Stability Analysis of Constrained Optimization Dynamics via Passivity Techniques

TL;DR

This paper introduces a passivity-based framework for analyzing the stability of constrained optimization algorithms, specifically primal-dual gradient methods, using hybrid systems and energy-based methods.

Contribution

It develops a novel passivity analysis approach for constrained optimization dynamics, combining Brayton Moser formulation and hybrid systems theory.

Findings

Proves convergence of the primal-dual method using passivity arguments.

Models inequality constraints as a state-dependent switching system.

Demonstrates effectiveness through energy management simulations.

Abstract

In this paper, we present passivity based convergence analysis of continuous time primal-dual gradient method for convex optimization problems. We first show that a convex optimization problem with only affine equality constraints admit a Brayton Moser formulation. This observation leads to a new passivity property derived from a Krasovskii type storage function. Secondly, the inequality constraints are modeled as a state dependent switching system. Using hybrid methods, it is shown that each switching mode is passive and the passivity of the system is preserved under arbitrary switching. Finally, the two systems, (i) one derived from the Brayton Moser formulation and (ii) the state dependent switching system, are interconnected in a power conserving way. The resulting trajectories of the overall system are shown to converge asymptotically, to the optimal solution of the convex optimization problem. The proposed methodology is applied to an energy management problem in buildings and simulations are provided for corroboration.