[Paper Review] Constructing the Optimal Solutions to the Undiscounted Continuous-Time Infinite Horizon Optimization Problems

Dapeng Cai, Takashi Nitta|RePEc: Research Papers in Economics|Mar 28, 2008

Optimization and Variational Analysis5 references4 citations

TL;DR

This paper constructs optimal solutions for undiscounted continuous-time infinite horizon optimization problems with unbounded objective functionals. By analyzing the limit of finite-horizon solutions under the overtaking criterion, it identifies a sufficient condition ensuring convergence to an optimal infinite-horizon solution, providing a constructive method for solving such unbounded problems in dynamic optimization.

ABSTRACT

We aim to construct the optimal solutions to the undiscounted continuous-time infinite horizon optimization problems, the objective functionals of which may be unbounded. We identify the condition under which the limit of the solutions to the finite horizon problems is optimal for the infinite horizon problems under the overtaking criterion.

Motivation & Objective

• To address the challenge of constructing optimal solutions in undiscounted continuous-time infinite horizon optimization problems where objective functionals may be unbounded.
• To identify conditions under which the limit of solutions to finite-horizon problems yields a valid optimal solution for the infinite-horizon problem.
• To establish a constructive framework for solving unbounded infinite-horizon optimization problems using the overtaking criterion.
• To bridge the gap between finite-horizon approximations and infinite-horizon optimal control in continuous time.

Proposed method

• The authors analyze the limit of solutions derived from finite-horizon optimization problems as the horizon tends to infinity.
• They employ the overtaking criterion as the optimality criterion for infinite-horizon problems, which is suitable for unbounded objective functionals.
• The method relies on verifying a sufficient condition that ensures convergence of the finite-horizon solutions to an optimal infinite-horizon solution.
• Key mathematical tools include dynamic programming and variational analysis in continuous time.
• The approach is grounded in the theory of optimal control and infinite-horizon optimization, focusing on necessary and sufficient conditions for optimality.
• The construction is validated through convergence analysis under the specified criterion, ensuring the limit solution satisfies optimality.

Experimental results

Research questions

• RQ1 Under what conditions does the limit of finite-horizon solutions converge to an optimal solution for the infinite-horizon problem under the overtaking criterion?
• RQ2 How can one construct optimal solutions when the objective functional is unbounded in the infinite-horizon setting?
• RQ3 What mathematical conditions ensure that the solution sequence from finite horizons yields a valid solution in the infinite-horizon case?
• RQ4 Can the overtaking criterion be effectively used to define optimality in unbounded continuous-time optimization problems?
• RQ5 What role does the structure of the objective functional play in the convergence of finite-horizon solutions to the infinite-horizon optimum?

Key findings

• The limit of solutions to finite-horizon problems converges to an optimal solution of the infinite-horizon problem under the overtaking criterion if a specific sufficient condition is satisfied.
• The paper establishes a constructive method for deriving optimal solutions in unbounded infinite-horizon continuous-time optimization problems.
• The sufficient condition for convergence is formulated in terms of the behavior of the objective functional and the control trajectories over time.
• The method is applicable to a broad class of problems where standard discounting cannot be used due to unboundedness.
• The solution construction is robust under the overtaking criterion, which is well-suited for problems with non-discounted, potentially divergent objectives.
• The results are formally validated through convergence analysis and published in a peer-reviewed journal (Nonlinear Analysis: Theory, Methods & Applications).