Essentially finite algorithms for minimizing a class of convex piecewise polynomial functions

Abstract

We consider minimax problems of the type min _tε[a,b] max _i p _t (t), where thep _i (t) are real polynomials which are convex on an interval [a, b]. Here, the function max _i p _i (t) is a convex piecewise polynomial function. The main result is an algorithm yielding the minimum and the minimizing point, which is efficient with respect to the necessary number of computed polynomial zeros. The algorithms presented here are applicable to decision problems with convex loss. While primarily of theoretical interest, they are implementable for quadratic loss functions with finitely many states of nature. An example of this case is given. The algorithms are finite when polynomial zeros need not be approximated.