Summary: In a recent work [M. Tawarmalani and N. V. Sahinidis (2000), Convex Extensions and Convex Envelopes of l.s.c. Functions. Math. Program
(http://archimedes.scs.uiuc.edu/papers/extensions.pdf)] introduced the concept of convex extensions for lower semi-continuous functions and studied their properties. In this work, we present new techniques for constructing convex and concave envelopes of nonlinear functions using the theory of convex extensions. In particular, we develop the convex envelope and concave envelope of \(z=x/y\) over a hypercube. We show that the convex envelope is strictly tighter than previously known convex underestimators of \(x/y\). We then propose a new relaxation technique for fractional programs which includes the derived envelopes. The resulting relaxation is shown to be a semidefinite program. Finally, we derive the convex envelope for a class of functions of the type \(f(x ,y)\) over a hypercube under the assumption that \(f\) is concave in \(x\) and convex in \(y\).