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A Unified Funnel Restoration SQP Algorithm

David Kiessling, Sven Leyffer, Charlie Vanaret

TL;DR

This work presents a unified double-loop framework for nonlinearly constrained optimization (NCOs) implemented in the Uno solver, focusing on a funnel-based globalization strategy to guarantee global convergence. It develops a unified funnel restoration SQP algorithm that can operate with either a trust-region or a line-search globalization mechanism, and it systematically couples this with a feasibility-restoration phase to handle infeasible subproblems. The paper proves global convergence of the trust-region funnel SQP method, analyzes convergence to feasibility and to stationary points of the constraint violation, and demonstrates competitive performance on CUTEst problems, with the funnel variant slightly outperforming the traditional filter approach in several metrics. The approach is modular, extensible, and open-source, offering practical impact for robust solving of challenging NCOs and enabling further research on funnel/ filter globalization strategies within a unified solver framework.

Abstract

We consider nonlinearly constrained optimization problems and discuss a generic double-loop framework consisting of four algorithmic ingredients that unifies a broad range of nonlinear optimization solvers. This framework has been implemented in the open-source solver Uno, a Swiss Army knife-like C++ optimization framework that unifies many nonlinearly constrained nonconvex optimization solvers. We illustrate the framework with a sequential quadratic programming (SQP) algorithm that maintains an acceptable upper bound on the constraint violation, called a funnel, that is monotonically decreased to control the feasibility of the iterates. Infeasible quadratic subproblems are handled by a feasibility restoration strategy. Globalization is controlled by a line search or a trust-region method. We prove global convergence of the trust-region funnel SQP method, building on known results from filter methods. We implement the algorithm in Uno, and we provide extensive test results for the trust-region line-search funnel SQP on small CUTEst instances.

A Unified Funnel Restoration SQP Algorithm

TL;DR

This work presents a unified double-loop framework for nonlinearly constrained optimization (NCOs) implemented in the Uno solver, focusing on a funnel-based globalization strategy to guarantee global convergence. It develops a unified funnel restoration SQP algorithm that can operate with either a trust-region or a line-search globalization mechanism, and it systematically couples this with a feasibility-restoration phase to handle infeasible subproblems. The paper proves global convergence of the trust-region funnel SQP method, analyzes convergence to feasibility and to stationary points of the constraint violation, and demonstrates competitive performance on CUTEst problems, with the funnel variant slightly outperforming the traditional filter approach in several metrics. The approach is modular, extensible, and open-source, offering practical impact for robust solving of challenging NCOs and enabling further research on funnel/ filter globalization strategies within a unified solver framework.

Abstract

We consider nonlinearly constrained optimization problems and discuss a generic double-loop framework consisting of four algorithmic ingredients that unifies a broad range of nonlinear optimization solvers. This framework has been implemented in the open-source solver Uno, a Swiss Army knife-like C++ optimization framework that unifies many nonlinearly constrained nonconvex optimization solvers. We illustrate the framework with a sequential quadratic programming (SQP) algorithm that maintains an acceptable upper bound on the constraint violation, called a funnel, that is monotonically decreased to control the feasibility of the iterates. Infeasible quadratic subproblems are handled by a feasibility restoration strategy. Globalization is controlled by a line search or a trust-region method. We prove global convergence of the trust-region funnel SQP method, building on known results from filter methods. We implement the algorithm in Uno, and we provide extensive test results for the trust-region line-search funnel SQP on small CUTEst instances.
Paper Structure (28 sections, 7 theorems, 30 equations, 7 figures, 3 tables, 5 algorithms)

This paper contains 28 sections, 7 theorems, 30 equations, 7 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction and Background
  2. Contributions.
  3. Outline.
  4. Unified Abstraction of Nonlinearly Constrained Optimization
  5. Subproblem Method: Sequential Quadratic Programming
  6. Globalization Mechanism: Trust Region or Line Search
  7. Trust-Region Methods
  8. Line Search Methods
  9. Constraint Relaxation Strategy: Feasibility Restoration
  10. Globalization Strategies
  11. Progress Measures
  12. Funnel Method
  13. A note on the funnel reduction mechanism.
  14. Filter Methods
  15. Connections between Funnel and Filter Methods
  16. ...and 13 more sections

Key Result

Theorem 1

Assume that the algorithm does not terminate finitely at a KKT point, and consider sequences $\{\tau^{(k)}\}$, $\{h^{(k)}\}$, and $\{f^{(k)}\}$ such that $h^{(k)} \geq 0$, $f^{(k)}$ is bounded below and $h^{(k+n)} \leq \beta \tau^{(k)}$ for all $k, n \in \mathbb{N}$. Furthermore, let constants $\bet or that In both cases, it follows that $h^{(k)} \to 0$ for $k \to \infty$.

Figures (7)

  • Figure 1: Funnel (in gray) around the feasible set $\mathcal{F}$. The frontier of the funnel envelope \ref{['eq:funnel_sufficient_decrease']} is shown as a dashed curve.
  • Figure 2: Illustration of the funnel method Samadi2018.
  • Figure 3: Performance profiles for all four algorithmic configurations with respect to constraint evaluations.
  • Figure 4: Distribution plot of constraint evaluations for funnel vs filter globalization strategies.
  • Figure 5: Comparison of the number of $f$-type (blue), $h$-type (orange), and restoration steps (green) for the trust-region funnel method and the trust-region filter method.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: KKT conditions Nocedal2006
  • Theorem 1
  • proof
  • Lemma 1: Lemma 2 in fletcher2002a
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 5 more