Finite horizon approximations of infinite horizon linear programs

A Shadow Simplex Method for Infinite Linear Programs

“…In addition, when (P ) has a unique optimal solution x * , a sequence of optimal solutions to finite-dimensional shadow problems converges to x * . Similar value convergence for finite-dimensional approximations of a different class of CILPs was earlier established in [28]. Finally, we remark that since V (P (N )) is a sequence of real numbers that converges to V (P ) as N → ∞, V (P (N k )) also converges to V (P ) as k → ∞ for any subsequence N k of positive integers.…”

“…CILPs often arise from infinite-horizon dynamic planning problems [27,28] in a variety of models in Operations Research, most notably, a class of deterministic or stochastic dynamic programs with countable states [17,32,41,45] whose special cases include infinite-horizon problems with time-indexed states considered in Sections 4 and 5. Other interesting special cases of CILPs include infinite network flow problems [42,47], infinite extensions of Leontief systems [50,51], and semi-infinite linear programs [5,25,26], i.e., problems in which either the number of variables or the number of constraints is allowed to be countably infinite.…”

“…From this perspective, since CILPs are characterized by infinite information, our first question is whether it is possible to design a procedure that uses finite information to perform finite computations in each iteration and yet approximates the optimal value of the original infinite problem. Any such implementable procedure could proceed by solving a sequence of finite-dimensional truncations of the original infinite problem [27,28,29]. However, constructing an appropriate finite truncation of countably infinite equality constraints in countably infinite variables is not straightforward.…”

A combined stochastic programming and optimal control approach to personal finance and pensions

“…Several studies investigated the methods that could overcome this issue. For example, Grinold (1977Grinold ( , 1983 presents and compares different methods of approximating the general multistage optimization problem upon horizon, and concludes that a dual equilibrium technique will give improving approximations of the optimal solution as the horizon increases, and perfect approximations in the limit. Canestrelli (2005, 2006) show that a dynamic portfolio problem written as a multistage stochastic program can be rewritten as a discrete time optimal control problem, whereas Barro and Canestrelli (2011) further extends this work to a broader class of multistage stochastic programming problems, and reformulates the problems as discrete time stochastic optimal control problems.…”

Approximate Receding Horizon Approach for Markov Decision Processes: Average Award Case

“…For example, it has been applied to planning problems (e.g., inventory control) that can be modeled as linear programs [14] and that can be represented as a shortest path problem in an acyclic network (see [13] for example problems and references therein), a routing problem in a communication network by formulating the problem as a nonlinear optimal control problem [2], dynamic games [8], aircraft tracking [31], the stabilization of nonlinear time-varying systems [21,26,28] in the model predictive control literature, and macroplanning in economics [20], etc. The intuition behind the approach is that if the horizon is "long" enough to obtain a stationary behavior of the system, the moving horizon control would have good performance.…”