Parabolic Target-Space Interior-Point Methods

• Parabolic target-space interior-point algorithms are a modern approach that leverages paraboloid-like geometry to track strictly feasible iterates and achieve strongly polynomial global complexity.
• They employ predictor-corrector mechanisms with universal tangent directions and higher-order predictors to deliver superlinear, quadratic, or cubic local convergence near optimal solutions.
• The framework ensures single-phase convergence from any strictly feasible starting point, integrating efficient basis detection and robust proximity measures for convex optimization and variational inequalities.

Parabolic target-space interior-point algorithms form a modern sub-class of path-following methods for solving high-dimensional convex optimization, variational inequalities, and monotone complementarity problems. These algorithms combine convex-analytic central-path tracking with “parabolic target-space” geometry, resulting in strongly polynomial global complexity bounds and—crucially—enhanced local superlinear or even higher-order convergence in neighborhoods of the solution. This article details the theoretical underpinnings, algorithmic structure, convergence properties, foundational mathematical tools, and practical implications of this class, consolidating the main developments drawn from recent research.

1. Conceptual Foundation and Parabolic Target-Space Framework

The parabolic target-space approach generalizes classical central-path interior-point methods by introducing explicit target parameterizations (e.g., an auxiliary “controller” variable $w$) that guide the iterate trajectory not only along affine lines but within a paraboloid-like geometry in the extended primal-dual space. In standard LP or LCP cases, each iteration targets a "parabolic target"—for example, a vector in the $(x,s)$ or augmented $(x, s, y, w)$ space—using a proximity function (e.g., a scaled logarithmic barrier) to maintain iterates in neighborhoods with controlled centrality measures.

The method is inherently “one-phase”: it accepts any strictly feasible initial primal-dual pair and transitions directly and smoothly along a parabolic trajectory toward the optimal set, obviating the need for preliminary centering phases or forcibly keeping the iterates on a classical central path. The fundamental structural property is that a strictly feasible strictly interior point always exists, and the proximity function can be defined relative to the parabolic target to maintain convergence guarantees.

2. Algorithmic Structure and Predictor Directions

The canonical parabolic target-space interior-point algorithm proceeds via a predictor–corrector mechanism at each iteration:

1. Predictor Step:
   • Compute a search direction $\Delta$ using the linearized KKT system:

   $$X\Delta s + S\Delta x = d,\qquad A\Delta x = 0,\qquad \Delta s + A^\top \Delta y = 0,$$

   where $X$ and $S$ are the diagonal matrices of current primal and dual variables, and $d$ is a right-hand side crafted to decrease the chosen proximity measure (e.g., functional “centrality” relative to the parabolic target). - The Universal Tangent Direction (UTD) is a widely employed predictor: it is cheap to compute, remains well-conditioned globally, and is defined independently of the local perturbation $d$. It is constructed using the “parabolic controller” variable $w$ and current residuals.

2. Enhanced Predictor Strategies:

   • An auto-correcting predictor adds a correction term to the standard UTD direction. The correction is determined to exactly counter the residual $x_is_i-\rho(w)$ (where $\rho(w)$ is the targeted complementarity product), thus strongly improving the local behavior and accelerating convergence near the solution.
   • A second-order predictor (or quadratic prediction) appends an additional term so that the update approximates a local segment of the central path by a quadratic curve, enabling even cubic local convergence.
3. Corrector Step (optional or implicit in some variants):
   • Update or project the current iterate back into the self-concordant neighborhood if the predictor step takes the point outside the prescribed centrality region, using the same or a related KKT system and proximity measure.
4. Proximity and Neighborhood Maintenance:
   • The proximity measure $\Psi(z)$ (often defined as $$\Psi(z) = -\sum_{i=0}^n \ln r_i(z) + (n+1)\ln \rho(w)$$, with $r_i(z)$ the component-wise centrality residuals) is monitored at each iteration to ensure iterates stay within a self-concordant or analytic center neighborhood.

This algorithmic pattern is parameterized by the choice of predictor direction and neighborhood control strategy—these underpin the complexity and convergence behavior.

3. Convergence Theory: Global Complexity and Local Acceleration

The parabolic target-space framework guarantees optimal or near-optimal global complexity:

• Worst-Case Complexity: The number of outer iterations required to achieve an $\epsilon$-accurate solution is $\mathcal{O}(\sqrt{n}\ln(1/\epsilon))$ for $n$ variables, matching the best-known theoretical bounds for path-following interior-point algorithms.
• Local Superlinear to Cubic Convergence:
  • The UTD-based algorithm achieves superlinear local convergence once the functional proximity is reduced below a threshold.
  • With auto-correcting predictors, this improves to quadratic convergence; specifically, for error $f_k$ near the solution, $f_{k+1} = \mathcal{O}(f_k^2)$.
  • Second-order (quadratic) prediction yields cubic convergence: $f_{k+1} = \mathcal{O}(f_k^3)$, a behavior rarely achieved with no significant increase in per-iteration cost.
• Finite Termination with Optimal Basis Detection:
  • The inclusion of optimal basis tests (via indicator vectors computed from $x$, $1/s$, or $xs^{-1}$) ensures finite termination in cases where the solution is sufficiently centered, with the threshold for finite termination sometimes lower than that for local quadratic convergence.

4. Mathematical Underpinnings and Proximity Measures

Parabolic target-space IPMs rely on precise convex-analytic constructs:

• The proximity (centrality) measure relative to the parabolic target is typically barrier-based, e.g.,

$$\Psi(z) = -\sum_{i=0}^n \ln r_i(z) + (n+1)\ln \rho(w),$$

with $r_i(z)$ representing the residual for each variable and $\rho(w)$ the targeted complementarity.

• Centrality control is achieved by forcing the iterate into a neighborhood defined by $\Psi(z)$ not exceeding a chosen threshold $\tau$.
• Convergence analysis utilizes decomposition of the search direction, explicit evaluation of residuals after each predictor–corrector step, and tight control of step sizes via scaling and proximity thresholds.

In the monotone LCP case, the algorithm further exploits the structure of the complementarity system by using matrix splitting and moving all iterates within strictly feasible regions, leveraging the monotonicity of the underlying operator for both global complexity and local acceleration.

5. Numerical Performance and Comparative Benchmarks

Empirical experiments performed on random LP and monotone LCP instances of various dimensions consistently affirm theoretical guarantees:

• Iteration Count: For reducing the duality gap (or complementarity product $x^\top s$) to $10^{-8}$, the required number of predictor steps grows only slowly with problem size and is minimized when using the second-order prediction.
• Local Acceleration: Auto-correcting and second-order predictors outperform pure UTD-based predictors near the solution; the latter sharply accelerates convergence once enclosure within a sufficiently small neighborhood is achieved.
• Corrector Steps: The total number of corrector steps is small compared to predictors even in large-scale instances, reflecting rapid neighborhood reentrance and reduced overhead.

Empirical results show that, although all methods converge globally, the higher-order predictor variants almost always achieve lower iteration counts and sharper error reductions per iteration.

6. Optimization Schemes and Practical Implementation

Practical implementations benefit from:

• Single-Phase Operation: Initialization from any strictly feasible point, no centering or path-following pre-processing phase.
• Efficient Basis Detection: Detection of a candidate optimal basis via sorted indicator vectors and explicit computation (e.g., $x^*(B) = A_B^{-1}b$), which triggers finite termination once a criterion based on proximity measures and barrier parameters is satisfied.
• Robustness to Parameter Choices: The step sizes, proximity thresholds, and neighborhood definitions require only minor tuning, making the schemes robust across a broad range of problem instances.
• Low-Cost Iterations: The cost per iteration is dominated by solution of the KKT system (matrix of size $n \times n$ or $m \times m$, as problem dictates) and basic vector operations.

The parabolic target-space methods are compatible with most standard enhancements such as adaptive neighborhood tightening, higher-order corrections, warm starts, or acceleration strategies based on inertia or dynamic trust regions.

7. Comparative Analysis, Extensions, and Research Implications

Parabolic target-space interior-point algorithms uniquely combine the following properties, as summarized by current research:

Property	Parabolic Target-Space Methods	Classical IPMs	Predictor–Corrector IPMs
Global Complexity Optimality	√	√	√
Local Superlinear/Quadratic/Cubic	√	(mostly superlinear, sometimes quadratic)	(rarely cubic)
Single-Phase Convergence	√	×	×
Efficient Basis/Termination Tests	√	×	×
Empirical Iteration Reduction	√	×	×

• Novelty: The combination of optimal (or near-optimal) global iteration complexity with provably superlinear, quadratic, or even cubic local convergence is rare among interior-point methods.
• Generalizability: The framework applies not only to classical linear programming but also readily adapts to monotone LCP, convex programming, and structured conic scenarios.
• Open Research Directions: Integrating dynamic parameter updates, exploiting higher-order structure for further acceleration, and extending optimal basis detection criteria to broader classes of optimization problems are under active exploration.

The above account derives from principal contributions in the recent literature, notably in the design and analysis of parabolic target-space path-following algorithms for LP and monotone LCP (Nesterov, 19 Dec 2024, -Nagy et al., 29 Jul 2025). These are complemented by empirical verification and comparative research vis-à-vis existing predictor–corrector and classical interior-point frameworks.

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Summary Table: Key Theoretical Properties and Algorithmic Variants

Algorithm Variant	Global Complexity	Local Rate	Basis Detection	Reference
UTD-based Parabolic Target-Space IP	$\mathcal{O}(\sqrt{n}\log(1/\epsilon))$	Superlinear	Yes	(Nesterov, 19 Dec 2024, -Nagy et al., 29 Jul 2025)
Auto-Correcting Predictor	$\mathcal{O}(\sqrt{n}\log(1/\epsilon))$	Quadratic	Yes	(Nesterov, 19 Dec 2024, -Nagy et al., 29 Jul 2025)
Second-Order Predictor	$\mathcal{O}(\sqrt{n}\log(1/\epsilon))$	Cubic	Yes	(Nesterov, 19 Dec 2024)

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These developments position parabolic target-space interior-point algorithms as a leading approach for large-scale, accurate, and efficiently terminating optimization in both theory and practice.