A smooth gradient approximation neural network for general constrained nonsmooth nonconvex optimization problems.

Area of Science:

• Optimization Theory
• Neural Networks
• Applied Mathematics

Background:

• Nonsmooth and nonconvex optimization problems are prevalent in engineering and complex systems.
• These problems pose significant challenges for algorithm design and convergence analysis due to function characteristics.
• Existing methods often struggle with the inherent difficulties of nonsmoothness and nonconvexity.

Purpose of the Study:

• To present a novel smooth gradient approximation neural network for nonsmooth nonconvex optimization.
• To address the challenges posed by nonsmooth nonregular objective functions and nonconvex inequality constraints.
• To establish theoretical convergence guarantees and demonstrate practical applicability.

Main Methods:

• A smooth approximation technique with a time-varying control parameter is employed for objective functions.
• A hard comparator function is utilized to enforce state solutions within nonconvex inequality constraint sets.
• Convergence analysis proves that accumulation points are stationary points of the optimization problem.

Main Results:

• The proposed neural network effectively handles nonsmooth nonregular objective functions and nonconvex constraints.
• Theoretical analysis confirms that accumulation points converge to stationary points.
• The network demonstrates capability in solving generalized convex optimization problems.
• The algorithm exhibits weaker convergence conditions and a simpler structure compared to related neural networks.

Conclusions:

• The developed neural network provides an effective and robust solution for nonsmooth nonconvex optimization.
• The algorithm's practical applicability is validated through simulations and an application in optimizing condition numbers.
• This work offers a promising advancement in computational optimization for complex systems.