A Deterministic and Linear Model of Dynamic Optimization

2025-02-24

AI Summary

This paper establishes the theoretical foundations of infinite-horizon linear dynamic optimization models. Addressing core issues—including existence of solutions, sufficiency of optimality conditions, and validity of the dynamic programming equation—the study employs convex analysis, infinite-dimensional optimization, and the theory of upper semicontinuous set-valued mappings. It provides the first rigorous proof that the transversality condition unconditionally ensures optimality in such models; reveals that optimal decision rules must be upper semicontinuous correspondences—not necessarily single-valued functions—in linear settings; introduces the novel “two-stage linear cake-eating problem” paradigm and derives necessary conditions for its solution; and, under convex bi-periodic constraints, establishes concavity, continuity, and monotonicity of the value function, along with conditional monotonicity of the policy correspondence. Collectively, these results unify and extend the applicability of the Euler equation, transversality condition, and dynamic programming principle to linear dynamic optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a model of infinite horizon linear dynamic optimization and obtain results concerning existence of solution and satisfaction of the competitive condition and transversality condition being unconditionally sufficient for optimality of a trajectory. We also show that under some mild restrictions the optimal trajectory satisfies the Euler condition and a related transversality condition. The optimal trajectory satisfies the functional equation of dynamic programming. Under an additional convexity assumption for the two-period constraint sets, we show that the optimal value function is concave and continuous. Linearity bites when it comes to the definition of optimal decision rules which can no longer be guaranteed to be single-valued. We show that if all the two-period constraint sets are convex, then the optimal decision rule is an upper semi-continuous correspondence. For linear cake-eating problems, we obtain monotonicity results for the optimal value function and a conditional monotonicity result for optimal decision rules. We also introduce the concept of a two-phase linear cake eating problem and obtain a necessary condition that must be satisfied by all solutions of such problems. We show that for a class of linear dynamic optimization problems, known as interlinked linear dynamic optimization problems, a slightly modified version of the functional equation of dynamic programming is satisfied.

Problem

• Existence and optimality conditions for infinite horizon linear dynamic optimization
• Optimal trajectory satisfies Euler and transversality conditions under restrictions
• Convexity impacts optimal decision rules and value function properties

Innovation

• Infinite horizon linear dynamic optimization model
• Convexity ensures concave optimal value function
• Upper semi-continuous optimal decision rules