Projected Neural Network for a Class of Non-Lipschitz Optimization Problems With Linear Constraints

Area of Science:

• Optimization Theory
• Machine Learning
• Applied Mathematics

Background:

• Nonsmooth, nonconvex, and non-Lipschitz optimization problems are prevalent in sparse optimization.
• Existing methods may struggle with the complexities of these problem classes.
• The need for robust and efficient solution methods is critical.

Purpose of the Study:

• To propose a projected neural network for solving nonsmooth, nonconvex, and non-Lipschitz optimization problems.
• To define and analyze a generalized stationary point with enhanced optimality.
• To investigate the convergence and properties of the proposed network for sparse optimization.

Main Methods:

• Generalization of the Clarke stationary point to a novel generalized stationary point.
• Development of a projected neural network utilizing a smoothing method.
• Analysis of global existence, boundedness, and uniqueness of network solutions.
• Investigation of accumulation points and asymptotic convergence properties.

Main Results:

• The proposed neural network guarantees globally existent and bounded solutions under bounded level set conditions.
• Any accumulation point of the network is a generalized stationary point.
• Under suitable conditions, network solutions asymptotically converge to a stationary point.
• Specific analysis for non-Lipschitz problems reveals lower bound properties and unified identification of nonzero elements.

Conclusions:

• The projected neural network effectively addresses challenging optimization problems in sparse applications.
• The defined generalized stationary point offers stronger optimality guarantees.
• Numerical results confirm the efficiency and applicability of the proposed method for sparse optimization models.