A contraction approach to dynamic optimization problems.

Area of Science:

• Optimization Theory
• Dynamic Systems Analysis
• Mathematical Economics

Background:

• Infinite-horizon optimization problems with time-varying parameters are common in economics and resource management.
• Classical Bellman problems often assume autonomous systems, limiting their application to time-dependent scenarios like seasonality.
• Existing methods struggle to rigorously handle periodic non-autonomy in discrete dynamic optimization.

Purpose of the Study:

• To develop a novel approach for solving infinite-horizon, multidimensional optimization problems with finite periodicity.
• To extend the framework of contraction problems to incorporate periodic non-autonomy.
• To provide a rigorous method for analyzing dynamic optimization problems with seasonality and certain dynamic games.

Main Methods:

• Formulating the periodic optimization problem as a set of coupled equations.
• Utilizing a vector-valued value function and an iterative process.
• Classifying the problem within a general framework of contraction problems with unique solutions.

Main Results:

• The periodic optimization problem is shown to be a special case of contraction problems.
• Solutions are obtained via an iterative approach using a vector-valued value function.
• An infinite-horizon resource management example with periodic pricing demonstrates differing optimal exploitation levels and convergence to a limit cycle.

Conclusions:

• The proposed contraction approach offers a rigorous extension of the Bellman problem for periodic dynamic optimization.
• This method effectively handles seasonality and certain dynamic games by addressing non-autonomy.
• The findings are applicable to resource management and other fields requiring dynamic decision-making under periodic conditions.