Class Of Continuous Linear Programs Research Articles

Symmetric Strong Duality for a Class of Continuous Linear Programs with Constant Coefficients

We consider Continuous Linear Programs over a continuous finite time horizon $T$, with linear cost coefficient functions and linear right hand side functions and a constant coefficient matrix, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the Separated Continuous Linear Programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. We present simple necessary and sufficient conditions for feasibility. We formulate a symmetric dual and investigate strong duality by considering discrete time approximations. We prove that under a Slater type condition there is no duality gap and there exist optimal solutions which have impulse controls at $0$ and $T$ and have piecewise constant densities in $(0,T)$. Moreover, we show that under non-degeneracy assumptions all optimal solutions are of this form, and are uniquely determined over $(0,T)$.

Solution Algorithms for a Class of Continuous Linear Programs with Fuzzy Valued Objective Functions

This paper discusses a class of continuous linear programs with fuzzy valued objective functions. A member of this class is called a fuzzy separated continuous linear program (FSCLP). Such problems have applications in a number of domains, including, production and inventory systems, communication networks, and pipeline systems for transportation. The discretization approach is used to construct two ordinary fuzzy linear programming problems, which give a lower and an upper bound on the optimal value of FSCLP. It is then shown how to construct an improved feasible solution for FSCLP starting from a nonoptimal one. This leads to the development of a class of algorithms based on a sequence of discrete approximations to FSCLP. Numerical examples in the context of continuous-time networks are presented to show the applicability of the proposed method.

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.

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.

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.

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.

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.

Some Properties of a Class of Continuous Linear Programs

This paper discusses a class of linear programs posed in a function space; a member of this class is called a separated continuous linear program (SCLP). Such problems occur, for example, in the planning of production and inventory. We characterize the $L_\infty $ extreme point solutions of SCLP in a manner analogous to the basic solutions of finite dimensional linear programming and give a sufficient condition for there to exist optimal extreme point solutions with finitely many constant-basis intervals. SCLP is to date the most general continuous linear program for which such strong characterizations have been found.