A note on error bounds for convex and nonconvex programs

Given a single feasible solution xF and a single infeasible solution xl of a mathematical program, we provide an upper bound to the optimal dual value. We assume that xF satisfies a weakened form of the Slater condition. We apply the bound to convex programs and we discuss its relation to Hoffman-like bounds. As a special case, we recover a bound due to Mangasarian on the distance of a point to a convex set specified by inequalities.