suppose that ( has inequality constraints only barrier functions are an important example of penalty functions that can handle inequality constraints the prototypical example is logarithmic barrier function which gives the unconstrained optimizatione (xf (xn ln(- ( )) = such that the minimizer of tends to the minimizer of as direct minimization of via an unconstrained minimization algorithm is frequently too difficult insteadit is common to combine the logarithmic barrier function with the lagrangian method as follows the idea is to introduce nonnegative auxiliary or slack variables sk that satisfy the equalities (xs for all these equalities ensure that the inequality constraints are maintainedg ( - for all theninstead of the unconstrained optimization of we consider the unconstrained optimization of the lagrangianl(xsbf (xn barrier functions ln = ( (xs )( = where and are the lagrange multipliers for the equalities ( )+ observe how the logarithmic barrier function keeps the slack variables positive in additionwhile the optimization of is over dimensions (recall that rn )the optimization of the lagrangian function is over dimensions despite this enlargement of the search space with the variables and bthe optimization of the lagrangian is easier in practice than the direct optimization of example (interior-point method for nonnegativityone of the simplest and most common constrained optimization problems can be formulated as the minimization of (xsubject to nonnegative xthat ismin > (xin this casethe lagrangian with logarithmic barrier ( isx (xsbf (xn ln sk ( xk the kkt conditions in theorem are necessary condition for minimizerand yield the nonlinear system for [xsb] (xb - / - where / is shorthand notation for column vector with components { / to solve this systemwe can use newton' method for root finding (seefor examplealgorithm )which requires formula for the matrix of jacobi of herethis ( nx ( nmatrix ish - jl (xsbo - slack variables