Some Properties of a Class of Continuous Linear Programs

Anderson, Edward; Nash, Peter; Perold, André F.

doi:10.1137/0321046

Cited by 59 publications

(41 citation statements)

References 5 publications

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“…Drews [10], Hartberger [20], and Segers [44] later followed him. Perold [37,38] developed the first simplexlike algorithm for CLP (see also Anderson, Nash, and Perold [1] and Anderson and Philpott [3]. Anstreicher [5] continued Perold's work in his Ph.D. thesis, even though both their algorithms were still incomplete.…”

Section: F Y(t) ≤ H(t) (4) U(t) ≥ 0 T∈ [0 T] Where B(t) C(t) G(mentioning

confidence: 99%

“…Note that the variables u(t) and y(t) are linked only through (1), in which u(t) appears only under the integration operator and y(t) does not appear under the integration operator. The problem (SCLP ) was first introduced by Anderson [4] in order to model job-shop scheduling problems (see also Avram, Bertsimas, and Ricard [6], Weiss [49]).…”

Section: Hu(t) ≤ B(t) Y(t) U(t) ≥ 0 T∈ [0 T]mentioning

confidence: 99%

“…Then the above relationship together with (35) gives (36)). 1 . By Lemma 4, the zero-length intervals can be located only at the breakpoints in DP k 1 .…”

mentioning

confidence: 99%

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A New Algorithm for State-Constrained Separated Continuous Linear Programs

Luo¹,

Bertsimas

1998

SIAM J. Control Optim.

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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.

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Section: F Y(t) ≤ H(t) (4) U(t) ≥ 0 T∈ [0 T] Where B(t) C(t) G(mentioning

confidence: 99%

Section: Hu(t) ≤ B(t) Y(t) U(t) ≥ 0 T∈ [0 T]mentioning

confidence: 99%

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A New Algorithm for State-Constrained Separated Continuous Linear Programs

Luo¹,

Bertsimas

1998

SIAM J. Control Optim.

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“…Some special cases of SCLP were solved by Anderson, Nash and Philpott [5,6,7,8] and Hajek and Ogier [24], and some general results were derived by Anderson, Nash and Perold [4]. This research is summarized in the book of Anderson and Nash [3].…”

Section: Hu(t) ≤ B(t) U(t) ≥ 0 T ∈ [0 T ]mentioning

confidence: 99%

A simplex based algorithm to solve separated continuous linear programs

Weiss

2008

Math. Program.

101

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“…This is defined as follows (we choose to minimize rather than maximize as this is how most optimization problems are now stated), SCLP: minimize cr(t)(t)dt (1) subject to Gx(s) ds + (t) a(t), (2) Hx(t) + z(t) b(t), x(t), y(t), z(t) :> O, t e [0, T]. Here x(t), z(t), b(t) and c(t) are bounded measurable functions and y(t) and a(t) are continuous functions.…”

mentioning

confidence: 99%

An Algorithm for a Class of Continuous Linear Programs

Pullan¹

1993

SIAM J. Control Optim.

101

113

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Abstract. 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.

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Some Properties of a Class of Continuous Linear Programs

Cited by 59 publications

References 5 publications

A New Algorithm for State-Constrained Separated Continuous Linear Programs

A New Algorithm for State-Constrained Separated Continuous Linear Programs

A simplex based algorithm to solve separated continuous linear programs

An Algorithm for a Class of Continuous Linear Programs

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