Lipschitz and Continuous Optimization

Abstract

In this chapter, we discuss global optimization problems where the functions in?volved are Lipschitz—continuous or have a related property on certain subsets M c ℝⁿ. Section 1 presents a brief introduction into the most often treated univariate case. Section 2 is devoted to branch and bound methods. First it is shown that the well-known univariate approaches can be interpreted as branch and bound methods. Next, several extensions of univariate methods to the case of n dimensional problems with rectangular feasible sets are discussed. Then it is recalled from Chapter IV that very general Lipschitz optimization problems and also very general systems of equations and (or) inequalities can be solved by means of branch and bound techniques. As an example of Lipschitz optimization, the problem of minimizing a concave function subject to separable indefinite quadratic constraints is discussed in some detail. Finally, the concept of Lipschitz functions is extended to so-called functions with concave minorants.