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A New Duality-Free Framework for Convex Optimisation with Superlinear Convergence and Effective Warm-Starting

Michael Cummins, Eric Kerrigan

TL;DR

This work addresses the warm-start challenge of second-order convex solvers by introducing a duality-free Proximal Value Method (PVM) that reformulates constrained problems as the unconstrained minimisation of a convex value function $r^*(t)$. It combines the Proximal Point Algorithm with a semismooth Newton inner loop to solve the resulting subproblems, and establishes sufficient conditions for superlinear convergence. For quadratic programs, it provides a tractable formulation and concrete convergence results, including feasibility recovery and infeasibility detection, with empirical MPC experiments showing competitive cold-start performance and up to a 70% reduction in Newton iterations when warm-started. The approach yields a certificate of infeasibility and supports effective warm-starting, making it a practical alternative to interior-point methods for a broad class of convex problems. Overall, PVM offers a robust, warm-start-friendly framework that can extend to wider convex settings and potentially nonconvex extensions with further development.

Abstract

Modern second order solvers for convex optimisation, such as interior point methods, rely on primal dual information and are difficult to warm start, limiting their applicability in real time control. We propose the PVM, a duality free framework that reformulates the constrained problem as the unconstrained minimisation of a value function. The resulting problem always has a solution, yields a certificate of infeasibility and is amenable to warm starting. We develop a second order algorithm for Quadratic Programming based on the PPA and semismooth Newton methods, and establish sufficient conditions for superlinear convergence to an arbitrarily small neighbourhood of the solution. Numerical experiments on a MPC problem demonstrate competitive performance with state of the art solvers from a cold start and up to 70\% reduction in Newton iterations when warm starting.

A New Duality-Free Framework for Convex Optimisation with Superlinear Convergence and Effective Warm-Starting

TL;DR

This work addresses the warm-start challenge of second-order convex solvers by introducing a duality-free Proximal Value Method (PVM) that reformulates constrained problems as the unconstrained minimisation of a convex value function . It combines the Proximal Point Algorithm with a semismooth Newton inner loop to solve the resulting subproblems, and establishes sufficient conditions for superlinear convergence. For quadratic programs, it provides a tractable formulation and concrete convergence results, including feasibility recovery and infeasibility detection, with empirical MPC experiments showing competitive cold-start performance and up to a 70% reduction in Newton iterations when warm-started. The approach yields a certificate of infeasibility and supports effective warm-starting, making it a practical alternative to interior-point methods for a broad class of convex problems. Overall, PVM offers a robust, warm-start-friendly framework that can extend to wider convex settings and potentially nonconvex extensions with further development.

Abstract

Modern second order solvers for convex optimisation, such as interior point methods, rely on primal dual information and are difficult to warm start, limiting their applicability in real time control. We propose the PVM, a duality free framework that reformulates the constrained problem as the unconstrained minimisation of a value function. The resulting problem always has a solution, yields a certificate of infeasibility and is amenable to warm starting. We develop a second order algorithm for Quadratic Programming based on the PPA and semismooth Newton methods, and establish sufficient conditions for superlinear convergence to an arbitrarily small neighbourhood of the solution. Numerical experiments on a MPC problem demonstrate competitive performance with state of the art solvers from a cold start and up to 70\% reduction in Newton iterations when warm starting.
Paper Structure (20 sections, 49 equations, 3 figures, 1 table)

This paper contains 20 sections, 49 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semismooth Newton
  4. The Proximal Point Algorithm
  5. The Proximal Value Method
  6. The Value Function
  7. Exact Proximal Iterations on the Value Function
  8. Inexact Proximal Iterations on the Value Function
  9. Solving a Quadratic Program
  10. Controlling the Overshoot
  11. Semi-Smooth Newton
  12. Convergence Analysis
  13. Numerical Evaluation
  14. Parameter Selection and Termination
  15. Warm-Starting
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Value function $r^\star$ for two problems with objective $f(x) = \frac{1}{2}x^2$ and equality constraint $x=1$. The inequality constraints are defined as $x \leq 0$ and $x \leq 2$ for the infeasible and feasible problems, respectively.
  • Figure 2: Median number of Newton iterations required to reach convergence when perturbing $t$ and $(x,s)$ warm start. All problems are solved to within $10^{-8}$ accuracy of the true solution.
  • Figure 3: Statistics for total number of Newton iterations required to reach convergence when perturbing $t$ and $(x,s)$ warm start.