Separated Continuous Linear Programs Research Articles

Polynomial Approximations for Continuous Linear Programs

Continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (nonseparated) problem instances. In this paper we propose a more generic approximation scheme for nonseparated continuous linear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuous linear program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Slater-type constraint qualifications. We also highlight the...

A simplex based algorithm to solve separated continuous linear programs

We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions.

Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions

We give an approximation scheme for separated continuous linear programming problems. Such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems. In a network with linear flow costs and linear, per-unit-time holding costs, our algorithm finds a drainage of the network that, for given constants ε > 0 and δ > 0, has total cost (1 + ε)OPT + δ, where OPT is the cost of the minimum cost drainage. The complexity of our algorithm is polynomial in the size of the input network, 1/ε, and log(1/δ). The fluid relaxation is a continuous problem. While the problem is known to have a piecewise constant solution, it is not known to have a polynomially sized solution. We introduce a natural discretization of polynomial size and prove that this discretization produces a solution with low cost. This is the first polynomial time algorithm with a provable approximation guarantee for fluid relaxations.

An extended algorithm for separated continuous linear programs

Separated continuous linear programs (SCLP) are a class of continuous linear programs which, among other things, can serve as a useful model for dynamic network problems where storage is permitted at the nodes. Recent work on SCLP has produced a detailed duality theory, conditions under which an optimal solution exists with a finite number of breakpoints, a purification algorithm, as well as a convergent algorithm for solving SCLP under certain assumptions on the problem data. This paper combines much of this work to develop a possible approach for solving a wider range of SCLP problems, namely those with fairly general costs. The techniques required to implement the algorithm are no more than standard (finite-dimensional) linear programming and line searching, and the resulting algorithm is simplex-like in nature. We conclude the paper with the numerical results obtained by using a simple implementation of the algorithm to solve a small problem.

Convergence of a General Class of Algorithms for Separated Continuous Linear Programs

Separated continuous linear programs (SCLP) are a type of infinite-dimensional linear program which can serve as a useful model for a variety of dynamic network problems where storage is permitted at the nodes. This paper proves the convergence of a general class of algorithms for solving SCLP under certain restrictions on the problem data. This is the first such proof for any nondiscretization algorithm for solving any form of continuous linear programs.

A New Algorithm for State-Constrained Separated Continuous Linear Programs

During the last few decades, significant progress has been made in solving large-scale 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.

Existence and duality theory for separated continuous linear programs

This paper surveys the recent developments in the theoretical study of separated continuous linear programs (SCLP). This problem serves as a useful model for various dynamic network problems where storage is permitted at the nodes. We demonstrate this by modelling some hypothetical problems of water distribution, transportation and telecommunications. The theoretical developments we present for SCLP fall into two main topics. The first of these is the existence of optimal solutions of various forms. These results culminate in one guaranteeing the existence of a piecewise analytic optimal solution, that is, having a finite number of breakpoints. The second topic we discuss is duality. Under this heading we develop a theory that closely resembles that for finite-dimensional linear programming. For instance, we define complementary slackness and give conditions under which there exist complementary slack primal and dual optimal solutions. Throughout the paper we observe that the main theorems are sufficiently general to include any reasonable practical problems

A Duality Theory for Separated Continuous Linear Programs

This paper presents a detailed duality theory for a class of continuous linear programs called separated continuous linear programs (SCLP), based on a particular dual problem SCLP*. Using weak duality, a notion of complementary slackness is introduced, and several sufficient conditions for optimality of SCLP are derived along with the existence of complementary slack variables for basic feasible solutions for SCLP. Following this, a fairly general condition for the absence of a duality gap between SCLP and SCLP* is given, as are several conditions for the existence of an optimal solution for SCLP*. Finally, using all these ingredients, a strong duality result between SCLP and SCLP* is proven when the problem data are piecewise analytic. A simple counterexample is presented to show that strong duality may not follow if the assumptions of piecewise analyticity do not hold.

Purification for separated continuous linear programs

This paper presents a purification algorithm for a class of infinite-dimensional linear programs called separated continuous linear programs (SCLP). This takes an initial feasible solution and produces an extreme point solution without a decrease in objective function value. The algorithm presented here for SCLP is also shown to be the best possible purification algorithm in a certain class.

Forms of Optimal Solutions for Separated Continuous Linear Programs

This paper discusses the nature of optimal solutions for a class of continuous linear programs called separated continuous linear Programs. It is shown that under various different assumptions on the problem data there exist optimal solutions that are piecewise constant, piecewise polynomial, or, more generally, piecewise analytic. These results are reminiscent of bang-bang results in linear optimal control.

An adaptive discretization algorithm for a class of continuous network programs

AbstractAn algorithm is derived for a class of continuous‐time minimum‐cost network flow problems. The algorithm is based on some recent results in the theory of separated continuous linear programs and works with a sequence of discrete approximations to the continuous‐time problem. We prove the convergence of the algorithm and present some numerical results which compare its performance against that of linear network flow algorithms applied to a uniform discrete approximation of the continuous‐time problem.

On the Solutions of a Class of Continuous Linear Programs

This paper discusses the form of solutions for a class of continuous linear programs called separated continuous linear programs. It is shown that under certain assumptions on the problem data the optimal solutions can be taken to be piecewise analytic functions. This yields a strong duality result as a corollary.

An Algorithm for a Class of Continuous Linear Programs

This paper discusses a class of continuous linear programs posed in a function space called separated continuous linear programs (SCLP). A dual linear program and a corresponding discrete approximation are introduced followed by a discussion of their properties. The discrete approximation gives rise to an improvement step which is constructed from any given feasible (nonoptimal) solution for SCLP. A strong duality result follows from this. There are a variety of possible implementations of an algorithm for solving SCLP problems using this improvement step. Finally some computational results are given from one possible implementation.