table some common classes of optimization problems name (xconstraints linear program (lpcx ax and inequality form lp cx ax quadratic program (qp ax bx dx dex convex qp ax bx dx dex convex program (xconvex {gi ( )convex{hi ( )of the form > bi ( recognizing convex optimization problems or those that can be transformed to convex optimization problems can be challenging howeveronce formulated as convex optimization problemsthese can be efficiently solved using subgradient [ ]bundle [ ]and cutting-plane methods [ lagrange function lagrangian method the main components of the lagrangian method are the lagrange multipliers and the lagrange function the method was developed by lagrange in for the optimization problem ( with only equality constraints in kuhn and tucker extended lagrange' method to inequality constraints given an optimization problem ( containing only equality constraints hi ( mthe lagrange function is defined as (xbf (xlagrange multipliers bi hi ( ) = where the coefficients {bi are called the lagrange multipliers necessary condition for point xto be local minimizer of (xsubject to the equality constraints hi ( mis (xb (xb lagrangian for some value bthe above conditions are also sufficient if (xbis convex function of given the original optimization problem ( )containing both the equality and inequality constraintsthe generalized lagrange functionor lagrangianis defined as (xabf (xk = ai gi (xm = bi hi (