Before progressing to complicated decision-making models, first consider the simple case of a single decision maker that must make the best decision. This leads to a familiar optimization problem, which is formulated as follows.
What does it mean to be the ``best'' action? If is finite, then the best action, is
There are two ways to fix this frustrating behavior. One is to require that is a closed set and is bounded (both were defined in Section 4.1). Since closed sets include their boundary, this problem will be avoided. The bounded condition prevents a problem such as optimizing , and . What is the best ? Smaller and smaller values can be chosen for to produce a lower cost, even though is a closed set.
The alternative way to fix this problem is to define and use the notion of an infimum, denoted by . This is defined as the largest lower bound that can be placed on the cost. In the case of and , this is
As a general rule, if you are not sure which to use, it is safer to write in the place were you would use . The infimum happens to yield the minimum whenever a minimum exists. In addition, it gives a reasonable answer when no minimum exists. It may look embarrassing, however, to use in cases where it is obviously not needed (i.e., in the case of a finite ).
It is always possible to make an ``upside-down'' version of an optimization problem by multiplying by . There is no fundamental change in the result, but sometimes it is more natural to formulate a problem as one of maximization instead of minimization. This will be done, for example, in the discussion of utility theory in Section 9.5.1. In such cases, a reward function, , is defined instead of a cost function. The task is to select an action that maximizes the reward. It will be understood that a maximization problem can easily be converted into a minimization problem by setting for all . For maximization problems, the infimum can be replaced by the supremum, , which is the least upper bound on over all .
For most problems in this book, the selection of an optimal in a single decision stage is straightforward; planning problems are instead complicated by many other aspects. It is important to realize, however, that optimization itself is an extremely challenging if and are complicated. For example, may be finite but extremely large, or may be a high-dimensional (e.g., 1000) subset of . Also, the cost function may be extremely difficult or even impossible to express in a simple closed form. If the function is simple enough, then standard calculus tools based on first and second derivatives may apply. It most real-world applications, however, more sophisticated techniques are needed. Many involve a form of gradient descent and therefore only ensure that a local minimum is found. In many cases, sampling-based techniques are needed. In fact, many of the sampling ideas of Section 5.2, such as dispersion, were developed in the context of optimization. For some classes of problems, combinatorial solutions may exist. For example, linear programming involves finding the min or max of a collection of linear functions, and many combinatorial approaches exist [259,264,664,731]. This optimization problem will appear in Section 9.4.
Given the importance of sampling-based and combinatorial methods in optimization, there are interesting parallels to motion planning. Chapters 5 and 6 each followed these two philosophies, respectively. Optimal motion planning actually corresponds to an optimization problem on the space of paths, which is extremely difficult to characterize. In some special cases, as in Section 6.2.4, it is possible to find optimal solutions, but in general, such problems are extremely challenging. Calculus of variations is a general approach for addressing optimization problems over a space of paths that must satisfy differential constraints ; this will be covered in Section 13.4.1.
Steven M LaValle 2020-08-14