A convex optimization problem can have only linear equality constraint functions. In some special cases, however, it is possible to handle convex equality constraint functions, i. e., constraints of the form g(x) = 0, where g is convex. We explore this idea in this problem. Consider the optimization problem minimize f0(x) subject to fi(x) 0; i = 1; : : : ;m h(x) = 0; (4.65) where fi and h are convex functions with domain Rn. Unless h is ane, this is not a convex optimization problem. Consider the related problem minimize f0(x) subject to fi(x) 0; i = 1; : : : ; m; h(x) 0; (4.66) where the convex equality constraint has been relaxed to a convex inequality. This problem is, of course, convex. Now suppose we can guarantee that at any optimal solution x? of the convex problem (4.66), we have h(x?) = 0, i. e., the inequality h(x) 0 is always active at the solution. Then we can solve the (nonconvex) problem (4.65) by solving the convex problem (4.66). Show that this is the case if there is an index r such that f0 is monotonically increasing in xr f1; : : : ; fm are nonincreasing in xr h is monotonically decreasing in xr.