In this work, we propose a predictor-corrector interior point method for linear programming in a primal-dual context, where the next iterate is chosen by the minimization of a polynomial merit function of three variables: the first is the steplength, the second defines the central path and the third models the weight of a corrector direction. … Read more
Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are (i) Does linear superiorization provide a feasible point whose linear … Read more
Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same … Read more
We consider a linear problem over a finite set of integer vectors and assume that there is a verification oracle, which is an algorithm being able to verify whether a given vector optimizes a given linear function over the feasible set. Given an initial solution, the algorithm proposed in this paper finds an optimal solution … Read more
A class of interior point methods using inexact directions is analysed. The linear system arising in interior point methods for linear programming is reformulated such that the solution is less sensitive to perturbations in the right-hand side. For the new system an implementable condition is formulated that controls the relative error in the solution. Based … Read more
Standard load profiles (SLPs) are used to calculate the natural gas demand of non-daily metered customers based on temperature forecasts. The most recent version of natural gas SLPs in Germany was published by the Federal Association of Energy and Water in June 2015. With the concept SigLinDE, a linearization of the old SLPs was carried … Read more
We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The employed Krylov subspace methods do not suffer from rank-deficiency and therefore no … Read more
In this paper we establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged. We also provide a tight bound on the achievable convergence rate. Most existing results establishing linear convergence in such methods require restrictive assumptions regarding … Read more
One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. … Read more
In a recent work, Muter et al. (2013a) identified and characterized a general class of linear programming (LP) problems – known as problems with column-dependent-rows (CDR-problems). These LPs feature two sets of constraints with mutually exclusive groups of variables in addition to a set of structural linking constraints, in which variables from both groups appear … Read more