Theorem 66 Assume that f and ci, i=1, ... ,p are continuous differentiable convex functions on R. If (;, ) ER" x R$ ~ R satisfies the KKT conditions for (25), then o is a solution of the primal problem (25); (, ) is a solution of the dual problem (26); there holds sup h(2, 4) = h(, ) = L (;, ) = f(7) = inf f(x). (2,4)ER? xRM XEN Consequently, the duality gap is zero. Theorem 67 Assume that of and ci, i=1, ... ,p are continuous differentiable convex functions on R, and the Slater's condition is satisfied; the primal problem (25) has a solution **; the dual problem (26) has a solution (, ) and infxern L (x;, ) is attained at a ox+L (x;, ) is a strictly convex function on R". Then 7 = ** is the unique solution of (25) and f(x) = L (7;, ). Problem 6 (16 points) Consider the optimization problem minimize " P2 subject to a?r