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Achieving violation-free distributed optimization under coupling constraints

Changxin Liu, Xiao Tan, Xuyang Wu, Dimos V. Dimarogonas, Karl H. Johansson

TL;DR

The paper addresses violation-free distributed optimization for problems with coupling constraints by reformulating the problem with auxiliary variables and a network-aware linear mapping, ensuring the projection of the reformulated feasible set matches the original. It derives a min-min structure where the outer gradient can be computed locally as affine transformations of KKT multipliers, and provides convergence guarantees with Lipschitz-gradient bounds. Two violation-free distributed algorithms are developed and analyzed, including an accelerated dual-averaging method, with extensions to general convex cases and a practical validation via a distributed control barrier function (CBF) controller. The approach enables safe, feasible operation of distributed systems even when optimization is interrupted, with demonstrated effectiveness in both numerical experiments and closed-loop safety-critical control.

Abstract

Constraint satisfaction is a critical component in a wide range of engineering applications, including but not limited to safe multi-agent control and economic dispatch in power systems. This study explores violation-free distributed optimization techniques for problems characterized by separable objective functions and coupling constraints. First, we incorporate auxiliary decision variables together with a network-dependent linear mapping to each coupling constraint. For the reformulated problem, we show that the projection of its feasible set onto the space of primal variables is identical to that of the original problem, which is the key to achieving all-time constraint satisfaction. Upon treating the reformulated problem as a min-min optimization problem with respect to auxiliary and primal variables, we demonstrate that the gradients in the outer minimization problem have a locally computable closed-form. Then, two violation-free distributed optimization algorithms are developed and their convergence under reasonable assumptions is analyzed. Finally, the proposed algorithm is applied to implement a control barrier function based controller in a distributed manner, and the results verify its effectiveness.

Achieving violation-free distributed optimization under coupling constraints

TL;DR

The paper addresses violation-free distributed optimization for problems with coupling constraints by reformulating the problem with auxiliary variables and a network-aware linear mapping, ensuring the projection of the reformulated feasible set matches the original. It derives a min-min structure where the outer gradient can be computed locally as affine transformations of KKT multipliers, and provides convergence guarantees with Lipschitz-gradient bounds. Two violation-free distributed algorithms are developed and analyzed, including an accelerated dual-averaging method, with extensions to general convex cases and a practical validation via a distributed control barrier function (CBF) controller. The approach enables safe, feasible operation of distributed systems even when optimization is interrupted, with demonstrated effectiveness in both numerical experiments and closed-loop safety-critical control.

Abstract

Constraint satisfaction is a critical component in a wide range of engineering applications, including but not limited to safe multi-agent control and economic dispatch in power systems. This study explores violation-free distributed optimization techniques for problems characterized by separable objective functions and coupling constraints. First, we incorporate auxiliary decision variables together with a network-dependent linear mapping to each coupling constraint. For the reformulated problem, we show that the projection of its feasible set onto the space of primal variables is identical to that of the original problem, which is the key to achieving all-time constraint satisfaction. Upon treating the reformulated problem as a min-min optimization problem with respect to auxiliary and primal variables, we demonstrate that the gradients in the outer minimization problem have a locally computable closed-form. Then, two violation-free distributed optimization algorithms are developed and their convergence under reasonable assumptions is analyzed. Finally, the proposed algorithm is applied to implement a control barrier function based controller in a distributed manner, and the results verify its effectiveness.
Paper Structure (19 sections, 6 theorems, 56 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 6 theorems, 56 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Our contribution
  4. Paper organization and notation
  5. Problem statement
  6. Basic setup
  7. Communication network
  8. Problem reformulation and algorithm
  9. Problem reformulation
  10. Perturbed function and minimization algorithm
  11. Convergence analysis
  12. Computing Lipschitz constants
  13. Rate of convergence
  14. Extension to non-strongly convex case
  15. Numerical experiment
  16. ...and 4 more sections

Key Result

Proposition 1

Suppose Assumption assump:network holds. The problems in eq:centralized_problem and transformed_P are equivalent in the sense that

Figures (7)

  • Figure 1: A graph with 4 nodes.
  • Figure 2: Convergence of objective error $\phi(y)-\phi(y^*)$.
  • Figure 3: Convergence of primal variable error $\sqrt{\sum_{i=1}^N \lVert x_i-x_i^* \rVert^2}$.
  • Figure 4: Vector values of coupling constraint function $\sum_{i=1}^N A_i^Tx_i+b_i$.
  • Figure 5: Consensus error of local dual variables $[\lVert(I-P^{[1]}) \mu^{[1]} \rVert, \lVert(I-P^{[2]}) \mu^{[2]} \rVert]$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1
  • Lemma 1
  • Remark 2
  • Lemma 2
  • Proposition 2
  • Proposition 3
  • Lemma 3
  • Definition 1: CBF