JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: VoL 46, No, l, MAY 1985 Lower Subdifferentiable Functions and Their Minimization by Cutting Planes 1 F. PLASTRIA 2 Communicated by O. L. Mangasarian Abstract. This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined. Key Words. Lower subgradients, boundedly lower subdifferentiable functions, quasiconvex functions, Lipschitz functions, cutting plane algorithm. 1. Introduction One of the earliest methods for nonlinear optimization was the cutting plane method of Kelley (Ref. 1) and Cheney and Goldstein (ReL 2). At each iteration of this algorithm, a linear program is solved obtained by the linearization of the nonlinear function(s) defining the problem. The exact- ness and convergence of these algorithms were only ensured for convex functions. It is indeed only possible to construct linear lower approximating functions at each point when the source function is convex. However, when a minimization problem is concerned, the only points of interest are those where the objective function is less than the values observed previously. Based on this idea, the notion of lower subdifferential functions arises naturally as those functions that can be approximated 1The author thanks the referees for several constructive remarks. z Assistant Lecturer, Centrum voor Statistiek en Operationeel Onderzoek, Vfije Universiteit Brussel, Brussels, Belgium. 37 0022-3239/85/0500-0037504,50/0 0 1985 Plenum Publishing Corporation