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Modified projected Gauss-Newton method for constrained nonlinear least-squares: application to power flow analysis

Yassine Nabou, Lucian Toma, Ion Necoara

TL;DR

The paper addresses constrained nonlinear least-squares by recasting the problem with a Euclidean merit function $f(x)=\\|F(x)\\| + I_{\\mathbf{C}}(x)$ and proposing a Modified Projected Gauss-Newton method. At each iteration, it linearizes $F$ at the current point and solves a strongly convex subproblem to obtain the next iterate, ensuring a descent via a line-search-style condition and enforcing feasibility with the convex set $\\mathbf{C}$. Global convergence to stationary points is established without requiring Jacobian nondegeneracy, and convergence rates are derived under the Kurdyka–Łojasiewicz (KL) property. The method is demonstrated on power-flow analysis problems using IEEE bus cases, where MPG-N often requires far fewer iterations than projected gradient descent, highlighting its practical utility for stabilized constrained nonlinear least-squares in power systems.

Abstract

In this paper, we consider a modified projected Gauss-Newton method for solving constrained nonlinear least-squares problems. We assume that the functional constraints are smooth and the the other constraints are represented by a simple closed convex set. We formulate the nonlinear least-squares problem as an optimization problem using the Euclidean norm as a merit function. In our method, at each iteration we linearize the functional constraints inside the merit function at the current point and add a quadratic regularization, yielding a strongly convex subproblem that is easy to solve, whose solution is the next iterate. We present global convergence guarantees for the proposed method under mild assumptions. In particular, we prove stationary point convergence guarantees and under Kurdyka-Lojasiewicz (KL) property for the objective function we derive convergence rates depending on the KL parameter. Finally, we show the efficiency of this method on the power flow analysis problem using several IEEE bus test cases.

Modified projected Gauss-Newton method for constrained nonlinear least-squares: application to power flow analysis

TL;DR

The paper addresses constrained nonlinear least-squares by recasting the problem with a Euclidean merit function and proposing a Modified Projected Gauss-Newton method. At each iteration, it linearizes at the current point and solves a strongly convex subproblem to obtain the next iterate, ensuring a descent via a line-search-style condition and enforcing feasibility with the convex set . Global convergence to stationary points is established without requiring Jacobian nondegeneracy, and convergence rates are derived under the Kurdyka–Łojasiewicz (KL) property. The method is demonstrated on power-flow analysis problems using IEEE bus cases, where MPG-N often requires far fewer iterations than projected gradient descent, highlighting its practical utility for stabilized constrained nonlinear least-squares in power systems.

Abstract

In this paper, we consider a modified projected Gauss-Newton method for solving constrained nonlinear least-squares problems. We assume that the functional constraints are smooth and the the other constraints are represented by a simple closed convex set. We formulate the nonlinear least-squares problem as an optimization problem using the Euclidean norm as a merit function. In our method, at each iteration we linearize the functional constraints inside the merit function at the current point and add a quadratic regularization, yielding a strongly convex subproblem that is easy to solve, whose solution is the next iterate. We present global convergence guarantees for the proposed method under mild assumptions. In particular, we prove stationary point convergence guarantees and under Kurdyka-Lojasiewicz (KL) property for the objective function we derive convergence rates depending on the KL parameter. Finally, we show the efficiency of this method on the power flow analysis problem using several IEEE bus test cases.

Paper Structure

This paper contains 8 sections, 6 theorems, 48 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Notations and preliminaries
  3. Modified Projected Gauss-Newton method
  4. Convergence analysis
  5. Better rates under KL
  6. Power flow analysis
  7. Numerical simulations
  8. Conclusion

Key Result

Lemma 1

Let Assumption eq:ass1 hold. At $k$th iteration of MPG-N algorithm, if $M_k - L_F \geq \delta$, then inequality eq:alg_desc holds.

Figures (2)

  • Figure 1: Representation of the IEEE 14-bus system ICEG:13.
  • Figure 2: Comparison between MPG-N and PGD methods in terms of $\|F(x)\|$ along iterations on several IEEE bus systems.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • ...and 3 more