Generic Approaches to Optimization

Abstract

A smuggler in the mountainous region of Profitania has n items in his cellar. If he sells an item i across the border, he makes a profit p i. However, the smuggler's trade union only allows him to carry knapsacks with a maximum weight of M. If item i has weight w i , what items should he pack into the knapsack to maximize the profit from his next trip? This problem, usually called the knapsack problem, has many other applications. The books [122, 109] describe many of them. For example, an investment bank might have an amount M of capital to invest and a set of possible investments. Each investment i has an expected profit p i for an investment of cost w i. In this chapter, we use the knapsack problem as an example to illustrate several generic approaches to optimization. These approaches are quite flexible and can be adapted to complicated situations that are ubiquitous in practical applications. In the previous chapters we have considered very efficient specific solutions for frequently occurring simple problems such as finding shortest paths or minimum spanning trees. Now we look at generic solution methods that work for a much larger range of applications. Of course, the generic methods do not usually achieve the same efficiency as specific solutions. However, they save development time. Formally, an optimization problem can be described by a set U of potential solutions , a set L of feasible solutions, and an objective function f : L → Ê. In a maximization problem, we are looking for a feasible solution x * ∈ L that maximizes the value of the objective function over all feasible solutions. In a minimization problem , we look for a solution that minimizes the value of the objective. In an existence problem, f is arbitrary and the question is whether the set of feasible solutions is nonempty.