Infinite Horizon Optimization

Abstract

We consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces. Examples include equipment replacement, production planning and, more generally, infinite stage mathematical programs. It is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity. Moreover, the existence of a unique algorithmically optimal (i.e. accumulation point) solution is shown to be a necessary and sufficient condition for convergence in the product topology (i.e. policy convergence) of all finite horizon optima. Under the weaker condition of Hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima, we show how to force policy convergence through a natural tie-breaking rule. Finally, a forward algorithm is presented which, in the presence of a unique infinite horizon optimum, is guaranteed to converge.

A Shadow Simplex Method for Infinite Linear Programs

“…3. Assumption 2.4 is motivated by a similar assumption in the general infinite-horizon optimization framework of [46] and other more specific work in this area [44]. It is conceptually similar to the ubiquitous assumption (see Chapter 3 of [5]) in infinite-dimensional linear programming that the costs and the variables are embedded in a (continuous) dual pair of vector spaces.…”

Duality in convex minimum cost flow problems on infinite networks and hypernetworks

“…Hypothesis H1 was inspired by a similar assumption in recent work on duality in CILPs . This type of hypotheses is standard in the literature on countably infinite mathematical programs as they ensure that the objective function in the primal problem is well‐defined and finite (). The goal is to find a flow that satisfies the supply, demand, and transshipment requirements at all nodes; abides by the arc flow capacities; and achieves this at minimum total cost.…”

Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory Costs

“…There is also a body of literature that deals with computational procedures for solving finite horizon versions of the problem Tzur 1991, Wagelmans et al 1989). Researchers have investigated forecast horizon issues for doubly infinite convex programming problems in the past (Bean and Smith 1984, Bean and Smith 1993, Bes and Sethi 1988, and Schochetman and Smith 1989, Schochetman and Smith 1992. A common assumption in most of these works is that the infinite horizon optimal solution is unique, which is difficult to verify in practice and not always met.…”