Nondifferentiable Optimization

Of recent coinage, the term "nondifferentiable optimization" (NDO) covers a spectrum of problems related to finding extremal values of nondifferentiable functions. Problems of minimizing nonsmooth functions arise in engineering applications as well as in mathematics proper. The Chebyshev approximation problem is an ample illustration of this. Without loss of generality, we shall consider only minimization problems. Among nonsmooth minimization problems, minimax problems and convex problems have been studied extensively ([31], [36], [57], [110], [120]). Interest in NDO has been constantly growing in recent years (monographs: [30], [81], [127] and articles and papers: [14], [20], [87]-[89], [98], [130], [135], [140]-[142], [152], [153], [160], all dealing with various aspects of non­ smooth optimization). For solving an arbitrary minimization problem, it is neces­ sary to: 1. Study properties of the objective function, in particular, its differentiability and directional differentiability. 2. Establish necessary (and, if possible, sufficient) condi­ tions for a global or local minimum. 3. Find the direction of descent (steepest or, simply, feasible--in appropriate sense). 4. Construct methods of successive approximation. In this book, the minimization problems for nonsmooth func­ tions of a finite number of variables are considered. Of fun­ damental importance are necessary conditions for an extremum (for example, [24], [45], [57], [73], [74], [103], [159], [163], [167], [168].