Structure of Solutions for Continuous Linear Programs with Constant Coefficients

Shindin, Evgeny; Weiss, Gideon

doi:10.1137/140971725

Cited by 5 publications

(17 citation statements)

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“…We refer to the coefficients of the matrix A as the structure parameters, to b, c as the rate parameters, and to β, γ, T as the boundary parameters of the M-CLP problem. This paper continues research described in [36,30,31]. The algorithm for the solution of M-CLP builds on and extends the algorithm [36] for solution of SCLP, and is somewhat similar in principle.…”

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confidence: 77%

“…Note: Theorem 4.5 explains why we choose the formulations (26), (27) of the Boundary-LP/LP * problems. This one to one correspondence of optimal solutions that is shown in the Theorem does not exist for the more natural pair of boundary problems (S13), (S14) in [31].…”

Section: Boundary Problemsmentioning

confidence: 91%

“…The optimal solution of M-CLP in the open interval t ∈ (0, T ) is likewise given by continuous piecewise linear U j (t) with piecewise constant derivatives u j (t) and continuous piecewise linear (here we define by convention x(0−) = 0, q(0−) = 0). These results were derived in [30,31].…”

Section: Overview and Contributionmentioning

confidence: 91%

“…Notes on presentation: Throughout the paper we quote results from [30,31]. We will refer to results from the duality paper [30] by appending D and from the structure paper [31] by appending S (e.g. D5.3, or (S4.2)).…”

Section: Outline or The Rest Of The Papermentioning

confidence: 99%

“…The set of equations (19) are formulated in such a way that they force 2K + 2J boundary variables, N − 1 variables that are left-hand sides of (13) and N (K + J) − N n=1 (|K n | + |J n |) of the x n k ∈ {n, k :ẋ n k = 0}, q n j ∈ {n, j :q n+1 j = 0} that correspond to non-basic variables in the 1 There are some typos in the definition of M in [31]. They were corrected in [29].…”

Section: The Vector Of Unknowns (Writing All Vectors As Row Vectors) Ismentioning

confidence: 99%

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A simplex-type algorithm for continuous linear programs with constant coefficients

Shindin

Weiss

2018

Math. Program.

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We consider continuous linear programs over a continuous finite time horizon T , with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions. Specifically, 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. In previous papers we formulated a symmetric dual and have shown strong duality. We also have presented a detailed description of optimal solutions and have defined a combinatorial analogue to basic solutions of standard LP. In this paper we present an algorithm which solves this class of problems in a finite bounded number of steps, using an analogue of the simplex method, in the space of measures.Here A is a K × J constant matrix, β, b, γ, c are constant vectors of corresponding dimensions, the integrals are Lebesgue-Stieltjes, and include the jumps at 0 in U, P . The unknowns are vectors of cumulative control functions U, P and vectors of non-negative slack or state functionsx, q, over the time horizon [0, T ]. It is convenient to think of dual time as running backwards, so that P (T − t) is the vector of dual variables that correspond to the constraints of (1) at time t, and U (t) correspond to the constraints of (2) at time T − t. The special feature here is that the controls are measures, or equivalently functions of bounded variation. We refer to the coefficients of the matrix A as the structure parameters, to b, c as the rate parameters, and to β, γ, T as the boundary parameters of the M-CLP problem. This paper continues research described in [36,30,31]. The algorithm for the solution of M-CLP builds on and extends the algorithm [36] for solution of SCLP, and is somewhat similar in principle. While some SCLP problems cannot be solved with the algorithm of [36], every SCLP problem can be solved as a special case of an M-CLP problem, by the algorithm presented in this paper. The problem (1) is solved parametrically. It starts from an artificial set of boundary parameters T 0 , β 0 , γ 0 that have a simple optimal solution, and moves in a finite bounded number of steps along a parametric straight line L(θ), 0 ≤ θ ≤ 1, to the boundary parameters β, γ, T of the original problem, where in each step an M-CLP pivot is performed.At this stage it is too early to estimate the efficiency of this algorithm. Like standard LP simplex its worst case performance is exponential. It is however worth mentioning here, that we are currently in the process of re-programming the SCLP algorithm, beyond the pilot inefficient MATLAB implementation that was reported in [36]. While we have not yet stabilized the improved code, we made some improvements to the MATLAB current code. In comparing our current implementation of the simplex based algorithm [36], against an optimized uniform discretization solved by CPLEX we have found that the simplex based algorithm can outperform CPLEX...

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confidence: 77%

Section: Boundary Problemsmentioning

confidence: 91%

Section: Overview and Contributionmentioning

confidence: 91%