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

Abstract

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 ). Key words. Continuous linear programming, symmetric dual, strong duality. AMS subject classifications. 34H99,49N15,65K99,90C48 P(t) ≥ 0, P(t) non-decreasing and right continuous on [0,T].with K unknown dual functions P with the same convention P(0−) = 0. It is convenient to think of dual time as running backwards, so that P(T − t) corresponds to U(t).The main feature to note here is that the objective as well as the left hand side of the constraints are formulated as Lebesgue-Stieltjes integrals with respect to a vector of monotone non-decreasing control function U(t), in other words our controls are in the space of measures. This is in contrast to most formulations in which the objective and left hand side of the constraints are Lebesgue integrals with respect to a measurable bounded control u(t), in other words controls which are in the space of densities. In particular, while in the usual formulation the left hand side of the constraints is an absolutely continuous function, our formulation allows the left hand side of the constraint to have jumps, as a result of jumps in U(t), which correspond to impulse controls.Our main results in this paper include the following: *

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

“…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].…”

“…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.…”

“…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)).…”

“…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].…”

“…We state that this can be done, but it will be more appropriate to deal with the details when one creates an implementation of the algorithm. For further discussion of the requirement of perturbations see [29] A.2 Discussion of λ and µ…”