Question

Find an optimization problem in which the principle of optimality does not apply and therefore that the optimal solution cannot be obtained using dynamic programming. Justify your answer.

Short answer

A chess game is an example where the Principle of Optimality doesn't apply. Here, every move is highly dependent not only on the current scenario but also on potential future scenarios. This dynamic nature of decision making in the game breaks the Principle of Optimality, making it not suitable for being solved by dynamic programming.

Step-by-step solution

Explanation of the Principle of Optimality

The Principle of Optimality applies when an optimal strategy can be pieced together from optimal solutions to subproblems. This effectively means: If an optimal solution contains within it optimal solutions to subproblems, then the Principle of Optimality applies. For example, the problem of finding the shortest path from A to B, assuming a path from A to B goes via C, the shortest path from A to B is the shortest path from A to C plus the shortest path from C to B.

Identify a problem where the Principle of Optimality does not apply

Contrarily, if an optimal solution contains within it subproblems for which there exist solutions that are not optimal, then the principle does not apply. A typical example of a situation where this principle is not working is a problem that involves a non-stationary decision, which means that the decision at a certain step depends not only on the state and control at that step, but also on the time or stage itself. Say, for example, in a chess game where the best move could worsens based on the subsequent decisions of the opponent.

Justification

In this example, the Principle of Optimality doesn't hold because making the best decision now doesn't necessarily contribute to an optimal solution overall. This is because the advantage of every possible move in chess isn’t just dependent on the current scenario, but also on every potential future scenario. Therefore, this breaks the Principle of Optimality and makes it incompatible with dynamic programming, as dynamic programming requires the Principle of Optimality to function.

Key concepts

Dynamic Programming

Dynamic programming is a problem-solving approach that breaks a complex problem into simpler subproblems. It solves each of these subproblems just once and stores their solutions. The main idea is that we can use the solutions to smaller problems to construct an optimal solution to the overall problem.

This technique is much like building a puzzle. We solve smaller pieces first and then combine these solutions to form a bigger picture. Key aspects include:

• Overlapping Subproblems: These are smaller problems that recur more than once.
• Optimal Substructure: This property ensures that optimal solutions of subproblems lead to an optimal solution for the overall problem.

Dynamic programming is primarily suited to problems that exhibit these properties, allowing us to solve them efficiently and effectively.

Optimal Solutions

An optimal solution is essentially the best possible solution to a problem, beating all other feasible solutions. It’s like the gold medal winner in solving a particular problem. In mathematical optimization, finding this optimal solution often involves several steps:

• Define the Objective: Clearly establish what needs to be optimized.
• Identify Constraints: Recognize any limitations or rules that must be adhered to in the solution.
• Analyze Problem Structure: Determine whether the problem exhibits an optimal substructure property suitable for techniques like dynamic programming.

However, reaching an optimal solution isn’t always straightforward. Sometimes, due to external factors, the best decision at a moment might not contribute to the overall best solution, particularly in non-stationary decision problems.

Non-stationary Decision Problems

Non-stationary decision problems are unique because the decision-making environment changes over time. Here, the best decision today might not remain the best in the future. These changes can be due to external factors like increased competition or internal factors like resource depletion.

In these problems:

• Decisions depend on current and future states of the system.
• Time plays a critical role as the optimal choice can vary with different stages.

A classic example is playing chess, where the best move doesn't just depend on the current state of the board but also on an unpredictable future influenced by the opponent's responses. Such problems don't allow the principle of optimality, which dynamic programming depends on, to apply effectively. Therefore, they often require different strategies and models for optimization, considering multiple timelines and scenarios.