Solution of the Stationarity Equations

The KKT conditions are a set of variables + Equations + inequai nonlinear equations (NLEs) in 

V is the vector of guessed values at the kth iteration (the initial guesses when k = 0), is the vector of residuals at is the Jacobian at is the vector of 

When the Newton-Raphson method is applied to solve the KKT Eqs. (18.24)-(18.26), and Eqs. (18.29) and (18.30) can be rewritten in terms of these variables. This was accomplished by Jirapongphan (1980) who showed that beginning with the vector of guesses, one iteration of the Newton-Raphson method is equivalent to solving the following quadratic program (QP)  

Algorithms for the solution of quadratic programs, such as the Wolfe (1959) algorithm, are very reliable and readily available. Hence, these have been used in preference to the implementation of the Newton-Raphson method. For each iteration, the quadratic objective function is minimized subject to linearized equality and inequality constraints. Clearly, the most computationally expensive step in carrying out an iteration is in the evaluation of the Lapla-cian of the Lagrangian, V xL x , X which is also the Hessian matrix of the La-grangian that is, the matrix of second derivatives with respect to X . 

To circumvent this calculation, Powell (1977) used the Broyden, Fletcher, Goldfarb, Shanno (BFGS) quasi-Newton method to approximate X . This saves con- 

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