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Accelerated first-order optimization under nonlinear constraints.

Michael Muehlebach1, Michael I Jordan2

  • 1Learning and Dynamical Systems, Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 Tuebingen, Baden-Wuerttemberg Germany.

Mathematical Programming
|January 13, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

New accelerated first-order algorithms for constrained optimization were developed using analogies with non-smooth dynamical systems. These methods offer improved efficiency and handle non-convex constraints, making them suitable for machine learning tasks.

Keywords:
Constrained optimizationGradient-based methodsMachine learningNonlinear programming

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Area of Science:

  • Optimization Theory
  • Dynamical Systems
  • Machine Learning

Background:

  • First-order algorithms are crucial for constrained optimization.
  • Existing methods like Frank-Wolfe and projected gradients can be computationally intensive.
  • Non-smooth dynamical systems offer novel perspectives for algorithm design.

Purpose of the Study:

  • To design a new class of accelerated first-order algorithms for constrained optimization.
  • To develop algorithms that avoid full feasible set optimization per iteration.
  • To address challenges in handling non-convex constraints efficiently.

Main Methods:

  • Exploiting analogies between first-order optimization algorithms and non-smooth dynamical systems.
  • Developing algorithms where constraints are expressed in terms of velocities.
  • Proving convergence in non-convex settings and deriving accelerated rates in convex settings.

    • Main Results:

    • Convergence to stationary points is proven, even for non-convex problems.
    • Accelerated convergence rates are derived for both continuous and discrete time convex settings.
    • Algorithms exhibit mild complexity growth with decision variables and constraints.
    • Efficient handling of non-convex L_p constraints (p<1) and state-of-the-art performance for L_1 constraints.

    Conclusions:

    • The new algorithms offer an efficient and scalable approach to constrained optimization.
    • The velocity-based constraint formulation enables practical application to machine learning problems.
    • These algorithms demonstrate strong performance on compressed sensing and sparse regression tasks.