ABSTRACT
In this chapter I take up again the state variable, addressing it here from a slightly different angle. Whereas in the previous chapter my main concern was to illustrate how to track down the state, here I focus on the question of how to fend off the adverse effects of a non-Markovian objective function and global constraints on it. As we have seen, such an objective function and/or constraints tend to enlarge the state space and thereby trigger the Curse of Dimensionality. So, in this chapter I delineate three schemes whose express aim is to avert this ill. The distinctive characteristic of these schemes is that they identify a structural feature in the dimensionality-prone problem of interest — the target problem — which allows tackling it through a surrogate problem, namely a simplified version of the former, which involves an exogenous parameter. The underlying thesis here is that for a certain value of the parameter in
question, the optimal solution obtained for the surrogate parametric problem is also optimal for the target problem. The resulting solution procedure consists of an algorithm which searches for the optimal solution for the target problem through an iterative solution of the parametric problem for various values of the exogenous parameter. These parametric schemes (hybrid algorithms) amount to a collaborative
effort between dynamic programming and
· Fractional programming (Appendix B). · Composite concave programming (Appendix C ). · Lagrange multiplier methods. The first two are designed to handle problems with non-Markovian ob-
jective functions, and the third is designed for problems subject to global constraints.
