theorem karush-kuhn-tucker (kktconditions necessary condition for point xto be local minimizer of (xin the optimization problem ( is the existence of an aand bsuch that (xab (xab gi ( * * gi ( ki ki for convex programs we have the following important results [ ] every local solution xto convex programming problem is global solution and the set of global solutions is convex ifin additionthe objective function is strictly convexthen any global solution is unique for strictly convex programming problem with objective and constraint functionsthe kkt conditions are necessary and sufficient for unique global solution duality the aim of duality is to provide an alternative formulation of an optimization problem which is often more computationally efficient or has some theoretical significance (see [ page ]the original problem ( is referred to as the primal problem whereas the reformulated problembased on lagrange multipliersis called the dual problem duality theory is most relevant to convex optimization problems it is well known that if the primal optimization problem is (strictlyconvex then the dual problem is (strictlyconcave and has (uniquesolution from which the (uniqueoptimal primal solution can be deduced the lagrange dual program (also called the wolfe dualof the primal program ( )ismax (aba, subject toa where lis the lagrange dual functionl(abinf (xab)xx ( giving the greatest lower bound (infimumof (xabover all possible it is not difficult to see that if is the minimal value of the primal problemthen (ab for any and any this property is called weak duality the lagrangian dual program thus determines the best lower bound on if dis the optimal value for the dual problem then the difference dis called the duality gap the duality gap is extremely useful for providing lower bounds for the solutions of primal problems that may be impossible to solve directly it is important to note that for primal dual lagrange dual program