Algorithms for large-scale nonlinear optimization.

摘要

We investigate two algorithmic approaches for the efficient and robust solution of large-scale, generally constrained, nonlinear optimization problems.; The first part of this thesis describes an active-set algorithm for large-scale non-linear programming. The step computation is performed in two stages. In the first stage a linear program (LP) is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the L₁ penalty function. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the LP. The EQP is solved by means of a projected conjugate gradient method. Sesqui-linear techniques are explored where the term sesqui-linear refers to the fact that quadratic information about the problem is used, together with a simplex iteration, for identifying the active set. Numerical experiments are presented illustrating the performance of the algorithm.; In the second part of this thesis, a slack-based feasible interior-point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust-region methods require special attention. It is shown how the Cauchy point, which is often computed in trust-region methods, must be modified so that the feasible method is effective for problems containing both equality and inequality constraints. The relationship between slack-based methods and traditional feasible methods is discussed. Numerical results showing the relative performance of feasible versus infeasible interior-point methods are presented.