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...

[1]  R Bellman,et al.  Bottleneck Problems and Dynamic Programming. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[2]  W. Tyndall.  A DUALITY THEOREM FOR A CLASS OF CONTINUOUS LINEAR PROGRAMMING PROBLEMS , 1965 .

[3]  R. Grinold.  Continuous programming part two: Nonlinear objectives , 1969 .

[4]  Jaromir Abrham,et al.  Numerical solutions to continuous linear programming problems , 1973, Z. Oper. Research.

[5]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[6]  E. Anderson.  A new continuous model for job-shop scheduling , 1981 .

[7]  André F. Perold.  Extreme Points and Basic Feasible Solutions in Continuous Time Linear Programming , 1981 .

[8]  E. Anderson,et al.  Some Properties of a Class of Continuous Linear Programs , 1983 .

[9]  Stanley B. Gershwin,et al.  Scheduling manufacturing systems with work-in-process inventory , 1990, 29th IEEE Conference on Decision and Control.

[10]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[11]  M. Pullan.  An algorithm for a class of continuous linear programs , 1993 .

[12]  Andy Philpott,et al.  An adaptive discretization algorithm for a class of continuous network programs , 1995, Networks.

[13]  D. Bertsimas,et al.  A New Algorithm for State-Constrained Separated Continuous Linear Programs , 1999 .

[14]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[15]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[16]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[17]  Lisa Fleischer,et al.  Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions , 2005, Math. Oper. Res..

[18]  Gideon Weiss,et al.  A simplex based algorithm to solve separated continuous linear programs , 2008, Math. Program..

[19]  Evgenia Smirni,et al.  Burstiness in Multi-tier Applications: Symptoms, Causes, and New Models , 2008, Middleware.

[20]  Yoni Nazarathy,et al.  Near optimal control of queueing networks over a finite time horizon , 2009, Ann. Oper. Res..

[21]  Evgenia Smirni,et al.  Injecting realistic burstiness to a traditional client-server benchmark , 2009, ICAC '09.

[22]  V. Kadirkamanathan,et al.  International Journal of Systems Science , 2014 .

[23]  Anton van den Hengel,et al.  Semidefinite Programming , 2014, Computer Vision, A Reference Guide.