Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constrained Optimization From a Control Perspective via Feedback Linearization

Runyu Zhang, Arvind Raghunathan, Jeff Shamma, Na Li

TL;DR

This work develops a control-theoretic framework for constrained optimization by applying feedback linearization (FL) to equality and inequality constraints. It establishes global convergence rates to first-order $KKT$ points, reveals a precise equivalence between FL discretization and SQP, and extends the approach with a momentum-accelerated FL variant that attains provable convergence guarantees in continuous time. The results unify FL with classical optimization methods, showing how FL-Newton and FL-proximal choices relate to Newton-type and proximal SQP updates, and demonstrate practical benefits in nonconvex and inequality-constrained scenarios. Empirically, FL-based methods, especially with momentum and second-order information, accelerate convergence on problems such as logistic regression with cross-client constraints and AC optimal power flow, illustrating potential for real-time and distributed constrained optimization.

Abstract

Tools from control and dynamical systems have proven valuable for analyzing and developing optimization methods. In this paper, we establish rigorous theoretical foundations for using feedback linearization (FL) -- a well-established nonlinear control technique -- to solve constrained optimization problems. For equality-constrained optimization, we establish global convergence rates to first-order Karush-Kuhn-Tucker (KKT) points and uncover the close connection between the FL method and the Sequential Quadratic Programming (SQP) algorithm. Building on this relationship, we extend the FL approach to handle inequality-constrained problems. Furthermore, we introduce a momentum-accelerated feedback linearization algorithm and provide a rigorous convergence guarantee.

Constrained Optimization From a Control Perspective via Feedback Linearization

TL;DR

This work develops a control-theoretic framework for constrained optimization by applying feedback linearization (FL) to equality and inequality constraints. It establishes global convergence rates to first-order points, reveals a precise equivalence between FL discretization and SQP, and extends the approach with a momentum-accelerated FL variant that attains provable convergence guarantees in continuous time. The results unify FL with classical optimization methods, showing how FL-Newton and FL-proximal choices relate to Newton-type and proximal SQP updates, and demonstrate practical benefits in nonconvex and inequality-constrained scenarios. Empirically, FL-based methods, especially with momentum and second-order information, accelerate convergence on problems such as logistic regression with cross-client constraints and AC optimal power flow, illustrating potential for real-time and distributed constrained optimization.

Abstract

Tools from control and dynamical systems have proven valuable for analyzing and developing optimization methods. In this paper, we establish rigorous theoretical foundations for using feedback linearization (FL) -- a well-established nonlinear control technique -- to solve constrained optimization problems. For equality-constrained optimization, we establish global convergence rates to first-order Karush-Kuhn-Tucker (KKT) points and uncover the close connection between the FL method and the Sequential Quadratic Programming (SQP) algorithm. Building on this relationship, we extend the FL approach to handle inequality-constrained problems. Furthermore, we introduce a momentum-accelerated feedback linearization algorithm and provide a rigorous convergence guarantee.

Paper Structure

This paper contains 32 sections, 9 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Notations:
  4. Feedback Linearization (FL) for solving equality constrained optimization
  5. Control perspective on equality-constrained optimization cerone_new_2024
  6. Feedback linearization (FL) for equality-constrained optimization cerone_new_2024
  7. FL control method: Convergence and Relationship to SQP
  8. Convergence Results
  9. Relationship with SQP
  10. Extension to inequality constraints
  11. Relationship with the SQP algorithm
  12. Convergence Result
  13. Momentum Acceleration for Constrained Optimization
  14. Convergence Analysis for FL-momentum: Nonconvex Case
  15. Numerical Verifications
  16. ...and 17 more sections

Key Result

Theorem 1

Let Assumption assump:lipschitz, assump:f-lower-bounded and assump:eq-JJ hold and let the control gain $K$ be a diagonal positive definite matrix, i.e., $K = \textup{diag}\{k_i\}_{i=1}^m$, where $k_i > 0$. Then we have that the dynamic of the feedback linearization method eq:feedback-linearization s

Figures (5)

  • Figure 1: Control Perspective for Constrained Optimization
  • Figure 2: Result for Heterogeneous Logistic Regression
  • Figure 3: Running different constrained optimization algorithm for Problem \ref{['eq:constrained-logistic']}
  • Figure 4: Comparison of Approximated FL and Approximated FL-PI algorithm
  • Figure 5: Result for AC OPF on IEEE-39 bus (left) and IEEE-118 bus (right) bus system

Theorems & Definitions (23)

  • Remark 1: Scalability and Computational Complexity
  • Remark 2: Extensions of the FL approach
  • Theorem 1
  • Theorem 2
  • Remark 3: Choice of $T(x)$
  • Remark 4: Comparison with other first-order methods
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 5: Limitation of the result
  • ...and 13 more