An active-set algorithm for solving large-scale nonsmooth optimization models with box constraints.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Computational Mathematics

Background:

• Active set algorithms are highly effective for smooth, box-constrained optimization problems.
• Nonsmooth optimization problems present unique challenges not addressed by standard smooth optimization techniques.

Purpose of the Study:

• To extend the active set method to handle nonsmooth box-constrained optimization problems.
• To improve computational efficiency for large-scale nonsmooth optimization.

Main Methods:

• Utilized Moreau-Yosida regularization to transform nonsmooth objective functions into smooth ones.
• Integrated a limited-memory BFGS method to reduce computational load.
• Developed an algorithm ensuring feasible iterates and a decreasing objective function sequence.

Main Results:

• The proposed algorithm allows for rapid changes in the active set.
• Subproblems are reduced to lower-dimensional linear systems.
• Demonstrated global convergence under specific conditions.
• Numerical results confirm effectiveness for large-scale nonsmooth problems (up to 5,000 variables).

Conclusions:

• The extended active set method effectively addresses nonsmooth box-constrained optimization.
• The integration of Moreau-Yosida regularization and limited-memory BFGS enhances performance for large-scale problems.