A Projection Neural Network to Nonsmooth Constrained Pseudoconvex Optimization.

Area of Science:

• Optimization Theory
• Computational Neuroscience
• Applied Mathematics

Background:

• Nonsmooth pseudoconvex optimization problems with various constraints pose significant computational challenges.
• Existing neural network approaches often require precise penalty parameters and initial point selection.
• There is a need for robust and computationally efficient neural network models for constrained optimization.

Purpose of the Study:

• To propose a single-layer projection neural network (PNN) for solving nonsmooth pseudoconvex optimization problems.
• To address linear equality, convex inequality, and bound constraints within the optimization framework.
• To eliminate the need for exact penalty parameters and initial point dependence.

Main Methods:

• Development of a PNN utilizing penalty functions and differential inclusions.
• Incorporation of a Tikhonov-like regularization method to avoid exact penalty parameter calculation.
• Application of nonsmooth analysis to prove theoretical properties of the network's state solution.

Main Results:

• The proposed PNN guarantees that the state solution is always bounded and globally exists.
• The network's state solution converges to the constrained feasible region in finite time and remains within it.
• The state solution is proven to converge to an optimal solution of the considered optimization problems.

Conclusions:

• The developed PNN offers a simple structure and low computational load compared to subgradient-based methods.
• The algorithm demonstrates independence from initial point selection, enhancing its practical applicability.
• Numerical experiments and application examples validate the global convergence and effectiveness of the proposed PNN.