Continuous Linear Programming Problems Research Articles

The Difference between Finite Dimensional Linear Programming Problems and Infinite Dimensional Linear Programming Problems

This paper studies the difference between finite-dimensional linear programming problems and infinite dimensional linear programming problems. We discuss a special class of continuous linear programming problems. We develop the structure of extreme points of feasible region for this problem. Under some conditions we can characterize all extreme points of this problem. We show that under some conditions the optimal value for this problem may be finite but there is no optimal solution for it. Finally, we construct an example for this problem which has an optimal solution but all extreme points for this problem are not optimal solutions.

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.

New descent rules for solving the linear semi-infinite programming problem

The algorithm described in this paper approaches the optimal solution of a continuous semi-infinite linear programming problem through a sequence of basic feasible solutions. The descent rules that we present for the improvement step are quite different when one deals with non-degenerate or degenerate extreme points. For the non-degenerate case we use a simplex-type approach, and for the other case a search direction scheme is applied. Some numerical examples illustrating the method are given.

Interior-point algorithms for semi-infinite programming

In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.

A practical anti-degeneracy row selection technique in network linear programming

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.

An optimality test for semi-infinite linear programming

In this paper we present a test to characterize the optimal solutions for the continuous semi-infinite linear programming problem. This optimality characterization is a condition of Kuhn–Tucker type. The resolution of a linear program permits to check the optimality of a feasible point,to detect the unboundedness of the problem and to find descent directions. We give some illustrative examples. We show that the local Mangasarian–Fromovitz constraint qualification is almost equivalent to Slater qualification for this problem. Furthermore, it follows from our study that this optimality condition is always necessary for a wide class of semi-infinite linear programming problems

Stability conditions on continuous dynamic linear programming

Many on-line processes can be optimized in the sense of continuous dynamic linear programming problem, i.e., opt [c(t), x(t)] A(t) x(t) = b(t), x ⩾ 0 (a CDLP problem). In order to decrease the frequency of use of the corresponding program we have to explore the stability problems associated with the CDLP approach to on-line control.

On Duality and the Maximum Principle for Continuous Linear Programming Problems

In this paper, the Grinold maximum principle for continuous linear programming problems is extended to the case where the known duality theorems do not ensure the existence of an optimal solution of the dual problem in the usual reflexive L_p -spaces. The extended maximum principle is then applied to the investigation of a parabolic boundary control proble in with constraints on the control and the state.

Capacity expansion of linked reservoir systems

A procedure is presented for the analysis of time phasing of reservoir system development. The formulation is based on a multi-purpose reservoir model with stochastic inflows. The object is to select the reservoir sizing, timing of expansion and to establish operating policies such that the total cost associated with the system of linked reservoirs is minimized. The capacity expansion aspect is formulated as a mixed integer continuous linear programming problem. Due to the resulting problem size and its general structure, Benders' decomposition is applied. A numerical example illustrating the procedure is given.

Duality theorems for a class of continuous linear programming problems in a space of bochneb-integrable abstract functions

In this paper the duality theory for continuous linear programming problems is extended to problems defined in spaces of -p-times BocHNEB-integrable abstract functions (1 < p < ∞). Two duality theorems are proved, the first one including the existence of a primal optimal solution, the second one the existence of a dual optimal solution. As introduction the paper contains a short survey on results obtained in the field of duality theory for continuous programs, too.

A Non-Terminal Control Approach to Dynamic Economic Modelling in Non-Compact Space

We discuss a principle of point optimization as a basis of a dynamic economic model. The continuous dynamic linear programming problems serve to study the behaviour of an economic system over time. We try to get the result of a nonterminal control problem to be as close as possible to the solution of a terminal control problem, which is supossed to be as given. To reach it we have to “replace” a non-terminal curve by some region so as to include the terminal control problem solution. If this happens then the “long run” planning target has been reached through a set of “short run” planning decisions.

Regional economic models and continuous programming

This paper is concerned with an optimal investment allocation problem in a simple N-regional economic model. The problem is described as a class of optimal control problem, and formulated into a continuous linear programming problem. Both the primal and dual problems are considered. The procedure finds an optimal regional allocation of investment derived in terms of continuous programming.

Numerical solutions to continuous linear programming problems

A method for finding approximate solutions for continuous linear programming problems is suggested. The required conditions to be met are: a) the matrix associated with the integrals in the constraints is constant; b) all functions involved are of bounded variation; c) the matrices involved satisfy certain boundedness conditions, and d) there exist feasible solutions.

Duality theorems for continuous linear programming problems

Duality theorems for continuous linear programming problems

An Extended Duality Theorem for Continuous Linear Programming Problems

An Extended Duality Theorem for Continuous Linear Programming Problems

Some sufficient conditions for continuouslinear-programming problems

This paper treats continuous-linear programming problems with time delays in the constraints, which are expressed in the form of difference-differential inequalities. Lagrange multiplier functions are used to adjoin the constraints to the cost functional and to compute, by explicit algorithm, a candidate solution. It is proved that if this solution is feasible, then it is also optimal. A similar method is developed for dealing with dual problems. When both the primal and the dual problems have solutions, the relationship between their solutions is exhibited and it is shown how, in some cases, solutions to the dual problem can be constructed from the solutions to the primal problem and vice versa. Finally, the differences between continuous linear programming problems and problems of classical calculus of variations and modern optimal control are discussed. The main advantage of the method proposed in this paper, for computing trial solutions, is that it is quite straightforward and simple and that it does not depend on the simultaneous existence of solutions to both the primal and dual problems.

A class of continuous linear programming problems

A class of continuous linear programming problems

A Duality Theorem for a Class of Continuous Linear Programming Problems

Introduction. In his book Dynamic Programming [1], R. Bellman discusses a class of continuous linear programming problems which he calls bottleneck The discussion introduces and analyzes several examples of dynamic linear models of production allocation. Moreover, it is shown that to each (primal) program can be associated a dual program. Such pairs of primal and dual programs are then shown to satisfy the optimality condition (compare Theorem 2) and the equilibrium conditions (compare Theorem 3) enjoyed by pairs of (finite) linear programming problems. Others who have discussed these problems are P. Wolfe [2], R. S. Lehman [3], and T. C. Koopmans [4]. In [1] and [3] the authors' principal concern was in finding methods of solutions. They did not treat the problems with complete mathematical rigor. In this paper under suitable hypotheses we shall prove an analogue of the fundamental duality theorem of linear programming which is valid for this class of continuous linear programming problems. Furthermore, several related results will be stated, and the duality theorem will be applied to a dynamic Leontief model of production. The reader is referred to [5] for a discussion of the fundamentals of the theory of linear programming and linear models of production. Moreover, the reader may wish to examine the economic model introduced in ?9 for a motivation of the definition of these continuous linear programming problems.