Filter SLP Method To Solve The Problem Of Large-scale Non-smooth Constraints

Nonsmooth optimization problems are conventionally solved by introducing penalty function, which simultaneously makes hard the choice of suitable penalty parameters and further big the condition number of the Hessian matrix that leads to instability in the computation. This paper takes new technique filter and new approaches for solving nonsmooth optimization problems without penalty function. A new kind of algorithms is established under consideration of following three developments.(1) Take SLP subproblem instead of SQP subproblem. This paper uses linear model to approach constrained conditions, further solves LP subproblems instead of QP subproblems of conventional computation. That leads an effective estimation for searching iterative direction, and especially inexpensive computation for large scale and complicated optimization.(2) Update the bundle set by restricting the number of subgradients. One side, the bundle is accumulated by taking a number of null-steps at which the current point remains unchanged but new subgradient information is added to the bundle. The null-steps can enhance the linear program at the next iteration and promote global convergence. On the other side, the bundle is updated by setting a suitable fixed angle between the subgradient of auxiliary points and the subgradient of the current serious point. This approach controls the numbers of subgradients in bundle and decreases invalidity about computation of subgradients in the bundle.(3) This paper discusses the details on the feasible restoration of SLP subproblems, which in theory supports the implementation of this algorithm.Combination of the three things above leads our new algorithm, and its global convergence is proved.