Projective Cutting-Planes for Robust Linear Programming and Cutting Stock Problems

Abstract

We explore the Projective Cutting-Planes algorithm proposed in Porumbel (2020) from new angles by applying it to two new problems, that is, to robust linear programming and to a cutting-stock problem with multiple lengths. Projective Cutting-Planes is a generalization of the widely used Cutting-Planes, and it aims at optimizing a linear function over a polytope [Formula: see text] with prohibitively many constraints. The main new idea is to replace the well-known separation subproblem with the following projection subproblem: given an interior point [Formula: see text] and a direction [Formula: see text], find the maximum steplength t such that [Formula: see text]. This enables one to generate a feasible solution at each iteration, a feature that does not exist built-in in a standard Cutting-Planes algorithm. The practical success of this new algorithm does not mainly come from the higher level ideas already presented in Porumbel (2020) . Its success is significantly more dependent on the computation time needed to solve the projection subproblem in practice. Thus, the main challenge addressed by the current paper is the design of new techniques for solving this subproblem very efficiently for different polytopes [Formula: see text]. We first address a well-known robust linear programming problem in which [Formula: see text] is defined as a primal polytope. We then solve a multiple-length cutting stock problem in which [Formula: see text] is a dual polytope defined in a column generation model. Numerical experiments on both these new problems confirm the potential of the proposed ideas. This enables us to draw conclusions supported by numerical results from both the current paper and Porumbel (2020) while also gaining more insight into the dynamics of the algorithm. Summary of Contribution: The well-known Cutting-Planes algorithm relies on the separation subproblem to cut off the current optimal solution. This paper belongs to a line of work whose goal is to “upgrade” the widely used separation subproblem to the following projection subproblem: given a feasible solution x inside some polytope P and a direction d, what is the maximum t value such that x + td ∈ P. In simple terms, one has to “shoot” from x along direction d up to the point where the boundary of the polytope is hit. The greatest challenge is to solve this projection subproblem rapidly enough to compete well in terms of computational speed with the separation subproblem algorithm. This paper shows how to achieve this on two new problems: robust linear programming and multiple length cutting stock. The numerical results confirm one important advantage offered by the projection logic: it enables one to generate a new feasible interior solution (i.e., x + td) at each iteration. The interior points thus generated actually guide the evolution of the overall algorithm, which is a feature that does not exist (built-in) in a traditional Cutting-Planes algorithm.