Infinite-dimensional Linear Programming Research Articles

The Fundamental Theorem of Asset Pricing Without Probabilistic Prior Assumptions

Motivated by recent discussions on Knightian uncertainty, we develop the fundamental theorem of asset pricing in a probability-free setup. The usual assumption of a prior probability is removed; a certain continuity property in the state variable is introduced instead. We show that one can still develop a meaningful and rich theory of asset pricing. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no arbitrage theory to economic equilibrium by establishing a variant of the Harrison-Kreps-Theorem on viability and no arbitrage. Finally, we consider (super)hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.

Bisimulation Metrics for Continuous Markov Decision Processes

In recent years, various metrics have been developed for measuring the behavioral similarity of states in probabilistic transition systems [J. Desharnais et al., Proceedings of CONCUR'99, Springer-Verlag, London, 1999, pp. 258–273; F. van Breugel and J. Worrell, Proceedings of ICALP'01, Springer-Verlag, London, 2001, pp. 421–432]. In the context of finite Markov decision processes (MDPs), we have built on these metrics to provide a robust quantitative analogue of stochastic bisimulation [N. Ferns, P. Panangaden, and D. Precup, Proceedings of UAI-04, AUAI Press, Arlington, VA, 2004, pp. 162–169] and an efficient algorithm for its calculation [N. Ferns, P. Panangaden, and D. Precup, Proceedings of UAI-06, AUAI Press, Arlington, VA, 2006, pp. 174–181]. In this paper, we seek to properly extend these bisimulation metrics to MDPs with continuous state spaces. In particular, we provide the first distance-estimation scheme for metrics based on bisimulation for continuous probabilistic transition systems. Our work, based on statistical sampling and infinite dimensional linear programming, is a crucial first step in formally guiding real-world planning, where tasks are usually continuous and highly stochastic in nature, e.g., robot navigation, and often a substitution with a parametric model or crude finite approximation must be made. We show that the optimal value function associated with a discounted infinite-horizon planning task is continuous with respect to metric distances. Thus, our metrics allow one to reason about the quality of solution obtained by replacing one model with another. Alternatively, they may potentially be used directly for state aggregation.

A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems

This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.

Thinning and harvesting in stochastic forest models

This paper analyzes a stochastic forest growth model in which the manager is able to first thin the forest to promote better growth before harvesting. Both Wicksell single thinning and harvesting cycle and Faustmann on-going rotation problems are considered. The Wicksell problem is analyzed by first restricting the class of decision times to (thinning, harvesting) pairs that bound the growth away from infinity and imbedding the problem in an infinite-dimensional linear program on a space of triplets of measures. These measures capture the thinning and harvesting decisions along with the behavior of the growth process prior to harvest. An auxiliary linear program then leads to a nonlinear optimization problem for which an optimal value and solution are determined. The values of all the problems are be related through a set of inequalities. The solution of the nonlinear problem determines (random) thinning and harvesting times for the single thinning and harvesting cycle which demonstrate the equality of the values of these various problems. Finally for the Wicksell problem, the unrestricted class of thinning and harvest times is shown to give the same value as the restricted class. The Faustmann on-going thinning and harvesting rotation problem is reduced to a Wicksell problem which then allows for the characterization of the value as the solution to a different nonlinear optimization problem. The effects of the opportunity to thin the forest are illustrated on a mean-reverting stochastic model.

On the Number of Mather Measures of Lagrangian Systems

In 1996, Ricardo Ricardo Mañé discovered that Mather measures are in fact the minimizers of a “universal” infinite dimensional linear programming problem. This fundamental result has many applications, of which one of the most important is to the estimates of the generic number of Mather measures. Mañé obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able, with Gonzalo Contreras, to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest.

Problems and Techniques

This issue features two papers involving optimization problems, one targeted at minimizing temperature differences across a plate, and the other at multidimensional integration. 1. Want to turn the graphics card in your laptop or desktop into a personal supercomputer? Eddie Wadbro and Martin Berggren show you how, in their paper “Megapixel Topology Optimization on a Graphics Processing Unit.” The advantage of graphics cards is that they are cheap, relatively speaking, and that they allocate more resources to data processing than a CPU would. Graphics cards are also becoming easier to program (especially for those of us who grew up with Connection Machines and FPS-164s). Graphics processing units are natural hardware platforms for problems that are highly data parallel. This includes the topology optimization problem considered here, where a limited amount of high-conductivity material is to be distributed across a heated plate so that its temperature field is as even as possible. The authors express this as an area-to-point flow optimization problem. A subsequent finite element discretization gives a symmetric positive-definite linear system that is solved by a diagonally preconditioned conjugate gradient method. As it turns out, the optimal distribution of the high-conductivity material emanates like a root from the heat sink, with increasing girth and finer branches as the discretization is refined. 2. In their paper “Approximate Volume and Integration for Basic Semialgebraic Sets,” Didier Henrion, Jean Bernard Lasserre, and Carlo Savorgnan are concerned with deterministic techniques for difficult multidimensional integration, of the type where only brute force Monte Carlo methods have a chance at producing acceptable approximations. The bodies can be disconnected or nonconvex, and are described by sets of polynomial inequalities (i.e., semialgebraic sets). The foundation for this work was laid more than a hundred years ago, when Chebyshev, Markov, and Stieltjes showed how to approximate one-dimensional integrals by sequences of moments. In this paper, the authors formulate the multidimensional integration as an infinite-dimensional linear programming problem, and approximate the required moments by a hierarchy of semidefinite programming problems. Numerical examples illustrate that the approach produces accurate approximations in two and three dimensions.

A numerical approach to infinite-dimensional linear programming in $L_1$ spaces

An infinite-dimensional linear programming formulated on $L_1$ spaces, problem (P), is studied in this paper. A related optimization problem, general capacity problem (GCAP), is also mentioned in this paper. But we find that the optimal solution does not exist in problem (P). Thus, we approach the optimal value for problem (P) via solving the problem (GCAP). A proposed algorithm is shown that we solve a sequence of semi-infinite subproblems to approach the optimal value of problem (P). The error bound for the difference between the optimal value for problem (P) and optimal value for semi-infinite subproblem is also given in this paper. Finally, numerical examples are implemented and compared with discretization method to show our computational efficiency.

Best domain for an elliptic problem in cartesian coordinates by means of shape‐measure

AbstractIn (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143–155) we introduced a method to determine the optimal domains for elliptic optimal‐shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure‐theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

We investigate relationships between the deterministic infinite time horizon optimal control problem with discounting, in which the state trajectories remain in a given compact set Y, and a certain infinite dimensional linear programming (IDLP) problem. We introduce the problem dual with respect to this IDLP problem and obtain some duality results. We construct necessary and sufficient optimality conditions for the optimal control problem under consideration, and we give a characterization of the viability kernel of Y. We also indicate how one can use finite dimensional approximations of the IDLP problem and its dual for construction of near optimal feedback controls. The construction is illustrated with a numerical example.

Optimal partition of a large labor force into working pairs

Kremer and Maskin (Wage inequality and segregation by skill. Working Paper 5718, National Bureau of Economic Research, http://www.nber.org/papers/w5718, 1996) introduced an idealized model for pairing workers and managers with different skill levels into small teams selected to maximize productivity. They used it to analyze the impact of technological change and widening skill gaps on labor market segregation. The present paper extends their model to a workforce with multidimensional skill types, continuously distributed, and gives a mathematical analysis of the extension. Pure and mixed notions of optimal pairing are introduced, which play an important role in the formulation and analysis of the model. The existence and uniqueness of such pairings are established using techniques from the theory of optimal transportation and infinite-dimensional linear programming.

On generic uniqueness of optimal solution in an infinite dimensional linear programming problem

On generic uniqueness of optimal solution in an infinite dimensional linear programming problem

Stochastic Control with Imperfect Models

We consider the problem of worst case performance estimation for a stochastic dynamic model in the presence of model uncertainty. This is cast as a nonclassical controlled diffusion problem. An infinite dimensional linear programming formulation is given and its dual is derived. The dual is successively approximated on a bounded domain by a semi-infinite and a finite linear program. This uses function approximation based on a reproducing kernel Hilbert space. Error analysis for the approximation is provided along with an estimate of the sample complexity.

Duality in Linear Programming Problems Related to Deterministic Long Run Average Problems of Optimal Control

It has been established recently that, under mild conditions, deterministic long run average problems of optimal control are “asymptotically equivalent” to infinite-dimensional linear programming problems (LPPs) and that these LPPs can be approximated by finite-dimensional LPPs. In this paper we introduce the corresponding infinite- and finite-dimensional dual problems and study duality relationships. We also investigate the possibility of using solutions of finite-dimensional LPPs and their duals for numerical construction of the optimal controls in periodic optimization problems. The construction is illustrated with a numerical example.

An Infinite-Dimensional Linear Programming Algorithm for Deterministic Semi-Markov Decision Processes on Borel Spaces

We devise an algorithm for solving the infinite-dimensional linear programs that arise from general deterministic semi-Markov decision processes on Borel spaces. The algorithm constructs a sequence of approximate primal-dual solutions that converge to an optimal one. The innovative idea is to approximate the dual solution with continuous piecewise linear ridge functions that naturally represent functions defined on a high-dimensional domain as linear combinations of functions defined on only a single dimension. This approximation gives rise to a primal/dual pair of semi-infinite programs, for which we show strong duality. In addition, we prove various properties of the underlying ridge functions.

Numerical solution of a long-term average control problem for singular stochastic processes

This paper analyzes numerically a long-term average stochastic control problem involving a controlled diffusion on a bounded region. The solution technique takes advantage of an infinite-dimensional linear programming formulation for the problem which relates the stationary measures to the generators of the diffusion. The restriction of the diffusion to an interval is accomplished through reflection at one end point and a jump operator acting singularly in time at the other end point. Different approximations of the linear program are obtained using finite differences for the differential operators (a Markov chain approximation to the diffusion) and using a finite element method to approximate the stationary density. The numerical results are compared with each other and with dynamic programming.

Linear programming approach to the optimal stopping of singular stochastic processes

Optimal stopping of stochastic processes having both absolutely continuous and singular behavior (with respect to time) can be equivalently formulated as an infinite-dimensional linear program over a collection of measures. These measures represent the occupation measures of the process (up to a stopping time) with respect to “regular time” and “singular time” and the distribution of the process when it is stopped. Such measures corresponding to the process and stopping time are characterized by an adjoint equation involving the absolutely continuous and singular generators of the process. This general linear programming formulation is shown to be numerically tractable through three examples, each of which seeks to determine the stopping rule for a perpetual lookback put option using different dynamics for the asset price. Exact solutions are determined in the cases that the asset prices are given by a drifted Brownian motion and a geometric Brownian motion. Numerical results for the more realistic model of a regime switching geometric Brownian motion are also presented, demonstrating that the linear programming methodology is numerically tractable for models whose theoretical solutions are very difficult to obtain.

An applicable method for solving the shortest path problems

A theorem of Hardy, Littlewood, and Polya, first time is used to find the variational form of the well known shortest path problem, and as a consequence of that theorem, one can find the shortest path problem via quadratic programming. In this paper, we use measure theory to solve this problem. The shortest path problem can be written as an optimal control problem. Then the resulting distributed control problem is expressed in measure theoretical form, in fact an infinite dimensional linear programming problem. The optimal measure representing the shortest path problem is approximated by the solution of a finite dimensional linear programming problem.

A new approach for solving of optimal nonlinear control problems

In this paper, we are going to consider a nonlinear optimal control problem (NOC). First we change the (NOC) problem to an optimal differential inclusion problem (ODI), then by defining new control variables, (ODI) problem is converted to an optimal control problem where it is linear in term of control variable and we determine the approximation of this control problem, then by using measure theory, we may convert the new optimal control problem to an infinite dimensional linear programming problem. We approximate the latest one to a finite linear programming problem, then by using of the optimal solution of the linear programming we obtain an approximation of the state and feedback control of the original (ONC) problem.

Linear programming solutions of periodic optimization problems: approximation of the optimal control

Deterministic long run average problems of optimal control are ''asymptotically equivalent" to infinite-dimensional linear programming problems ( LPP ) and the latter are approximated by finite dimensional LPP. The solutions of this finite dimensional LPP can be used for numerical analysis of periodic optimization problems. In the present paper we establish the convergence of controls constructed on the basis of the solution of the finite dimensional LPP to the optimal control of a periodic optimization problem. Results are illustrated with a numerical example.

Existence of Optimal Policies for Semi-Markov Decision Processes Using Duality for Infinite Linear Programming

Semi-Markov decision processes on Borel spaces with deterministic kernels have many practical applications, particularly in inventory theory. Most of the results from general semi-Markov decision processes do not carry over to a deterministic kernel since such a kernel does not provide "smoothness." We develop infinite dimensional linear programming theory for a general stochastic semi-Markov decision process. We give conditions, general enough to allow deterministic kernels, for solvability and strong duality of the resulting linear programs. By using the developed linear programming theory we give conditions for the existence of a stationary deterministic policy for deterministic kernels, which is optimal among all possible policies.