Infinite-dimensional Linear Programming Research Articles

Linear Programming Approach to Deterministic Long Run Average Problems of Optimal Control

We establish that deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finite-dimensional LPPs, the solutions of which can be used for construction of the optimal controls. General results are illustrated with numerical examples.

The minimization of the fuel costs in the train transportation

This paper presenting case studies in modern large scale constrained optimization, the purpose of which is to illustrate how recent advances in algorithms and modelling languages have made it easy to solve difficult optimization problems using of software. In this paper we use measure theory technique and iterative dynamic programming algorithm for solving of a trajectory optimization problem: how to drive a train so as to minimize fuel costs. In the first method, we optimal control problem to using of atomic measures change this one to an infinite dimensional linear programming problem and then we approximate the latter one to a finite dimensional linear programming problem, then by the optimal solution of the final problem we obtain sub-optimal controls and then by these controls we obtain the approximate solution of the original problem. In the second method, we will using dynamic programming as iteration for solving optimal control problem.

Duality and Existence of Optimal Policies in Generalized Joint Replenishment

We establish a duality theory for a broad class of deterministic inventory control problems on continuous spaces that includes the classical joint replenishment problem and inventory routing. Using this theory, we establish the existence of an optimal policy, which has been an open question. We show how a primal-dual pair of infinite dimensional linear programs encode both cyclic and noncyclic schedules, and provide various results regarding cyclic schedules, including an example showing that they need not be optimal.

Slot nozzle design with specified pressure in a given subregion by using embedding method

In this paper we study a shape optimization problem connected with controlled pressure in a subregion of a 2D slot nozzle. Feasible shapes for this problem are characterized only with upper walls. This enables us to seek for the optimum in a class of functions and their derivatives instead of a class of shapes. The shape optimization problem can be written as an optimal control problem, then the resulting distributed control problem is expressed in a measure theoretical form, in fact an infinite dimensional linear programming problem. The optimal measure representing optimal shape is approximated by the solution of a finite dimensional linear programming problem. A brief sensitivity analysis on model parameters is given.

The Scalarization Approach to Multiobjective Markov Control Problems: Why Does It Work?

This paper concerns discrete-time multiobjective Markov control processes on Borel spaces and unbounded costs. Under mild assumptions, it is shown that the usual “scalarization approach” to obtain Pareto policies for the multiobjective control problem is in fact equivalent to solving the dual of a certain multiobjective infinite-dimensional linear program. The latter program is obtained from a multiobjective measure problem which is also used to prove the existence of strong Pareto policies, that is, Pareto policies whose cost vector is the closest to the control problem’s virtual minimum.

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.

Duality in procurement design

Finding an optimal mechanism in a standard adverse selection model is equivalent to solving an infinite dimensional linear program. We begin with certain feasible mechanisms—those implemented by auctions, take-it-or-leave-it offers, and combinations of these polar mechanisms—and search for the environments that make them optimal. We prove the optimality of each mechanism using the dual program.

Filter design: a finite dimensional convex optimization approach

AbstractWe consider the design of transfer functions (filters) satisfying upper and lower bounds on the frequency response magnitude or on phase response, in the continuous and discrete time domains. The paper contribution is to prove that such problems are equivalent to finite dimensional convex optimization problems involving linear matrix inequality constraints. At now, such optimization problems can be efficiently solved. Note that this filter design problem is usually reduced to a semi infinite dimensional linear programming optimization problem under the additional assumption that the filter poles are fixed (for instance, when considering FIR design). Furthermore, the semi infinite dimensional optimization is practically solved, using a gridding approach on the frequency. In addition to be finite dimensional, our formulation allows to set or not the filter poles. These problems were mainly considered in signal processing. Our interest is to propose an approach dedicated to automatic control problems. In this paper, we focus on the following problems: design of weighting transfers for H∞ control and design of lead‐lag networks for control. Numerical applications emphasize the interest of the proposed results. Copyright © 2003 John Wiley & Sons, Ltd.

多目的<i>l</i><sub>1</sub>制御における双対問題

In this paper, an l1-optimal control problem with frequency and time domain constraints on a closed-loop response is considered. The problem is formulated as an infinite dimensional linear programming problem, and the primal and dual approach is exploited. This sometimes results in non-zero gap between the primal and dual costs. This paper proposes a new method to approximate the primal and dual problems without the duality gap. As a result, the method yields a sub-optimal solution with arbitrary accuracy.

-domain four-blocks l1-optimal control design for SISO plants

The multiblock l1-optimal control problem for single-input single-output (SISO) plants is considered. It is shown that it can be converted via polynomial equation techniques to an infinite-dimensional linear programming (LP) problem. Finite dimensional sub/super approximations can be determined by considering two sequences of modified finite-dimensional linear programming problems derived directly from the YJBK parameterisation by exploiting the underlying algebraic structure. This approach induces the application of a consistent truncation strategy that leads to a redundancy-free constraint formulation and, as a consequence, to linear programming problems less affected by degeneracy. Further, more insight on the algebraic structure of the problem and on the achievement of exact rational solutions is provided, allowing the development of a simple and conceptually attractive theory.

Infinite DimEnsionAl State-space approach to reactor network synthesis: application to attainable region construction

In this work, we present a methodology for the construction of the attainable region (AR) and for the synthesis of globally optimal reactor networks employing isothermal, steady-state, plug flow reactors (PFR), continuous-stirred tank reactors (CSTR) as well as mixing. The problem is solved using the novel Infinite DimEnsionAl State-space (IDEAS) approach. Within the IDEAS framework, reactor network synthesis is formulated as an infinite dimensional convex (linear) optimization problem. Pointwise identification of the reactor network's attainable region is shown to be equivalent to the solution of an infinite dimensional linear program. A finite dimensional approximation strategy is presented and illustrated on an example from the literature.

Linear Programming Formulation for Optimal Stopping Problems

Optimal stopping problems for continuous time Markov processes are shown to be equivalent to infinite-dimensional linear programs over a space of pairs of measures under very general conditions. The measures involved represent the joint distribution of the stopping time and stopping location and the occupation measure of the process until it is stopped. These measures satisfy an identity for each function in the domain of the generator which is sufficient to characterize the stochastic process. Finite-dimensional linear programs obtained using Markov chain approximations are solved in two examples to illustrate the numerical accuracy of the linear programming formulation.

Solving General Capacity Problem by Relaxed Cutting Plane Approach

This paper studies a class of infinite dimensional linear programming problems known as the general capacity problem. A relaxed cutting plane algorithm is proposed. A convergence proof together with some analysis of the results produced by the algorithm are given. A numerical example is also included to illustrate the computational procedure.

A Polynomial Approach to ℓ1 Optimal Control Problems

In this paper, the scalar multi-block ℓ1-optimal control problem is considered. It is shown that it can be converted via polynomial equation techniques to an infinite dimensional linear programming (LP) problem. Finite dimensional sub/super approximations can be determined by considering two sequences of modified finite dimensional linear programming problems derived directly from the YJBK parameterization by exploiting the underlying algebraic structure. This approach induces the application of a consistent truncation strategy that leads to a redundancy-free constraint formulation and, as a consequence, to linear programming problems less affected by degeneracy. Further, more insight on the algebraic structure of the problem and on the achievement of exact rational solutions is provided, allowing the development of a simple and conceptually attractive theory.

Convergence of a General Class of Algorithms for Separated Continuous Linear Programs

Separated continuous linear programs (SCLP) are a type of infinite-dimensional linear program which can serve as a useful model for a variety of dynamic network problems where storage is permitted at the nodes. This paper proves the convergence of a general class of algorithms for solving SCLP under certain restrictions on the problem data. This is the first such proof for any nondiscretization algorithm for solving any form of continuous linear programs.

Application of infinite dimensional linear programming to IIR filter design with time domain constraints

Previously developed infinite dimensional linear programming techniques are used for the design of recursive filters with complex Chebyshev error criteria and time-domain constraints. The approach is applied to a numerical example that deals with the design of a Laguerre network as an equalization filter for a digital transmission channel.

Erratum: "Existence of Markov Controls and Characterization of Optimal Markov Controls"

A compactness condition used to establish the equivalence of infinite-dimensional linear programming problems with stochastic control problems in [SIAM J. Control Optim., 36 (1998), pp. 609--653.] was misstated. The correct statement of the condition is given, as well as an example to show that the erroneous condition is not sufficient to obtain equivalence.

Approximation of Infinite-Dimensional Linear Programming Problems which Arise in Stochastic Control

We study a general approximation scheme for infinite-dimensional linear programming (LP) problems which arise naturally in stochastic control. We prove that the optimal value of the approximating problems converges to the value of the original LP problem. For the controls, we show that if the approximating optimal controls converge, the limiting control is an optimal control for the original LP problem. As an application of this theory, we present numerical approximations to the LP formulation of stochastic control problems in continuous time. We study long-term average and discounted control problems. For the example for which the theoretical solution is known, our approximation results are very accurate.

The Global Optimization of Variational Problems with DiscontinuousSolutions

We consider variational problems in which the slope of the admissible curves is not necessarily bounded, so that they admit discontinuous solutions. A problem is first reformulated as one consisting of the minimization of an integral in a space of functions satisfying a set of integral equalities; this is then transfered to a nonstandard framework, in which Loeb measures take the place of the functions and a near-minimizer can always be found. This is mapped back to the standard world by means of the standard part map; its image is a minimizer, so that the optimization is global. The minimizer is shown to be the solution of an infinite dimensional linear program and by well-proven approximation procedures a finite dimensional linear program is found by means of which nearly-optimal curves can be constructed for the original problem. A numerical example is given.

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.