strong duality linearly constrained problemsif the primal is infeasible (does not have solution satisfying the constraints)then the dual is either infeasible or unbounded converselyif the dual is infeasible then the primal has no solution of crucial importance is the strong duality theoremwhich states that for convex programs ( with linear constrained functions hi and gi the duality gap is zeroand any xand (absatisfying the kkt conditions are (globalsolutions to the primal and dual programsrespectively in particularthis holds for linear and convex quadratic programs (note that not all quadratic programs are convexfor convex primal program with objective and constraint functionsthe lagrangian dual function ( can be obtained by simply setting the gradient (with respect to xof the lagrangian (xabto zero one can further simplify the dual program by substituting into the lagrangian the relations between the variables thus obtained furtherfor convex primal problemif there is strictly feasible point (that isa feasible point satisfying all of the inequality constraints with strict inequality)then the duality gap is zeroand strong duality holds this is known as slater' condition [ page the lagrange dual problem is an important example of saddle-point problem or minimax problem in such problems the aim is to locate point (xyx that satisfies sup inf (xyinf (xyf (xysup (xyinf sup (xyyy xx the equation xx yy sup inf (xyinf sup (xyyy xx minimax xx yy xx yy is known as the minimax equality other problems that fall into this framework are zerosum games in game theorysee also [ for number of combinatorial optimization problems that can be viewed as minimax problems numerical root-finding and minimization in order to minimize function rn one may solve ( -norm which gives stationary point of as consequenceany technique for root-finding can be transformed into an unconstrained optimization method by attempting to locate roots of the gradient howeveras noted in section not all stationary points are minimaand so additional information (such as is contained in the hessianif is needs to be considered in order to establish the type of stationary point alternativelya root of continuous function rn rn may be found by minimizing the norm of (xover all xthat isby solving min ( )with ( :kg( ) where for the -norm of [ yn ]is defined as kyk :  = |yi  / henceany (un)constrained optimization method can be transformed into technique for locating the roots of function