What are the advantages and disadvantages of using dynamic programming for pricing subproblems?

1 What is column generation?

Column generation is a decomposition technique that solves a linear program by generating only a subset of the variables that are relevant for the optimal solution. The original problem is reformulated as a master problem and a pricing problem. The master problem contains a restricted set of variables and constraints, while the pricing problem generates new variables that can improve the objective function value. The process iterates until no more profitable variables can be found.

2 What is dynamic programming?

Dynamic programming is a method for solving optimization problems that have a recursive structure, meaning that the optimal solution can be obtained by combining the optimal solutions of smaller and overlapping subproblems. Dynamic programming uses a table or a graph to store and reuse the results of subproblems, avoiding repeated computations and reducing the time and space complexity.

3 How can dynamic programming be used for pricing subproblems?

Pricing subproblems in column generation often have a recursive structure, especially when they involve network optimization problems such as shortest path, maximum flow, or vehicle routing. Dynamic programming can be used to find the most profitable variables to enter the master problem, by solving the subproblems efficiently and systematically. For example, in the shortest path problem, dynamic programming can find the minimum cost path from a source node to a destination node, by using the optimal costs of the subpaths from the source node to the intermediate nodes.

4 What are the advantages of using dynamic programming for pricing subproblems?

Dynamic programming for pricing subproblems offers several advantages, such as exploiting the structure and properties of the network optimization problems, reducing computational complexity and memory requirements, and providing exact solutions or lower bounds for the pricing problem. This can help optimize the pricing problem by avoiding redundant calculations and storing only the relevant information, while also improving the convergence and quality of the column generation method.

5 What are the disadvantages of using dynamic programming for pricing subproblems?

Dynamic programming for pricing subproblems has some drawbacks that may make it inapplicable or inefficient for certain types of network optimization problems, such as those with nonlinear or stochastic features, or those with multiple objectives or constraints. Additionally, this method requires a large amount of data and parameters to define and solve the subproblems, which can make implementation and debugging more complex. Furthermore, dynamic programming is susceptible to the curse of dimensionality, meaning that the number of subproblems and the size of the table or graph can increase exponentially with the size of the network or the number of variables.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?