13.1 Principle of optimality and recursive equations

Dynamic programming is a powerful optimization technique that breaks complex problems into simpler subproblems. It uses the principle of optimality to construct optimal solutions by combining optimal subproblem solutions, making it efficient for solving various optimization challenges.

The method employs recursive equations and multi-stage decision-making processes. It's characterized by optimal substructure, overlapping subproblems, and stage-wise decomposition, enabling efficient problem-solving in fields like resource allocation, inventory management, and path finding.

Understanding Dynamic Programming

Principle of optimality in dynamic programming

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• Optimal policy property ensures remaining decisions form optimal policy regardless of initial state and decision
• Breaks down complex problems into simpler subproblems enables efficient problem-solving
• Recursive algorithms solve optimization problems by leveraging subproblem solutions
• Constructs optimal solutions combining optimal subproblem solutions (knapsack problem, shortest path)

Recursive equations for optimization

• General structure: $f_n(s_n) = \text{opt} \{c_n(s_n, x_n) + f_{n+1}(s_{n+1})\}$ where $f_n(s_n)$ is optimal value function, $s_n$ is state, $x_n$ is decision variable, $c_n(s_n, x_n)$ is immediate cost/reward
• Formulation steps:

1. Identify stages, states, and decision variables (time periods, inventory levels, production quantities)
2. Define state transition function (how system evolves between stages)
3. Determine immediate cost/reward function (profits, costs associated with decisions)
4. Construct recursive equation using principle of optimality

Multi-stage decision problem solving

• Backward induction method starts from final stage, works backwards solving subproblems to build optimal solution (retirement planning, equipment replacement)
• Forward induction method starts from initial stage, uses previously computed optimal solutions for subsequent subproblems (project scheduling, production planning)
• Applications include shortest path problems (GPS navigation), resource allocation (budget distribution), inventory management (supply chain optimization)

Characteristics of dynamic programming problems

• Optimal substructure allows problem breakdown into smaller subproblems, incorporates optimal subproblem solutions (matrix chain multiplication)
• Overlapping subproblems encountered multiple times, solutions stored and reused (Fibonacci sequence calculation)
• Stage-wise decomposition divides problem into sequence of interrelated decisions (multi-period investment strategies)
• Bellman's curse of dimensionality increases computational complexity exponentially with state variables (robotics path planning)
• Deterministic problems have known decision outcomes (fixed production costs)
• Stochastic problems involve uncertainty or probability distributions (weather-dependent crop yields)