Introduction to minimax
We consider the problem of minimizing a nondifferentiable function, defined by the maximum of an inner function. We refer to this objective function as the max-function. In practical applications of minimax, the max-function takes the form of a maximized error, or disutility, function. For example, portfolio selection models in finance can be formulated in a scenario-based framework where the max-function takes the form of a maximized risk measure across all given scenarios. To solve the minimax problem, algorithms requiring derivative information cannot be used directly and the usual methods that do not require gradients are inadequate for this purpose. Instead of gradients, we need to consider generalized gradients or subgradients to formulate smooth methods for nonsmooth problems.
The minimax notation is introduced with relevant concepts in convex analysis and nonsmooth optimization. We consider the basic theory of continuous minimax, characterized by continuous values of maximizing and minimizing variables, and associated optimality conditions. These need to be satisfied at the solution generated by all algorithms. The problem of discrete minimax, with continuous minimization but discrete maximization variables, and related conditions are considered in Chapters 6 and 7.
Equation and section numbering follow the following rule: (1.2.3) refers to Equation 3 in Chapter 1, Section 2. In Chapter 1 only, this is referred to as (2.3), elsewhere as (1.2.3). Chapter 1, Section 2 is referred to in Chapter 1 only as Section 2, elsewhere as Section 1.2.
In this book, we consider strategies, algorithms, properties and applications of worst-case design problems. When taking decisions under uncertainty, it is desirable to evaluate the best policy in view of the worst-case uncertain effect. Essentially, this entails minimax formulations in which the best decision and the worst case is determined simultaneously. In this sense, optimality is defined over all possible values of the uncertain effects as opposed to certain likely realizations. Worst-case design is useful in all disciplines with rival representations of the same system. For example, in economics Chow