We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is … Read more
An attractive feature of the face method \cite{pan14} for solving LP problems is that it uses the orthogonal projection of the negative objective gradient on the related null space as the search direction. However, the method would not be amenable for solving large sparse problems, since it handles the involved normal system by orthogonal transformations. … Read more
The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows … Read more
We consider the vehicle routing problem with resource constraints motivated by a home health care application. We propose a branch and price algorithm to solve the problem. In our problem, we consider different types of patients that require a nurse or a health aid or both. The patients can be serviced by the appropriate vehicles … Read more
The implementation of a linear programming interior point solver is described that is based on iterative linear algebra. The linear systems are preconditioned by a basis matrix, which is updated from one interior point iteration to the next to bound the entries in a certain tableau matrix. The update scheme is based on simplex-type pivot … Read more
Given a point x inside a polytope P and a direction d, the projection of x along d asks to find the maximum step length t such that x+td is feasible; we say x+td is a pierce point because it belongs to the boundary of P. We address this projection sub-problem with arbitrary interior points … Read more
Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years for its potential to quantify more effectively the risk of large losses than conditional value at risk. In this paper we consider the case that the true loss function is unavailable either because it is difficult to be identified or the … Read more
This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is … Read more
We introduce a new variant of Chubanov’s method for solving linear homogeneous systems with positive variables. In the \BP\ we use a recently introduced cut in combination with Nemirovski’s Mirror-Prox method. We show that the cut requires at most $O(n^3)$ time, just as Chabonov’s cut. In an earlier paper it was shown that the new … Read more
This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems find $x\in L\cap\mathbb{R}^n_{+} $ and find $x\in L^\perp\cap\mathbb{R}^n_{+},$ where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed … Read more