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Summary
This summary is machine-generated.

We developed a method to confirm quadratic Lyapunov inequalities for first-order optimization algorithms. This approach identifies conditions for convergence analysis in convex optimization problems.

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Area of Science:

  • Optimization Theory
  • Convex Analysis
  • Control Theory

Background:

  • First-order methods are crucial for solving large-scale convex optimization problems.
  • Lyapunov inequalities are essential for analyzing the convergence of dynamical systems, including optimization algorithms.
  • Existing methods often lack a unified framework for verifying convergence properties.

Purpose of the Study:

  • To establish a general methodology for proving the existence of quadratic Lyapunov inequalities for first-order convex optimization methods.
  • To provide a necessary and sufficient condition for the existence of such inequalities.
  • To extend the applicability of convergence analysis to a broader range of algorithms and parameters.

Main Methods:

  • Formulating first-order methods as linear systems in state-space form.
  • Analyzing the feedback interconnection with subdifferentials of objective functions.
  • Deriving a condition for quadratic Lyapunov inequality existence via semidefinite programming.

Main Results:

  • A novel methodology for establishing quadratic Lyapunov inequalities is presented.
  • A necessary and sufficient condition for the existence of these inequalities is derived.
  • The methodology is demonstrated on various first-order methods, including the Chambolle-Pock algorithm.
  • Parameter regions for duality gap convergence in the Chambolle-Pock method are significantly extended.

Conclusions:

  • The proposed methodology offers a systematic way to analyze the convergence of first-order optimization algorithms.
  • The findings provide deeper insights into the convergence behavior of optimization methods.
  • This work facilitates the design and selection of more efficient optimization algorithms for convex problems.