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 c...

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Published in:SIAM journal on optimization Vol. 22; no. 2; pp. 628 - 648
Main Authors: Bampou, Dimitra, Kuhn, Daniel
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects:
Algorithms
Approximation
Linear programming
Optimization
Polynomials
ISSN:1052-6234, 1095-7189
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Abstract 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 potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models. [PUBLICATION ABSTRACT]
AbstractList 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 potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models. [PUBLICATION ABSTRACT]
Author Kuhn, Daniel
Bampou, Dimitra
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Title Polynomial Approximations for Continuous Linear Programs
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