Learning To Scale Mixed-Integer Programs

Many practical applications require the solution of numerically challenging linear programs (LPs) and mixed-integer programs (MIPs). Scaling is a widely used preconditioning technique that aims at reducing the error propagation of the involved linear systems, thereby improving the numerical behavior of the dual simplex algorithm and, consequently, LP-based branch-and-bound. A reliable scaling method often makes … Read more

Affine Decision Rule Approximation to Immunize against Demand Response Uncertainty in Smart Grids’ Capacity Planning

Generation expansion planning (GEP) is a classical problem that determines an optimal investment plan for existing and future electricity generation technologies. GEP is a computationally challenging problem, as it typically corresponds to a very large-scale problem that contains several sources of uncertainties. With the advent of demand response (DR) as a reserved capacity in modern … Read more

Simple Iterative Methods for Linear Optimization over Convex Sets

We give simple iterative methods for computing approximately optimal primal and dual solutions for the problem of maximizing a linear functional over a convex set $K$ given by a separation oracle. In contrast to prior work, our algorithms directly output primal and dual solutions and avoid a common requirement of binary search on the objective … Read more

A New Face Algorithm Using LU Factorization for Linear Programming

The unique feature of the face algorithm \cite{pan14} is that it moves from face to face, rather than from vertex to vertex as the simplex algorithm. It uses the orthogonal projection of the negative objective gradient on the related null space as its search direction. Nevertheless, the algorithm is based on QR factorization, which would … Read more

Bound Propagation for Linear Inequalities Revisited

In 2011, Korovin and Voronkov (Proceedings of the 23rd International Conference on Automated Deduction, vol. 6803 of Lecture Notes in Computer Science, pp. 369-383) proposed a method based on bound propagation for solving systems of linear inequalities. In this paper, an alternate description of their algorithm which also incorporates an addition that returns a certificate … Read more

Column-Randomized Linear Programs: Performance Guarantees and Applications

We propose a randomized method for solving linear programs with a large number of columns but a relatively small number of constraints. Since enumerating all the columns is usually unrealistic, such linear programs are commonly solved by column generation, which is often still computationally challenging due to the intractability of the subproblem in many applications. … Read more

The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems

We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures … Read more

Projection and rescaling algorithm for finding most interior solutions to polyhedral conic systems

We propose a simple projection and rescaling algorithm that finds {\em most interior} solutions to the pair of feasibility problems \[ \text{find} x\in L\cap \R^n_{+} \text{ and } \text{find} \; \hat x\in L^\perp\cap\R^n_{+}, \] where $L$ is a linear subspace of $\R^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that … Read more

A New Preconditioning Approach for an Interior Point-Proximal Method of Multipliers for Linear and Convex Quadratic Programming

In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a … Read more

Short simplex paths in lattice polytopes

We consider the problem of optimizing a linear function over a lattice polytope P contained in [0,k]^n and defined via m linear inequalities. We design a simplex algorithm that, given an initial vertex, reaches an optimal vertex by tracing a path along the edges of P of length at most O(n^6 k log k). The … Read more