Predictor–Corrector Path Tracking

• Predictor–corrector path tracking is a numerical technique that splits iterations into a predictor phase using extrapolation and a corrector phase employing Newton-type methods.
• It leverages semismooth Newton refinements, sensitivity analysis, and adaptive step-size control to ensure fast, reliable convergence under parameter changes.
• These methods are applied in real-time optimization, numerical algebraic geometry, and model predictive control, supporting efficient tracking in high-dimensional, time-varying systems.

Predictor–Corrector Path Tracking is a family of numerical continuation algorithms designed to track solution trajectories of parametrized nonlinear equations, optimization problems, and polynomial systems under smooth or time-varying changes in parameters. Central to these methods is the decomposition of each tracking iteration into a predictor phase—which extrapolates the current solution along the estimated tangent or via higher-order local information—and a corrector phase—which refines the predicted guess using local root-finding or optimization, typically Newton-type methods. Predictor–corrector schemes are foundational to real-time parametric optimization, numerical algebraic geometry, stochastic process tracking, model predictive control, and time-varying semidefinite programming. Their convergence and robustness are facilitated by semismooth analysis, adaptive step-size selection, certified interval arithmetic, and active-set-aware generalized Jacobians.

1. Mathematical Principles and General Framework

Predictor–corrector path tracking addresses equations or optimality systems of the form

$F(x,p)=0,$

where $x \in \mathbb{R}^d$ (or $\mathbb{C}^d$) is the solution variable and $p \in \mathbb{R}^l$ is a parameter vector varying along a prescribed path or time interval. In parametric optimization, this encompasses KKT systems or nonsmooth root-finding for constrained nonlinear programs. In homotopy continuation, $F$ is a polynomial system depending on a homotopy parameter $t \in [0,1]$.

At each iteration, the method maintains $(x_k,p_k)$ close to the exact solution path $x^*(p_k)$ via:

• Predictor: Linear or higher-order extrapolation, using sensitivity equations, Taylor/Padé expansions, or derivative-based projections.
• Corrector: Local Newton-type or semismooth refinement, ensuring $F(x,p)=0$ (up to tolerance) and quadratic convergence in suitable neighborhoods.

A key analytical tool is the local Lipschitz (strong regularity) property of the solution map $p \mapsto x^*(p)$, ensured by constraint qualifications and second-order sufficient conditions for optimization, or by regularity and nonsingularity in algebraic systems (Liao-McPherson et al., 2018, Timme, 2019, Guillemot et al., 2024). Semismoothness and Clarke's generalized Jacobian extend classical convergence theory to nonsmooth reformulations and enable automated handling of active-set transitions.

2. Predictor Strategies and Sensitivity Analysis

The predictor step is constructed based on local or global information around $(x_{k-1},p_{k-1})$:

• Semismooth Sensitivity (PNLP): Linearize $F(x,p)$, solve

$$x_k^- = x_{k-1} - B_{k-1}^{-1}(F_{k-1} + V_{k-1}\Delta p_k)$$

where $B_{k-1} \in \partial_x F$, $V_{k-1} \in \partial_p F$ (Liao-McPherson et al., 2018).

• Power Series/Taylor/Padé: For algebraic systems, use truncated Taylor or rational Padé approximants; higher-order methods provide faster convergence and sharper singularity detection (Timme, 2019, Telen et al., 2020).
• ODE-Driven Drift (Stochastic Optimization): Integration of the continuous ODE

$$\theta^*(t)' = - [\nabla^2_{\theta\theta}\mathcal{R}]^{-1}\nabla_{\theta t}\mathcal{R}$$

gives discrepancy-minimizing predictor for time-varying optima (Maity et al., 2022).

• Linearized KKT (TV-SDPs): Solve the differentiated KKT system under Burer-Monteiro factorization, enforcing horizontal-space constraints to guarantee injectivity (Bellon et al., 2022).
• Robust Polynomial Tracking: Estimate step size $h$ by detecting nearest singularity (Fabry ratio test) and distance to branching via Hessian curvatures (Telen et al., 2020).

The local order of the predictor and explicit singularity estimates are crucial for adaptive step-size control and reliability, especially near ill-conditioned or singular loci.

3. Corrector Methods: Semismooth, Newtonian, and Certified Refinement

The corrector step refines the predictor's output to satisfy the target system $F(x,p) = 0$:

• Semismooth Newton (PNLP): Iteratively solve

$x_{k} = x_{k}^- - [E_k]^{-1}F(x_{k}^-,p_k)$

where $E_k\in\partial_x F$ is built from Clarke's generalized Jacobian. Correction continues until residual norms are sub-threshold, leveraging quadratic convergence (Liao-McPherson et al., 2018).

• Affine-Covariant Newton Iteration (Algebraic): With contraction factor tests (Smale-style), accept only approximate zeros that guarantee local convergence; reject otherwise (Timme, 2019).
• Interval Newton/Krawczyk Refinement (Certified Tracking): For interval arithmetic-based approaches, ensure box contraction in the Krawczyk operator, guaranteeing unique zeros within validated regions (Guillemot et al., 2024).
• Sequential Convex Subproblem (Optimization): Use adjoint-based sequential convex programming, solving a convex QP/SOCP for improved primal-dual pairs (Dinh et al., 2011).
• Gradient-Corrector (Stochastic): Apply stochastic gradient descent toward instantaneous risk minimization (Maity et al., 2022).
• TV-SDP Newton/Gauss–Newton: One linear correction step on the full KKT conditions at each new parameter value, with injectivity enforced by the gauge constraint (Bellon et al., 2022).

Rigor in contraction estimates and regularization are paramount for ensuring that the corrector both maintains proximity to the true solution trajectory and recovers from predictor inaccuracies.

4. Step-Size Adaptation, Homotopy, and Active-Set Transitions

Dynamic step-size control is essential for maintaining stability and accuracy:

• Homotopy Partitioning: When parameter variation is too large, interpolate in $M$ sub-steps ($h=1/M$), keeping local changes within robust bounds ($\kappa$), and adapting based on homotopy progress (Liao-McPherson et al., 2018).
• Singularity and Curvature Criteria: For algebraic tracking, select $h = \eta \cdot \min(R, C)$, where $R$ is the estimated singularity proximity and $C$ is curvature-based collision threshold (Telen et al., 2020).
• Contraction-Based Restriction: In Newton-corrector, adapt step size according to the contraction factor observed in iterations; shrink upon rejection and expand after robust convergence (Timme, 2019, Guillemot et al., 2024).
• Adaptive Precision: Increase numerical precision when effective step size falls below machine roundoff; revert when contraction is certifiable in standard precision (Timme, 2019, Guillemot et al., 2024).

Active-set changes in inequality-constrained problems are absorbed naturally by semismooth/MCP frameworks—Clarke's generalized Jacobian switches structure automatically, obviating explicit branch tracking (Liao-McPherson et al., 2018).

5. Application Domains and Algorithmic Instantiations

Predictor–corrector schemes have been developed, analyzed, and tested across diverse disciplines:

Problem Class	Predictor–Corrector Variant	Main Reference
Parametric Constrained Optimization	Semismooth Euler–Newton, NCP reformulation	(Liao-McPherson et al., 2018)
Polynomial Homotopy Continuation	Power-series Newton, Padé, Mixed Precision	(Timme, 2019, Telen et al., 2020, Guillemot et al., 2024)
Stochastic Optimization w/ Drift	ODE-based sensitivity, SGD corrector	(Maity et al., 2022)
Time-Varying SDP (Low-Rank Tracking)	Linearized KKT with horizontal-space constraint	(Bellon et al., 2022)
Sequential Convex Programming	Adjoint-based SCP predictor & convex corrector	(Dinh et al., 2011)
Nonlinear MPC (Spacecraft, Hydro)	Semismooth PC, APCSCP, reachability controllers	(Liao-McPherson et al., 2018, Dinh et al., 2011, Pham et al., 2017)

Empirical performance consistently shows superior solution tracking, bounded errors, and computational efficiency compared to non–predictor–corrector or one-step methods. Certified path trackers with interval arithmetic and Taylor models have further closed the gap with noncertified solvers, yielding machine-validated roots with minimal overhead (Guillemot et al., 2024).

6. Theoretical Guarantees and Convergence Analysis

Robustness and error bounds for predictor–corrector path tracking have been rigorously established under strong regularity, semismoothness, and Lipschitz conditions. Typical guarantees include:

• Local error recursion: Predictor error scales with previous iterate and parameter increment, typically of the form

$$\|e_k^-\|\leq \alpha\|e_{k-1}\|^2+\beta\|e_{k-1}\|\|\Delta p_k\|+\sigma\|\Delta p_k\|^2,$$

while corrector Newton steps provide quadratic error reduction

$\|e_k\|\leq \eta\|e_k^-\|^2.$

• Step-size contraction theorems: Explicit upper bounds on admissible $h$ for guaranteed convergence, e.g.,

$$h\leq t^*=\left(\frac{\sqrt{1+2\bar{h}}-1}{\omega\eta_p}\right)^{1/p}$$

(Timme, 2019).

• Certified interval box contraction: If operator norm is below threshold, unique zero enclosed (Guillemot et al., 2024).
• Local tube stability: Predictor–corrector iterates remain in a small tube around the exact solution path for sufficiently small parameter steps (Bellon et al., 2022, Dinh et al., 2011).
• Active-set adaptivity: No explicit branch detection required under the semismooth framework.

In stochastic settings, asymptotic tracking error (ATE) exhibits improved scaling: $O(h)$ for predictor–corrector and $O(\sqrt{h})$ for vanilla SGD (Maity et al., 2022).

7. Practical Implementation and Computational Considerations

Practically, each predictor–corrector iteration reduces to solving a sequence of linear systems (Newton steps, linearized sensitivity equations, Taylor or Padé evaluations), with complexity comparable to a single QP/SOCP or Newton solve per step. For high-dimensional algebraic systems, robust tracking methods leverage multithreaded computation on shared-memory architectures to amortize Hessian and SVD evaluations (Telen et al., 2020). Mixed-precision switching dynamically balances speed and accuracy (Timme, 2019, Guillemot et al., 2024).

In real-time constrained optimization (e.g., nonlinear MPC), semismooth predictor–corrector algorithms have demonstrated robust handling of rapid active-set changes and preserved computational efficiency, outperforming warm-started SQP and classical corrector-driven path tracking (Liao-McPherson et al., 2018, Dinh et al., 2011). Certified tracking workflows now integrate adaptive Taylor models, interval arithmetic, and rigorous contraction checks at each step, yielding validated roots with performance competitive to noncertified codes (Guillemot et al., 2024).

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Predictor–corrector path tracking thus encompasses a suite of rigorously analyzed techniques for trajectory following in nonlinear, nonsmooth, stochastic, and high-dimensional frameworks, driven by advances in semismooth analysis, interval certification, and computational optimization.