Estimates of the Duality Gap in Nonconvex Optimization

We associate with every real-valued function a number which measures its lack of convexity. This number is used to estimate the duality gap in optimization problems where the criterion and/or the constraints are nonconvex. It is shown that when the number of variables is very great with respect to the number of constraints, this duality gap is small in relative value. Approximating in this way problems where the criterion and constraints are given as integrals, we show that the duality gap vanishes.