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Constrained optimal impulse control and inventory model

A. Piunovskiy

TL;DR

This paper develops a constrained optimal impulse-control framework for deterministic systems over an infinite horizon with discounted costs, reformulated as a Markov decision process via occupation measures. It builds two pairs of convex programs (primal/dual) whose solvability and lack of duality gap are established under positivity and semicontinuity assumptions, with a Slater-type condition ensuring strong duality. The approach yields necessary and sufficient optimality conditions and connects occupation-measure solutions to induced stationary policies. The inventory-model example demonstrates a constraint-driven strategy that may involve waiting at zero stock before ordering, and recovers EOQ-type behavior in the undiscounted limit. Overall, the work provides a rigorous, LP-based toolkit for solving constrained impulsive control problems with practical applicability to inventory and similar systems.

Abstract

In this article, we consider the deterministic impulsively controlled system with infinite horizon and several discounted objective functionals. The constructed optimal control problem with functional constraints is reformulated as a Markov decision process, leading to (primal) convex and linear programs in the space of so-called occupation measures. We construct the dual programs and investigate the solvability of all the programs. Example of an inventory model illustrates the developed theory.

Constrained optimal impulse control and inventory model

TL;DR

This paper develops a constrained optimal impulse-control framework for deterministic systems over an infinite horizon with discounted costs, reformulated as a Markov decision process via occupation measures. It builds two pairs of convex programs (primal/dual) whose solvability and lack of duality gap are established under positivity and semicontinuity assumptions, with a Slater-type condition ensuring strong duality. The approach yields necessary and sufficient optimality conditions and connects occupation-measure solutions to induced stationary policies. The inventory-model example demonstrates a constraint-driven strategy that may involve waiting at zero stock before ordering, and recovers EOQ-type behavior in the undiscounted limit. Overall, the work provides a rigorous, LP-based toolkit for solving constrained impulsive control problems with practical applicability to inventory and similar systems.

Abstract

In this article, we consider the deterministic impulsively controlled system with infinite horizon and several discounted objective functionals. The constructed optimal control problem with functional constraints is reformulated as a Markov decision process, leading to (primal) convex and linear programs in the space of so-called occupation measures. We construct the dual programs and investigate the solvability of all the programs. Example of an inventory model illustrates the developed theory.
Paper Structure (8 sections, 16 theorems, 121 equations, 1 figure)

This paper contains 8 sections, 16 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary model description
  3. Reformulation as an MDP and auxiliary results
  4. First pair of convex programs
  5. Second pair of convex programs
  6. Inventory model
  7. Conclusion
  8. Statements and Declarations

Key Result

Proposition 1

Under Conditions con1 and ConstrainedPPZcondition01, the MDP is positive semicontinous in the sense that the action space $\bf B$ is compact, $Q$ is a continuous stochastic kernel, and the cost functions $\bar{C}_j(\cdot,\cdot)$ are non-negative lower semicontinuous for all $j=0,1,\ldots, J$.

Figures (1)

  • Figure 1: Optimal inventory strategy: Theorem \ref{['t8']}(a).

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Lemma 1
  • Definition 3
  • Definition 4
  • Lemma 2
  • Lemma 3
  • Corollary 1
  • Definition 5
  • ...and 17 more