This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ` 1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program.

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [9]. The step computation is performed in two stages. In the rst stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the `1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active atthesolution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1] test set.

Standard global convergence proofs are examined to determine why some algorithms perform better than other algorithms. We show that relaxing the conditions required to prove global convergence can improve an algorithm's performance. Further analysis indicates that minimizing an estimate of the distance to the minimum relaxes the convergence conditions in such a way as to improve an algorithm's convergence rate. A new line-search algorithm based on these ideas is presented that does not force a reduction in the objective function at each iteration, yet it allows the objective function to increase during an iteration only if this will result in faster convergence. Unlike the nonmonotone algorithms in the literature, these new functions dynamically adjust to account for changes between the influence of curvature and descent. The result is an optimal algorithm in the sense that an estimate of the distance to the minimum is minimized at each iteration. The algorithm is shown to be well defi...