Accelerated quadratic penalty dynamic approaches with applications to distributed optimization.

Area of Science:

• Optimization Theory
• Dynamic Systems
• Numerical Analysis

Background:

• Linearly constrained convex optimization is a fundamental problem.
• Existing methods may face challenges with constraint handling and convergence speed.
• Accelerated dynamic systems offer a promising alternative for complex optimization tasks.

Purpose of the Study:

• To develop and analyze accelerated continuous-time dynamic approaches for linearly constrained convex optimization.
• To investigate the convergence properties and robustness of these dynamic methods.
• To adapt the dynamic approach for solving distributed optimization problems.

Main Methods:

• Utilizing a quadratic penalty function to incorporate linear constraints into the objective function.
• Analyzing dynamic systems with vanishing damping (α/t).
• Establishing convergence rates for objective residual and feasibility violation.
• Applying the method to distributed constrained consensus, distributed extended monotropic optimization, and distributed optimization with separated equations.

Main Results:

• Achieved convergence rates of O(1/t^min{2α/3,2}) for objective residual and feasibility violation when α>0.
• Demonstrated robustness of convergence rates against external perturbations.
• Developed three variant distributed dynamic approaches for specific distributed optimization problems.
• Numerical examples confirmed the effectiveness of the proposed methods.

Conclusions:

• The proposed accelerated dynamic approaches with quadratic penalties provide an effective way to solve linearly constrained convex optimization problems.
• The methods exhibit robust and theoretically sound convergence properties.
• The dynamic framework is adaptable and effective for various distributed optimization scenarios.