Studies On Nonsmooth Constrained Optimization Algorithms Based On Bundle Modification Strategy

Nonsmooth optimization as an important branch of the optimization re-search, not only has important theoretical significance, but also is widely used in fields of optimal control,engineering design and image processing ect. The core issue on nonsmooth optimization research is aimed at designing various of fast and efficient numerical methods. This thesis studies new nu-merical algorithms for solving nonsmooth constrained optimization problem-s by combining the bundle modification strategy, feasible direction method, phase I-phase II method and subgradient aggregation technique.Firstly, a feasible direction algorithm for solving nonsmooth constrained optimization problem is proposed by designing a suitable new bundle modifi-cation strategy and combining with the idea of feasible direction method. The algorithm can generate feasible iteration points and ensure that the objective function value is monotonically nonincreasing. When the stability center is updated, the bundle modification strategy is executed, and the algorithm will generate auxiliary iteration points which descend faster or have better feasi-bility as alternatives to the corresponding points in bundle, aiming at getting a better bundle set. In addition, global convergence of the algorithm is proven.Secondly, with the number of iterations increases, in order to avoid the numerical calculation difficulties caused by the gradually expanding of the search direction finding sub-problem, this thesis puts forward a feasible di-rection algorithm with subgradient aggregation technique. By introducing the subgradient aggregation technique, the algorithm aggregates the gradients in bundle set, thus the number of constraints of the direction finding subproblem is greatly reduced, followed by significant reduction of the computation. The algorithm still has global convergence.Once again, to overcome the weakness of feasible direction method which requires a feasible initial point, with combination of the idea of phase I-phase II method, this thesis further works on feasible direction algorithm to promote a phase I-phase II algorithm for solving nonsmooth constrained optimization problem. The algorithm can accept infeasible initial points, and in phase I, the algorithm can automatically generate a feasible iteration point, once a feasible iteration point is found, the algorithm enters phase II, executing the feasible direction algorithm to get an optimal solution. The algorithm pos-sesses global convergence.Finally, the numerical experiments are carried out to show that the pro-posed algorithms in this thesis are stable and effective.