[Paper Review] FBstab: A Stabilized Semismooth Quadratic Programming Algorithm with Applications in Model Predictive Control

Dominic Liao‐McPherson, Ilya Kolmanovsky|arXiv (Cornell University)|Jan 13, 2019

Advanced Optimization Algorithms Research85 references34 citations

TL;DR

FBstab is a novel, proximally stabilized semismooth Newton method for convex quadratic programming that combines the primal-dual proximal point algorithm with a semismooth Newton solver. It achieves robust convergence, handles degenerate and infeasible problems, enables efficient warmstarting, and detects both primal and dual infeasibility—making it highly suitable for real-time model predictive control with minimal assumptions beyond positive semidefinite Hessian.

ABSTRACT

This paper introduces the proximally stabilized Fischer-Burmeister method (FBstab); a new algorithm for convex quadratic programming that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. We outline the algorithm, provide convergence and convergence rate proofs, report some numerical results from model predictive control benchmarks, and also include experimental results. We show that FBstab is competitive with and often superior to, state of the art methods, has attractive scaling properties, and is especially promising for model predictive control applications.

Motivation & Objective

• Address the need for numerically robust, real-time quadratic programming solvers in model predictive control (MPC) applications.
• Overcome limitations of existing methods such as poor warmstarting (active set), restrictive assumptions (e.g., strong convexity, LICQ), or lack of infeasibility detection.
• Develop a method that maintains fast convergence and reliability under degenerate solutions and primal/dual infeasibility.
• Enable efficient implementation on embedded systems by supporting warmstarting and structure exploitation.
• Provide theoretical guarantees on convergence and infeasibility detection under minimal assumptions (positive semidefinite Hessian only)

Proposed method

• Combines the primal-dual proximal point algorithm (proximal method of multipliers) with a semismooth Newton-type method to solve convex QPs.
• Solves regularized subproblems using a semismooth Newton method that converges rapidly due to automatic satisfaction of regularity conditions from proximal stabilization.
• Uses a semismooth reformulation of the KKT conditions to apply Newton's method to a nonsmooth system.
• Employs a proximal regularization that stabilizes Newton iterations and enables detection of primal infeasibility and dual infeasibility (unboundedness).
• Supports warmstarting by initializing the Newton solver from the previous solution, crucial for MPC applications.
• Leverages structured linear algebra routines similar to interior-point methods, enabling linear scaling with horizon length in MPC

Experimental results

Research questions

• RQ1 Can a semismooth Newton method be stabilized via proximal regularization to handle degenerate and infeasible QPs while maintaining fast convergence?
• RQ2 Does the proposed method detect both primal and dual infeasibility robustly under only positive semidefinite Hessian assumptions?
• RQ3 How does FBstab compare in performance and robustness to state-of-the-art QP solvers (e.g., IP, AS, DN, FO) in real-time MPC benchmarks?
• RQ4 Can FBstab be efficiently warmstarted and scaled linearly with receding horizon length in MPC problems?
• RQ5 Does the method maintain numerical robustness and low residual even under early termination or model mismatch?

Key findings

• FBstab achieves convergence to the optimal solution or detects infeasibility (primal or dual) under only the assumption that the Hessian is positive semidefinite and a solution exists.
• In the servo motor MPC benchmark, FBstab achieved an average solution time of 0.2 ms (normalized to 4.5 ms) with warmstart, outperforming all other methods in average time.
• For the spacecraft relative motion MPC problem, FBstab achieved an average warm-start time of 0.1 ms (normalized to 63.9 ms), significantly faster than FBRS and PDIP, and detected infeasibility correctly.
• In the copolymerization MPC problem, FBstab maintained an average warm-start time of 0.4 ms (normalized to 97.1 ms), outperforming all competitors including DAS and NNLS.
• In experimental validation on an inverted pendulum, FBstab maintained the QP residual below 10−4 with a maximum execution time of 4.3 ms, safely below the 10 ms sampling period.
• FBstab successfully detected dual infeasibility (unboundedness) in a test problem with c = −1 and b = ∞, returning δx = (0, 671) × 10⁵ after 3 proximal and 8 Newton iterations.