Finite dimensional approximation in infinite dimensional mathematical programming

Abstract

We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the first N variables and N constraints of (P). Viewing the surplus vector variable associated with the Nth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value of N sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.

Existence and Discovery of Average Optimal Solutions in Deterministic Infinite Horizon Optimization

“…Solutions that have the property of optimally reaching each of the states through which they pass are called efficient solutions (Schochetman and Smith [1992], Ryan, Bean, and Smith [1992]). Such a solution offers little opportunity for retrospective regret in that at every state along its path, there was no better way to reach that state.…”

“…In the terminology of Schochetman and Smith [1989], θ is a uniqueness point for F and {λ(F n )} is a nearest-point selection from the F n defined by θ. The result then follows from Theorem 3.4 of Schochetman and Smith [1989].…”

“…Infinite horizon optimization at its most fundamental level is the problem of selecting an infinite sequence of decisions which promises to minimize the associated costs incurred over an unbounded horizon (Bean and Smith [1984], Schochetman and Smith [1989]). One of the key difficulties inherent in this task is the dilemma of evaluating an infinite stream of costs whose cumulative value will typically diverge.…”

“…However, stationary strategies need no longer be optimal here. Instead, we generalize by restricting consideration to efficient strategies (see Ryan, Bean, and Smith [1992], Schochetman and Smith [1992] for a similar concept in the discounted case). A strategy is efficient if it is average-cost (or equivalently undiscounted total cost) optimal to each of the states through which it passes.…”

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.…”