Mathematical Programming 37 (1987) 169-183 North-Holland NONDIFFERENTIABLE REVERSE CONVEX PROGRAMS AND FACETIAL CONVEXITY CUTS VIA A DISJUNCTIVE CHARACTERIZATION S. SEN Department of Systems and lndustiral Engineering, University o/'Arizona, Tucson, A Z 85721, USA Hanif D. SHERAII Department o[" Industrial Engineering and Operations Research, Virginia Poh,technic Institute and State University, Blacksburg, VA 24061, USA Received 15 September 1985 Revised manuscript received 8 September 1986 Disjunctive Programs can often be transcribed as reverse convex constrained problems with nondilterentiable constraints and unbounded feasible regions. We consider this general class of nonconvex programs, called Reverse Convex Programs I RCPt, and show that under quite general conditions, the closure of the convex hull of the feasible region is polyhedral. This development is then pursued from a more constructive standpoint, in that, for certain special reverse convex sets, we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set. When interpreted in the context of convexity/intersection cuts, this provides the capability of generating any (negative edge extensionl facet cut. Although this characterization is more clarifying than computationally oriented, our development shows that if certain bounds are available, then convexity/intersection cuts can be strengthened relatively inexpensively. Key words: Disjunctive programming, convexity/intersection cuts, facet inequalities, reverse convex programs. 1. Introduction The paper considers mathematical programs in the presence of reverse convex constraints. By a reverse convex constraint, we mean a constraint g(x)<~ O, where g(.) is a real valued concave function. The earliest work on problems with such constraints may be traced to Rosen (1966), Avriel and Williams (1970), Dembo (1972) and Avriel (1973). Most of this work was motivated by complementary geometric programs. Specialized reverse convex programs have been investigated by Bansal and Jacobsen (1975a, b) and by Hillestad (1975). The motivation for these papers were economic models with budget constraints in the presence of economies of scale. Perhaps the most signficant contributions for this class of problems appear in Hillestad and Jacobsen (1980a, b), wherein a characterization of optimal solutions and global algorithms for single and multiple reverse convex constrained programs are provided. Assuming differentiability of constraints and 169