Continuous-time Linear Programming Problems Research Articles

Extended Form of Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices

An extended form of robust continuous-time linear programming problem with time-dependent matrices is formulated in this paper. This complicated problem is studied theoretically in this paper. We also design a computational procedure to solve this problem numerically. The desired data that appeared in the problem are considered to be uncertain quantities, which are treated according to the concept of robust optimization. A discretization problem of the robust counterpart is formulated and solved to obtain the ϵ-optimal solutions.

Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices

The uncertainty for the continuous-time linear programming problem with time-dependent matrices is considered in this paper. In this case, the robust counterpart of the continuous-time linear programming problem is introduced. In order to solve the robust counterpart, it will be transformed into the conventional form of the continuous-time linear programming problem with time-dependent matrices. The discretization problem is formulated for the sake of numerically calculating the ϵ-optimal solutions, and a computational procedure is also designed to achieve this purpose.

Numerical method for solving the continuous-time linear programming problems with time-dependent matrices and piecewise continuous functions

The numerical method is proposed in this paper to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients and the time-dependent matrices are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities of the involved functions, a methodology for not equally partitioning the time interval is proposed. The main issue of this paper is to obtain an analytic formula of the error bound, where the strong duality theorem for the primal and dual pair of continuous-time linear programming problems with time-dependent matrices and piecewise continuous functions is a by-product. We shall propose two kinds of computational procedure to evaluate the error bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. The detailed differences between these two computational procedures will be also presented. Finally we present a numerical example to demonstrate the usefulness of the numerical method.

Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems

A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the ϵ -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and ϵ -optimal solution.

Solving Continuous-Time Linear Programming Problems Based on the Piecewise Continuous Functions

ABSTRACTThe numerical method is proposed in this article to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients of this problem are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities, a different methodology for not equally partitioning the time interval is proposed. The main issue of this article is to obtain an analytic formula of error upper bound. In this article, we shall propose two kinds of computational procedure to evaluate the error upper bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. Finally, we present a numerical example to demonstrate the usefulness of the numerical method.

Parametric continuous-time linear fractional programming problems

The main purpose of this paper is to establish the strong duality theorem for the parametric formulation of continuous-time linear fractional programming problems. We also consider the so-called extended form of continuous-time linear fractional programming problems that assume the vector-valued functions to be measurable and bounded on the time interval $[0,T]$ . The by-product of the main result is the establishment of the strong duality theorem for the extended form of continuous-time linear fractional programming problems.

Note: continuous-time linear programming problems revisited

This note is aimed to correct the strong duality theorem of previous paper regarding the continuous-time linear programming problems. The argument presented in the previous paper can only be used to prove the case of piecewise continuous functions in which the discontinuities are the left-continuities.

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0, T] is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0, T] as subintervals of [0, T] . In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0, T] .

AN INTERVAL-TYPE ALGORITHM FOR CONTINUOUS-TIME LINEAR FRACTIONAL PROGRAMMING PROBLEMS

An interval-type computational procedure by combining the parametric method and discretization approach is proposed in this paper to solve a class of continuous-time linear fractional programming problems (CLFP). Using the different step sizes of discretization problems, we construct a sequence of convex, piecewise linear and strictly decreasing upper and lower bound functions. The zeros of upper and lower bound functions then determine a sequence of intervals shrinking to the optimal value of (CLFP) as the size of discretization getting larger. By using the intervals we can find corresponding approximate solutions to (CLFP). We also establish upper bounds of lengths of these intervals, and thereby we can determine the size of discretization in advance such that the accuracy of the corresponding approximate solution can be controlled within the predefined error tolerance. Moreover, we prove that the searched sequence of approximate solution functions weakly-star converges to an optimal solution of (CLFP). Finally, we provide some numerical examples to implement our proposed method.

Approximate solutions and error bounds for a class of continuous-time linear programming problems

In this article, we discuss a class of continuous-time linear programming (CLP) problems. We provide a discrete approximation procedure to find numerical solutions of CLP, establish the estimation for the error bound and prove that the searched sequence of approximate solution functions weakly star converges to an optimal solution of CLP. Finally, we provide some numerical examples to implement our proposed method and to show the quality of the proposed error bound.

AN INTERVAL-TYPE ALGORITHM FOR CONTINUOUS-TIME LINEAR FRACTIONAL PROGRAMMING PROBLEMS

An interval-type computational procedure by combining the parametric method and discretization approach is proposed in this paper to solve a class of continuous-time linear fractional programming problems (CLFP). Using the different step sizes of discretization problems, we construct a sequence of convex, piecewise linear and strictly decreasing upper and lower bound functions. The zeros of upper and lower bound functions then determine a sequence of intervals shrinking to the optimal value of (CLFP) as the size of discretization getting larger. By using the intervals we can find corresponding approximate solutions to (CLFP). We also establish upper bounds of lengths of these intervals, and thereby we can determine the size of discretization in advance such that the accuracy of the corresponding approximate solution can be controlled within the predefined error tolerance. Moreover, we prove that the searched sequence of approximate solution functions weakly-star converges to an optimal solution of (CLFP). Finally, we provide some numerical examples to implement our proposed method.

Approximate Solutions and Duality Theorems for Continuous-Time Linear Fractional Programming Problems

This article proposes a practical computational procedure to solve a class of continuous-time linear fractional programming problems by designing a discretized problem. Using the optimal solutions of proposed discretized problems, we construct a sequence of feasible solutions of continuous-time linear fractional programming problem and show that there exists a subsequence that converges weakly to a desired optimal solution. We also establish an estimate of the error bound. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non...

On a recurrence algorithm for continuous-time linear fractional programming problems

In this paper, we develop a discrete approximation method for solving continuous-time linear fractional programming problems. Our method enables one to derive a recurrence structure which shall overcome the computational curse caused by the increasing numbers of decision variables in the approximate decision problems when the subintervals are getting smaller and smaller. Furthermore, our algorithm provides estimation for the error bounds of the approximate solutions. We also establish the convergence of our approximate solutions to the continuous-time linear fractional programming problems. Numerical examples are provided to illustrate the quality of the approximate solutions.

A recurrence method for a special class of continuous time linear programming problems

This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.

Some remarks concerning duality for continuous-time programming problems

This paper attempts to present a new methodological approach to the study of duality for continuous-time programming, based on the use of the Lebesgue-Stieltjes integral. This approach allows for a unified treatment, including cases when—in more classical approach—either the primal or the dual problem has no feasible or no optimal solution, while the other one has an optimal solution. The continuous-time linear programming problem is discussed in more detail.

Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Frechet differentiability of all operators involved, and the existence of the topological complement of the null space of the Frechet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.

A duality theorem for continuous-time linear programming problems

In dealing with dynamic economic policy models one encounters optimization problems whose objective function is an integral of a linear function of a finite number of continuous variables and whose constraints are linear integral inequalities. A set of intertemporal efficiency conditions (equilibrium conditions) yielding the optimal policy are given. By approximating the continuous problem by a set of discrete problems and appealing to a well known convergence theorem in functional analysis a continuous analog of the duality theorem is proved.