Structure of solutions for continuous linear programs with constant coefficients
Abstract
We consider Continuous Linear Programs over a continuous finite time horizon , with linear cost coefficient functions, linear right hand side functions, and a constant coefficient matrix, as well as their symmetric dual. 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 a recent paper we have shown that under a Slater type condition, these problems possess optimal strongly dual solutions. In this paper we give a detailed description of optimal solutions and define a combinatorial analog to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.
ontinuous linear programming, symmetric dual, strong duality, structure of solutions, optimization in the space of measures, optimal sequence of bases
34H99,49N15,65K99,90C48
1 Introduction
This paper continues our research in a recent paper [13] on continuous linear programs of the form
MCLP  s.t.  
and their symmetric dual,
MCLP  s.t.  
Here is a constant matrix, are constant vectors of corresponding dimensions and the integrals are LebesgueStieltjes. The unknowns are vectors of cumulative control functions and , over the time horizon . By convention we take . It is convenient to think of the dual as running in reversed time, so that corresponds to . We will denote by and the slacks in (1), (1), and refer to them as the primal and dual states.
These problems are special cases of continuous linear programs (CLP) formulated by Bellman in 1953 [4], and further discussed and investigated in [2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14]. The mnemonic MCLP is used to stress that we look for solutions in the space of measures.
In [13] it was shown that if the primal and dual satisfy a Slater type condition, then optimal solutions exist, and strong duality holds. Furthermore, the optimal solutions can be chosen to be absolutely continuous on with jumps, i.e. impulse controls only at and . It was also pointed out that the MCLP formulation generalizes the SCLP (separated continuous linear programs) of [1, 8, 14]
In this paper we focus on a detailed description of the solutions of MCLP and MCLP, and we present detailed examples that illustrate the surprising richness and complexity of these solutions.
Our contributions in this paper are:

Identify an extreme point (vertex) of MCLP with a finite sequence of bases of an associated LP.

Give a detailed description of the optimal solution as it is computed for a given basesequence.

Derive the validity region of a basesequences as a convex polyhedral cone of model parameters for which the basesequence is optimal.

Give a simple nondegeneracy criterion, and prove uniqueness of the solution when it holds.

Show that strong duality holds for any feasible pair MCLP/MCLP, without requiring the Slater type condition of [13].

Point out the relation of MCLP to SCLP, and illustrate by examples how MCLP can solve SCLP problems which are not strongly dual.

Analyze in detail all one dimensional MCLP problems, and illustrate a complete solution of a 2 dimensional MCLP.
Further research to develop a simplextype algorithm for MCLP, based on our identification of vertices as base sequences, is in progress.
2 Preliminaries
In this section we state some definitions and briefly summarize the results of [13] on MCLP/MCLP.
Complementary slackness
{definition}Solutions of MCLP and MCLP are said to be complementary slack if
(3) 
Optimal solutions are always complementary slack, and any feasible complementary slack solutions are optimal.
Feasibility and nondegeneracy
Throughout the paper we will make following assumption:
Assumption \thetheorem (Feasibility)
Both MCLP and MCLP are feasible for target time horizon .
Assumption \thetheorem (Nondegeneracy I)
The vector is in general position to the matrix (it is not a linear combination of any less than columns), and the vector is in general position to the matrix .
Assumption \thetheorem (Nondegeneracy II)
The vector is in general position to the matrix , and the vector is in general position to the matrix .
Criterion for feasibility
{theorem}[Theorem 3.2 in [13]] MCLP is feasible if and only if the following standard linear program TestLP with unknown vectors is feasible.
TestLP  s.t.  
Slater type condition
{definition}We say that MCLP satisfies Slater type condition, if (2) has a feasible solution such that and for some .
Existence of strongly dual optimal solutions
{theorem}Partial characterization of the solutions
{theorem}[Proposition 5.2 and Theorem 5.5. in [13]] If MCLP/MCLP have optimal solutions , then:
(i) is concave piecewise linear on with a finite number of breakpoints.
(ii) There exists an optimal solution which is continuous piecewise linear on .
(iii) Under the nondegeneracy assumption 2, every optimal solution is of this form, and furthermore, is unique over .
3 Detailed description of the solution
In this section we give a detailed description of the solution. We start by discussing the proposed form of the solution, and define base sequences in Section 3.1. These base sequences play a role analogous to bases in standard linear programming. We then formulate and prove in Section 3.2 the structure theorem (Theorem 3.2), stating that a solution is optimal if and only if it is obtained from a base sequence. Two useful corollaries on boundary values then follow in Section 3.3. In Section 3.4 we show that under nondegeneracy assumption the solution is unique. Finally in Section 3.5 we define validity regions for base sequences and show that they are given by a convex polyhedral cone of the parameters of the problem.
3.1 Base sequences
We consider solutions of MCLP/MCLP that consist of impulse controls at and , piecewise constant control rates , and continuous piecewise linear states with possible discontinuities at . We refer to the set of values of controls and states at and as the boundary values. The time horizon is partitioned by which are the breakpoints in the rates and in the slopes of . We denote the values of the states at the breakpoints by , and , . Because there may be a discontinuity at we denote the values at itself by , , and let , be the values of the limit as . The constant slopes of the states and the constant values of the control rates for each interval are denoted and . Recall that the dual is running in reverse time, though we have labelled the intervals in the direction of the primal problem.
For solutions of this form the solution is fully described by giving the boundary values, the rates for each interval, and the lengths of the intervals . We will now discuss equations to calculate these.
The rates are complementary slack basic solutions of the following pair of LP problems
(5) 
(6) 
where by we denote the following sign restrictions: is zero, is nonnegative and is unrestricted. The sign restrictions, which are determined by the sets of indexes , vary from interval to interval.
We say that a basis of RatesLP and the corresponding complementary slack dual basis of RatesLP are admissible if . We say that two bases are adjacent if we can go in a single pivot operation from one to the other. The piecewise constant values of in the solution are given by a sequence of bases which are admissible and adjacent. We let: , , so that the basic variables of the primal RatesLP for the th interval are where . In the pivot , leaves the basis and enters the basis. The values of can then be calculated from (5, 6) once are given.
Next we have time interval equations. If then , and if then . We then get the following equations for :
(7)  
An additional equation is:
(8) 
It remains to find the boundary values. We let , , and , . We say that and are compatible if , and similarly, and are compatible if . The solution needs to satisfy these compatibility conditions.
By the definition of and by complementary slackness,
(9) 
This determines the value for of the boundary variables. We note however that in contrast to and for , all that we can say for the boundary values is that and similarly . One can see that by assumption 2 we always have .
The remaining boundary values need to satisfy two sets of equations. The first boundary equations relate to the constraints at time for the primal and the dual problem:
(10) 
The second boundary equations relate to the discontinuities of and at :
(11) 
We can replace in (11) by and . In this form the second boundary equations (11) together with first boundary equations (10) and time interval equations (7), (8) provide a set of equations determining the boundary values and the time interval lengths.
We refer to the sequence together with , or equivalently to the index sets as the base sequence.
3.2 The structure theorem
{theorem}[Structure Theorem]
(i) Consider the MCLP, MCLP problems (1), (1) and assume NonDegeneracy Assumption 2.
Let be admissible adjacent bases with rates , and let
be compatible with .
Let , and be a solution of (7)–(11), and let be constructed from these boundary values, time intervals, and rates. If all the boundary values, all the intervals, all the values at the breakpoints, and the limit values
are , then this is an optimal solution.
(ii) Conversely, if problems (1), (1) are feasible, then there exists an optimal solution given by a sequence of admissible adjacent bases and compatible boundary sets .
{proof}
(i) The proof is very similar to the proof of part (i) of the structure theorem in [14].
We have and by (8) they add up to . Hence is a partition of , are well defined (at all but the breakpoints) piecewise constant and are well defined continuous piecewise linear on with a possible discontinuity at .
To show optimality we need to show that , satisfy the constraints of (1), (1) as equalities, they are nonnegative, and they are complementary slack as in (3).
The primal and dual constraints hold at by (10), hold for all by integrating both sides of the constraints which involve in (5, 6) from to and hence hold also at by (11).
Since are admissible . That follows from and .
Next we show complementary slackness at all but the breakpoints. This is where we need to use the NonDegeneracy Assumption (2), as a result of which the following strict complementary slackness holds for all except the breakpoints:
(12) 
The proof of (12) is given in the proof of Theorem 3 in [14].
Finally, complementary slackness holds at time and by (9).
(ii) At this point we assume the NonDegeneracy Assumption 2 and Slatertype condition 2. We complete the proof, without these assumptions in Section 4. Let be a pair of optimal solutions, as described in Theorem 2, with piecewise constant rates and piecewise linear slacks , with breakpoints . We will construct an optimal base sequence from these solutions.
In each interval are optimal solutions of RatesLP/LP (5, 6). By feasibility, complementary slackness and nondegeneracy, must be basic solutions of the RatesLP/LP problems (5, 6), with admissible bases . If these are adjacent, the proof is complete. Else, if are not adjacent we can go from to in a sequence of pivots, preserving the admissibility. In this way we will have a new sequence of bases which are feasible and adjacent. The boundary values of will then also determine , where are compatible with , because are rightcontinuous functions.
It is seen from the structure theorem that the solution is determined by the base sequence , and if the conditions of the structure theorem are satisfied we call it an optimal base sequence. We refer to the constructed boundary values , the time partition , and the control rates and states as the optimal solution corresponding to the optimal base sequence.
3.3 Boundary values
The following corollaries determine some properties of the boundary values, which are used later in the proof of uniqueness in Section 3.4. {corollary} Let , and . Then and are optimal primal and dual solutions of the pair of dual LP problems:
(13) 
(14) 
Clearly, in the optimal solution are feasible solutions to (13), (14), with slacks for the primal and for the dual. Furthermore, if so that the corresponding constraint is not tight, then by (9) we have , and vice versa, if , then by (9) , and the corresponding constraint of the primal is tight. The same argument works for the , as well as for the dual slacks. Hence, are feasible complementary slack solutions, and so they are optimal.
Let the vectors and be defined by
Then the same boundary values and are also optimal solutions of the following pair of (not dual) LP problems:
(15) 
(16) 
Let be an optimal solution of MCLP/MCLP, as in the structure Theorem 3.2. Substitute the optimal solution in the objective and constraints of MCLP, to obtain:
Since satisfy the constraints of the rates LP we have
Also
Keeping all the other optimal values, we see that are optimal solutions of
s.t.  
which is equivalent to modBondaryLP (15). The proof for the dual boundary values is the same.
3.4 Uniqueness
We now show that optimal solutions of the form described in the structure theorem are unique optimal solutions. We need the following uniqueness condition: {definition} Uniqueness Condition We say that satisfies the uniqueness condition if:
Comment We believe that under the nondegeneracy assumptions 2, 2 the uniqueness condition is satisfied at all but a finite number of values of . We do not have a proof for this.
Let be an optimal solution as described in the Structure Theorem 3.2. Assume the nondegeneracy assumption 2, and assume that satisfy the Uniqueness Condition 3.4. Then are the unique optimal solution to MCLP. {proof} By Theorem 2 is unique on interval . Hence, all we need to show is that the boundary values are unique. Assume otherwise. Let be an optimal solution with another set of impulses . By Corollary 3.3, are optimal solutions of (15), with exactly the same constants and . Consider then the dual of (15), which has r.h.s . Under the Uniqueness Assumption 3.4 all feasible basic solutions of this dual are nondegenerate. This implies that the optimal solution of (15) is unique, and we have shown that . As a result also are also unique.
3.5 Validity regions
{definition}Let be a base sequence. Let be the set of all for which this base sequence is optimal. Then is called the validity region of . {theorem} The validity region of a basesequence is a convex polyhedral cone. {proof} Let be an optimal base sequence for at least one and let
be the corresponding optimal solution.
For a given a column index set we use the following notations:
is the matrix composed of the corresponding columns of the matrix .
is the matrix obtained from the unit matrix by replacing the diagonal elements with indexes in by the value 0.
We also define following matrices:
is the matrix of the coefficients of the time interval equations (7).
is the matrix composed of: if for the time interval equations of : and , otherwise .
is the matrix composed of: if for the time interval equations of : and , otherwise .
is the matrix where the left block is a zero matrix and the right block is a unit matrix.
is the matrix where the left block is a zero matrix and the right block is a unit matrix.
Finally, we define the matrix
Given , the base sequence determines all the coefficients of the matrix . It follows from the Structure Theorem 3.2 that:
(18) 
Any combination of and that solves (18) presents an optimal solution with the base sequence , and with in the validity region of the base sequence.
One can immediately see that for any , is the optimal base sequence for the boundary values , with the optimal solution . Similarly, if is in the validity region, with solution , then is in the validity region, with the solution . Hence is a convex cone.
Furthermore, the image under the linear transformation presented by of the convex nonnegative orthant, intersected with the planes at all the coordinates except and intersected with , is the validity region of the base sequence. This is obviously a convex polyhedral cone.
The following corollary follows immediate from Theorem 3.5. {corollary} Let be a straight line of boundary parameters. As changes, within the validity region of a single basesequence, each of the interval lengths , each of boundary values and each of are affine functions of . {proof} The proof for is the same as in the proof of Theorem 5 in [14]. The proof for the boundary values follows from the proof of convexity in Theorem 3.5.
4 Strong duality without Slatertype condition
{theorem}If MCLP and MCLP are feasible, then both have optimal solutions, and there is no duality gap. {proof} We denote by MCLP the MCLP problem with parameters . Assume MCLP/MCLP are feasible. Assume the Slater type condition 2 does not hold, otherwise there is nothing to prove. We will now construct an optimal solution for this problem. We choose some . For , let , , where , are vectors of 1’s of appropriate dimension. If NonDegeneracy Assumption 2 does not hold we choose vectors such that for all and are in general position to the matrices and respectively, and also , , otherwise . It is clear that MCLP/MCLP are feasible and satisfy the Slater type condition, and therefore have strongly dual optimal solutions by Theorem 2. Furthermore, NonDegeneracy Assumption 2 holds for , and hence by part (ii) of the proof of the Structure Theorem 3.2 the solution can be represented by an optimal base sequence . Consider now a parametric family of problems MCLP/MCLP . For every MCLP/MCLP will still be strictly feasible and will satisfy NonDegeneracy Assumption 2, and hence have optimal solutions represented by an optimal base sequence. Such a base sequence will be valid for some interval by Theorem 3.5. By part (i) of Theorem 2 all bases in an optimal base sequence are distinct, and so the total number of optimal base sequences is finite. Therefore we have a finite partition , where each interval belongs to a validity region of some optimal base sequence. The base sequence of the last interval is an optimal base sequence for MCLP, with an optimal solution.
Completion of proof of Theorem 3.2(ii)
This now follows immediately from Theorem 4 and its proof.
One of the following statements about MCLP and MCLP is true:
(i) both are feasible and have optimal solution without duality gap, or
(ii) both are infeasible, or
(iii) one of the problems is infeasible and the other is unbounded.
{proof}
Consider the TestLP problem (2) and its dual TestLP. For TestLP/TestLP we have the following three cases:
(i) TestLP and TestLP are feasible. Then by Theorem 2 MCLP and MCLP are also feasible and hence by Theorem 4 have optimal solutions without duality gap.
(ii) TestLP and TestLP are both infeasible. Then by Theorem 2 MCLP and MCLP