Nonconvex Distributed Composite Optimization With Coupled Inequality Constraints.

Area of Science:

• Optimization Theory
• Distributed Systems
• Network Science

Background:

• Distributed composite optimization problems involve minimizing sums of local functions across multiple agents.
• Nonconvexity and nonsmoothness in objective and constraint functions present significant algorithmic challenges.
• Time-varying multiagent networks add complexity to coordination and convergence.

Purpose of the Study:

• To develop an efficient distributed algorithm for nonconvex distributed composite optimization.
• To address challenges posed by nonconvex and nonlinear objective and constraint functions.
• To guarantee convergence to an epsilon-Karush-Kuhn-Tucker (KKT) point.

Main Methods:

• A novel distributed linearized augmented primal-dual algorithm is proposed.
• The algorithm incorporates distributed tracking and dynamic consensus techniques.
• Theoretical analysis is used to establish convergence properties.

Main Results:

• The proposed algorithm finds an epsilon-KKT point under appropriate parameter choices.
• Average optimality, constraint violation, and complementary slackness converge to zero at sublinear rates.
• Numerical application validates the algorithm's effectiveness.

Conclusions:

• The developed algorithm offers an effective solution for a challenging class of optimization problems.
• The theoretical guarantees provide confidence in the algorithm's performance.
• The approach is validated through a practical numerical example.