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 , 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 and and have piecewise constant densities in . Moreover, we show that under nondegeneracy assumptions all optimal solutions are of this form, and are uniquely determined over .
Key words. Continuous linear programming, symmetric dual, strong duality.
AMS subject classifications. 34H99,49N15,65K99,90C48
1 Introduction
We consider problems of the form:
MCLP  s.t.  
where is a constant matrix, are constant vectors of corresponding dimensions, the integrals are LebesgueStieltjes, are unknown functions over the time horizon , and by convention we take .
We formulate a symmetric dual problem
MCLP  s.t.  
with unknown dual functions with the same convention . It is convenient to think of dual time as running backwards, so that corresponds to .
The main feature to note here is that the objective as well as the left hand side of the constraints are formulated as LebesgueStieltjes integrals with respect to a vector of monotone nondecreasing control function , 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 , 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 , which correspond to impulse controls.
Our main results in this paper include the following:

We discuss how this formulation relates to and generalizes previous continuous linear programs.

We show weak duality and present a simple necessary and sufficient test for feasibility of MCLP. We also present a Slater type condition which is easily checked, using the same test.

We show that under this Slater type condition there is no duality gap between MCLP and MCLP, by considering discrete time approximations. We also show that in this case MCLP and MCLP posses optimal solutions.

We further show that in that case there exist optimal solutions for which and have impulse controls at and are absolutely continuous inside , with piecewise constant densities.

Finally, under appropriate simple nondegeneracy assumptions we show that all optimal solutions are of this form, and that the absolutely continuous part on is uniquely determined.
Further research to develop a simplextype algorithm that constructs solutions of this form is in progress.
We note that the question of existence of strong duality, and whether symmetric dual formulations are useful is far from simple when dealing with linear programs in infinite dimensional spaces [7, 20]. Our results in this paper furnish an example where indeed strong duality can hold with a symmetric dual, if a Slater type condition is satisfied.
2 Background and motivation
Continuous linear programs were introduced by Bellman in 1953 [8, 9] to model economic processes: find a bounded measurable which
BellmanCLP  s.t.  
Where are given matrix functions. These problems were investigated by Dantzig and some of his students, to model continuous time Leontief systems, and by several other early authors [10, 11, 13, 21, 22], with many publications since, but up to date no efficient algorithms or coherent theory have emerged, and these problems are considered very hard.
Separated continuous linear programs (SCLP) were introduced by Anderson [1, 2] in the context of jobshop scheduling:
AndersonSCLP  s.t.  
where are constant matrices, and are given vector functions. Some special cases of SCLP were solved by Anderson and Philpott [4, 5], and this research and related earlier work were summarized in the 1987 book of Anderson and Nash [3], which also contains many references to work on CLP up to that date.
Major progress in the theory of SCLP was achieved by Pullan [6], [14]–[19]. Pullan considered SCLP problems with , and piecewise analytic, and formulated a nonsymmetric dual to (LABEL:eqn.ASCLP) (here we modify Pullan’s original version by letting the dual run in reversed time, as in (LABEL:eqn.mdclp)):
PullanSCLP  s.t.  
Pullan showed that when the feasible region of is bounded strong duality holds between (LABEL:eqn.ASCLP) and (LABEL:eqn.pullandsclp). In the special case that are piecewise linear and piecewise constant Pullan provided an infinite but convergent algorithm to solve the problems and observed that was absolutely continuous, except for atoms at the breakpoints of .
The results of Pullan raised several questions:

Is the boundedness restriction necessary?

Can one formulate a symmetric dual?

Do solutions of the form observed by Pullan always exist?
More recently Weiss [24] considered the following SCLP problem
SCLP  s.t.  
and the symmetric dual
SCLP  s.t.  
with constant vectors and matrices . In contrast to previous work Weiss developed a simplex type algorithm which solves this pair of problems exactly, in a finite bounded number of steps, without using discretization.
The simplex type algorithm of Weiss can solve any pair of problems (LABEL:eqn.PWSCLP), (LABEL:eqn.DWSCLP) which possess optimal solutions that are bounded measurable functions. It produces solutions with piecewise constant, and continuous piecewise linear. However, there exist problems for which both (LABEL:eqn.PWSCLP) and (LABEL:eqn.DWSCLP) are feasible but either (LABEL:eqn.PWSCLP) or (LABEL:eqn.DWSCLP) or both do not possess optimal solutions in the space of bounded measurable functions. Moreover, one can construct examples, where (LABEL:eqn.PWSCLP) possess optimal solutions in the space of bounded measurable functions, but (LABEL:eqn.DWSCLP) is infeasible. Such problems cannot be solved by the algorithm of Weiss. This raises the question whether they can be solved in the space of measures, and motivates our formulation of MCLP, MCLP problems (LABEL:eqn.mpclp), (LABEL:eqn.mdclp).
Definition 2.1
Consider the SCLP problem (LABEL:eqn.PWSCLP). Then the MCLP problem with the following data:
is called the MCLP extension of SCLP.
Theorem 2.2
MCLP/MCLP are generalizations of SCLP/SCLP in the following sense:
(i) if SCLP (LABEL:eqn.PWSCLP) and SCLP (LABEL:eqn.DWSCLP) possess optimal solutions, then these solutions determine optimal solutions of the corresponding MCLP/MCLP extensions with the same objective value.
(ii) If the MCLP/MCLP extensions of the SCLP/SCLP have optimal solutions with no duality gap which are absolutely continuous, then this solution determines optimal solutions of the
SCLP/SCLP, with the same objective value.
(iii) If SCLP is feasible and the Slater type condition LABEL:def.slater holds for MCLP/MCLP extensions, then the supremum of the objective of SCLP is equal to the objective value of the optimal solution of the MCLP extension.
Proof. (i) Consider an optimal solution of (LABEL:eqn.PWSCLP). By the Structure Theorem (Theorem 3 in [24]) is absolutely continuous and hence of bounded variation. Therefore we can write as the difference of two nondecreasing functions . Let be the slacks of the constraints , and let , . Then the resulting satisfies the constraints of the MCLP extension, with the same objective value.
A similar argument applies to an optimal solution of (LABEL:eqn.DWSCLP), which determines a feasible solution of the MCLP extension, which is dual to MCLP, and has the same objective values.
Weak duality of MCLP and MCLP (see Proposition LABEL:thm:weakduality below) then shows that these solutions are the optimal solutions of MCLP and MCLP.
(ii) If the solution of the MCLP extension is absolutely continuous then taking and we get a feasible solution of SCLP, with the same objective value. The same holds for SCLP, and by weak duality these are optimal solutions.
(iii) The proof of this part is postponed to Section LABEL:sec.solform, after Theorem LABEL:thm.solform.
It is not hard to see that (LABEL:eqn.mpclp), (LABEL:eqn.mdclp) generalize also Anderson and Pullan’s problems (LABEL:eqn.ASCLP), (LABEL:eqn.pullandsclp) restricted to affine, and constant.
3 Weak duality, complementary slackness and feasibility
Proposition 3.1
Weak duality holds for MCLP, MCLP (LABEL:eqn.mpclp),(LABEL:eqn.mdclp).
Proof. Let be feasible solutions for (LABEL:eqn.mpclp), (LABEL:eqn.mdclp), and compare their objective values:
The first inequality follows from the primal constraints at , and from nondecreasing. The equality follows by changing order of integration, using Fubini’s theorem. The second inequality follows from the dual constraints at , and from nondecreasing.
Equality of the primal (MCLP) and dual (MCLP) objective will occur if and only if the following holds:
Complementary slackness condition. Let and be the slacks in (LABEL:eqn.mpclp), (LABEL:eqn.mdclp). The complementary slackness condition for MCLP, MCLP is
\hb@xt@.01(\theequation) 
In the following propositions and theorems in this and following sections we present results for MCLP. By symmetry these results hold for MCLP, with the obvious modifications.
We present now a simple necessary and sufficient condition for feasibility. This is similar to a condition derived by Wang, Zhang and Yao [23]. It involves the standard linear program TestLP and its dual TestLP.
TestLP  s.t.  
Theorem 3.2
MCLP is feasible if and only if TestLP is feasible.
Proof. (i) Sufficiency: Let be a solution of TestLP (LABEL:eqn.ptestLP), with slacks , . Then is a feasible solution of MCLP (LABEL:eqn.mpclp), with nonnegative slacks . To check this we have for :
(ii) Necessity: Let be a feasible solution of MCLP (LABEL:eqn.mpclp) with slack . Then with slack is a feasible solution for TestLP (LABEL:eqn.ptestLP), as is seen immediately.
We use the following definition:
Definition 3.3 (Slater type condition)
We say that the TestLP problem (LABEL:eqn.ptestLP) is strictly feasible at if there exists a feasible solution of (LABEL:eqn.ptestLP) and a constant such that and . We say that MCLP is strictly feasible at if there exists a feasible solution of (LABEL:eqn.mpclp) and a constant such that for all .
Corollary 3.4
MCLP is strictly feasible if and only if TestLP is strictly feasible.
Proof. Simply define and recall Theorem LABEL:the.feas for problems with replaced by .
4 Discrete time approximations and strong duality
In this section we consider a pair of MCLP/MCLP problems which are feasible, and use time discretization to solve them approximately. We prove that if MCLP and MCLP are strictly feasible, then there is no duality gap and an optimal solution exists. We use a discretization approach similar to [14].
4.1 General discretizations
For a partition we define the discretization of MCLP to be:
s.t.  
dCLP  
and for the same time partition the discretization of MCLP is defined as:
\hb@xt@.01(\theequation)  
s.t.  
dCLP  
Note that these two problems are not dual to each other.
Following Pullan [14], for a partition and values we define the piecewise linear extension:
and the piecewise constant extension:
The following proposition is an easy extension of Theorem LABEL:the.feas.
Proposition 4.1
All discretizations dCLP (LABEL:eqn.dCLP1) are feasible if and only if MCLP is feasible.
Proof. (i) Let be a feasible solution of MCLP (LABEL:eqn.mpclp) with slacks . Then (in these integrals we take and for ), , and is a feasible solution for dCLP (LABEL:eqn.dCLP1). To check this we have for :
(ii) Let be a feasible solution of dCLP. Define , let be the piecewise constant extension of , and let , . Then is a feasible solution of MCLP.
Proposition 4.2
Any feasible solution of dCLP can be extended to a feasible solution of MCLP with equal objective value.
Proof. We set to be the piecewise constant extension of and take to be the measure with density on and impulses . We also set to be the piecewise linear extension of , and take to be the same for both problems. It is immediate to see that this gives a feasible solution to MCLP. Furthermore, it is immediate to see that the objective of dCLP equals the objective of MCLP for this extended solution.
Proposition 4.3
The optimal values of the various problems satisfy:
Proof. The first and last inequalities follow from Proposition LABEL:thm.feasext and the middle inequality follows from weak duality.
4.2 Discretizations with equidistant partitions
Similar to Wang, Zhang and Yao [23] and to Pullan [14] we use even equidistant partitions, denoted which divides the interval into equal segments, each of length , i.e. . With this partition we introduce the notations:

Given a matrix we define the matrix , the matrix , and the matrix as follows:

We define the fold vectors, each with vector components:
Using this notation we rewrite problems (LABEL:eqn.dCLP1), (LABEL:eqn.dCLP2) for even equidistant partitions, as:
s.t.  
dCLP  
s.t.  
dCLP  
The reader may notice that in (LABEL:eqn.dCLPpi2) we have for convenience reversed the order of variables and the order of the constraints in the middle part of the problem relative to (LABEL:eqn.dCLP2)
To quantify the discretization error for time partition we define a modified pair of problems mdCLP, mdCLP:
mdCLP  
s.t.  Constraints of (LABEL:eqn.dCLPpi1) 
mdCLP  
s.t.  Constraints of (LABEL:eqn.dCLPpi2) 
We note that they are dual to each other. They are both feasible if (LABEL:eqn.dCLPpi1), (LABEL:eqn.dCLPpi2) are feasible. Moreover, since (LABEL:eqn.dCLPpi1), (LABEL:eqn.dCLPpi2) are (LABEL:eqn.dCLP1), (LABEL:eqn.dCLP2) rewritten, then problems mdCLP and mdCLP are feasible if and only if MCLP, MCLP are feasible, by Proposition LABEL:thm.discrete. In this case mdCLP and mdCLP also posses optimal solutions. Denote by and an optimal solution of mdCLP and mdCLP.
Proposition 4.4
If MCLP and MCLP are feasible then by solving mdCLP and mdCLP the following bounds holds:
where
Proof. The first inequality follows from Proposition LABEL:thm.dclpopt. To evaluate the second inequality we note that the optimal solutions of mdCLP and mdCLP are feasible but suboptimal solutions of dCLP and dCLP. Calculating the objective values of dCLP, dCLP for the solutions , we have:
\hb@xt@.01(\theequation) 
On the other hand, because mdCLP and mdCLP are dual problems:
\hb@xt@.01(\theequation) 
Combining (LABEL:eqn.dopt1) and (LABEL:eqn.dopt2), after easy manipulations we get:
Proposition 4.5
The sequence of optimal values of the dual problems mdCLP and mdCLP has finite lower and upper bounds, .
Proof. We consider the single interval partition , where we have the problem:
s.t.  
An optimal solution to (LABEL:eqn.pLP1) can be extended to a feasible solution of mdCLP as follows: . Hence the following inequality holds:
where , and we recall that .
Similarly, by considering the dual, an upper bound is obtained in terms of the solution of the dual test problem:
4.3 Bounding the discrete solutions
In this section we assume that MCLP as well as MCLP satisfy the Slater type condition LABEL:def.slater. Under this assumption we will show that all the optimal solutions of mdCLP and mdCLP are uniformly bounded.
We consider first the sequence of primal problems . We use the following notations:
Proposition 4.6
If MCLP is strictly feasible then all elements of have a uniform finite upper bound.
Proof. Take any , we will show that is bounded by a constant for all . Recall that are non decreasing, so this bound will hold for all .
We choose large enough and corresponding small enough so that:
where is a small constant, to be determined later. We will find a uniform bound for .
We use the following notation: