basis pursuit (for the case in which the measurements contain no errors), and the fused Dantzig selector (for the case in which the underlying signal is piecewise constant). … Read more
Linear energy system models are often a crucial component of system design and operations, as well as energy policy consulting. Such models can lead to large-scale linear programs, which can be intractable even for state-of-the-art commercial solvers—already the available memory on a desktop machine might not be sufficient. Against this backdrop, this article introduces an … Read more
In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware implementations of presolving as part of a parallel interior-point method to solve linear programs with block-diagonal structure, including both linking variables and linking constraints. While presolving … Read more
This paper explores the idea that two-stage worst-case regret minimization problems with either objective or right-hand side uncertainty can be reformulated as two-stage robust optimization problems and can therefore benefit from the solution schemes and theoretical knowledge that have been developed in the last decade for this class of problems. In particular, we identify conditions … Read more
We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a … Read more
Dantzig-Wolfe decomposition (DWD) is a classical algorithm for solving large-scale linear programs whose constraint matrix involves a set of independent blocks coupled with a set of linking rows. The algorithm decomposes such a model into a master problem and a set of independent subproblems that can be solved in a distributed manner. In a typical … Read more