A New Algorithm for State-Constrained Separated Continuous Linear Programs

Abstract

Abstract. During the last few decades, significant progress has been made in solving largescale finite-dimensional and semi-infinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinite-dimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving large-scale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving large-scale realistic problems (with several hundred continuous variables) arising in manufacturing systems.

The Continuous Assignment Problem and Its Application to Preemptive and Non-Preemptive Scheduling with Irregular Cost Functions

“…More interestingly, it also belongs to the class of the separated continuous linear programs that can be used to model a variety of optimal control problems but also some scheduling problems [6,7] and that was extensively studied. In particular, Pullan [18] presented a duality theory for these problems and practical algorithms exist to solve them [14,17].…”

A simplex based algorithm to solve separated continuous linear programs

“…Luo and Bertsimas [33] have used quadratic programming techniques in conjunction with discretization, and implemented these to a more general problem, namely state constrained SCLP's. An interior point method for solving CLP is suggested by Ito, Kelley and Sachs [28].…”

From fluid relaxations to practical algorithms for job shop scheduling: the makespan objective

“…However, based on several structural properties for this class of problems (see Anderson and Nash 1987), Luo and Bertsimas (1999), based on earlier work by Pullan (1993), propose provably convergent discretization-based methods that are able to quickly solve large-scale instances in practice. The algorithm of Luo and Bertsimas (1999) is used in our computational study.…”

“…Solve the fluid-control problem, and obtain the optimal fluid relaxation. This can be accomplished by applying the algorithm of Luo and Bertsimas (1999). The optimal fluid relaxation has R pieces, and breakpoints 0 T 1 T R and the corresponding lengths of the pieces are L r = T r+1 − T r .…”

“…However, the description of the optimal fluid solution, while insightful for the original problem, involves the enumeration of an exponential number of cases. Luo and Bertsimas (1999), building upon the work of Pullan (1993), use the theory of continuous linear programming to propose a convergent numerical algorithm for the problem that is able to efficiently solve problems involving hundreds of machines and job types; we use this algorithm in solving the fluid relaxations throughout this work.…”