Barrier SQP for Nonlinear Constrained Optimization

• Barrier SQP is an advanced nonlinear optimization method combining sequential quadratic programming with barrier functions to enforce strict feasibility.
• It constructs quadratic programming subproblems with logarithmic barriers to manage inequality constraints and uses higher-order corrections to mitigate the Maratos effect.
• The method achieves global and superlinear convergence, proving effective in process optimization, optimal control, and large-scale economic dispatch.

Barrier Sequential Quadratic Programming (Barrier SQP) is an advanced class of methods for nonlinear constrained optimization, which integrate sequential quadratic programming (SQP) and barrier function techniques to handle general nonlinear programs with equality and inequality constraints. The key distinction of Barrier SQP algorithms lies in the incorporation of interior-point/barrier mechanisms—typically logarithmic barrier terms—into the SQP framework, ensuring that iterations remain strictly within the feasible region defined by inequalities, while maintaining rapid convergence properties and handling active constraints robustly.

1. Core Principles and Formulation

Barrier SQP extends classical SQP methods by embedding barrier functions directly into the QP subproblem solved at each iteration. For a general nonlinear program:

$$\min_{x} f_0(x) \quad \text{subject to} \quad f_i(x) \leq 0, \ i \in I_1, \quad f_j(x) = 0, \ j \in I_2$$

SQP techniques linearize the constraints and quadratic-approximate the Lagrangian, producing at each iteration a QP subproblem restricting movement to feasible directions. To enforce strict feasibility, Barrier SQP augments the reformulated objective (often with equality constraints embedded via penalties, such as $F_c(x) = f_0(x) - c\sum_{i \in I_2} f_i(x)$) with barrier terms for the inequalities:

$$\min_d \ \nabla F_c(x^k)^T d + \frac12 d^T H_k d - \mu \sum_{i \in I_1} \log \left( -\left[ \bar{f}_i(x^k) + g_i(x^k)^T d \right] \right)$$

where $\mu > 0$ is the barrier parameter, iteratively decreased to drive the solution towards the boundary, and $\bar{f}_i(x)$ are shifted constraints ensuring feasibility for the expansion point.

The approach ensures iterates stay in the interior of the feasible region, where the unconstrained minimization of the merit function is well-posed. This inclusion of barrier terms is crucial in handling optimization landscapes with multiple active constraints or poorly scaled Jacobians.

2. Algorithmic Structure and Search Direction Generation

Barrier SQP algorithms typically deploy multi-step direction generation strategies to address limitations of pure-QP steps. As observed in (Guo et al., 2012), the process involves:

• Solving an always-feasible QP subproblem (with barrier terms as above) to obtain a primary direction $d_0$.
• Computing correction directions via one or multiple systems of linear equations (SLEs), constructed from constraint gradients and Hessian approximations, to 'repair' constraint violations or enhance descent properties.
• Aggregating directions using convex combinations (e.g., $\hat{d}^k = (1-\beta_k)d_0^k + \beta_k d_1^k$), with $\beta_k$ selected to satisfy descent criteria or to enforce interior feasibility.
• Including higher-order correction directions via additional SLEs to overcome the Maratos effect, namely the phenomenon where full-step acceptance by the merit function fails due to higher-order Taylor expansion discrepancies.

This structured directional refinement is carried forward into the barrier variant, ensuring that both the interiority and the sufficient decrease in merit functions are maintained, which is essential for convergence, especially in the presence of tight or conflicting constraints.

3. Barrier Parameter Update and Line Search

The barrier parameter $\mu$ is updated multiplicatively (commonly by factors $\sigma<1$): $\mu^+ = \sigma \mu$ at suitable iterations, driving solution iterates closer to the feasible boundary as optimization progresses. Line search mechanisms, such as Armijo or non-monotonic (watchdog) schemes, are implemented to safeguard sufficient decrease in a specially designed merit function that encodes both KKT stationarity and barrier-enforced feasibility. For example, the merit function may take the form:

$$\phi(x, s; \mu) = \frac12 \|\nabla \mathcal{L}(x, \lambda)\|_2^2 + \mu \|C(x) - s\|_1$$

with $s = \min(0, C(x))$ handling the slack/violation component, and the direction selection and step acceptance governed by the decrease in $\phi$ and satisfaction of directional derivatives.

Higher-order correction (cf. (Guo et al., 2012), Equation (5)) is used to defeat the Maratos effect, with corrections of the form:

$$\Gamma_k d_2^k = -(\|d_0^k\|^\tau + [\phi(x^k)]^\sigma)\omega - \varphi(x^k + d_0^k)$$

Parameters $\tau \in (2,3)$, $\sigma \in (0,1)$ calibrate the correction scaling.

4. Convergence and Performance Analysis

Barrier SQP algorithms can achieve both global and superlinear convergence under mild assumptions, without requiring strong conditions such as strict complementarity (per (Guo et al., 2012)). If the Hessian approximation $H_k$ is sufficiently accurate along computed directions (cf. superlinear: $$\|\nabla_{xx}^2 L_c(x^*, \lambda^*) - H_k)d_0^k\| = o(\|d_0^k\|)$$), then iteration error contracts superlinearly:

$\|x^{k+1} - x^*\| = o(\|x^k - x^*\|)$

Global convergence is assured provided the merit function is decreasing and the line search conditions are maintained. Numerical results on standard test sets (Hock–Schittkowski, Svanberg problems in CUTE) indicate Barrier SQP algorithms require fewer iterations and constraint evaluations compared to infeasible-SQP or SLE-based algorithms, and are competitive with established solvers such as SNOPT.

5. Implementation Considerations and Extensions

Key implementation details for Barrier SQP:

• The transformation of equality constraints into penalized objectives permits reformulation into inequality-only (barrier-ready) form. This systematic embedding ensures compatibility with the barrier terms and is essential for interior-point style convergence.
• Search direction refinement using QP–SLE combinations and higher-order corrections is required for robust step acceptance, particularly as iterates approach boundaries or near-tight constraints.
• Interior-point modifications, such as structured modified Newton methods employing low-rank Jacobian updates (Ek et al., 2020), reduce computational expense while maintaining sparsity and solution accuracy in the barrier subproblems.

Modern extensions include stochastic optimization variants (Na et al., 2021, Wang et al., 2023, Berahas et al., 26 May 2025), where stochastic objectives are handled by adaptive sampling and merit functions are constructed using exact augmented Lagrangians or Clarke subgradients, adapted for barrier constraints and robust line search. Complexities are optimal (e.g., O(ε⁻⁴) gradient evaluations and O(ε⁻²) subproblem solves) and batch size selection is driven adaptively by error/variance norm tests.

Quantum-assisted frameworks (Dehaghani et al., 20 Oct 2025) further enhance the scalability of Barrier SQP. Block encoding and quantum singular value transformation (QSVT) are used to invert the barrier-augmented Hessian and solve the Schur complement step in polylogarithmic time (complexity O(polylog(n_z/ε) * κ * log(1/ε′))), with explicit input-to-state stability guarantees in terms of barrier parameters and quantum error bounds.

6. Application Domains and Comparative Analysis

Barrier SQP methods are applied in diverse domains:

• Process optimization under numerical noise (Ma et al., 16 Feb 2024): Hybrid relaxation within QP/LSQ subproblems and rapid “repair” mechanisms.
• Optimal control and dynamic games (Abhijeet et al., 3 Oct 2025, Zhu et al., 2022, Zhu et al., 29 Mar 2024): Unified SQP frameworks relate to iLQR/DDP, yielding descent guarantees and robust line search-based globalization; merit functions for GNE include KKT stationarity and l₁ penalties or can be augmented by barrier terms.
• Economic dispatch in large-scale power systems (Jian et al., 2023): Splitting SQP decompositions reduce computational cost by handling box and linear constraints via slack variable projections.

Comparison with standard interior-point methods, robust-SQP, and augmented Lagrangian solvers indicates that Barrier SQP integrates sophisticated merit function, constraint penalization, and higher-order corrections, outperforming in iteration count and solution accuracy for both exact and stochastic variants.

7. Open Problems and Future Research Directions

Areas of continuing research involve:

• Integration of barrier SQP with adaptive sampling and dynamic batch size control for zero-order and stochastic optimization.
• Refinement of higher-order correction techniques to extend superlinear convergence without strict complementarity, particularly in tightly constrained or ill-conditioned scenarios.
• Efficient low-rank updating and structured modified Newton methods for barrier subproblems in large-scale settings, including quantum subroutine acceleration.
• Application of barrier SQP to multi-agent dynamic games with complex constraints, leveraging advanced merit functions and non-monotone globalization schemes.
• Robustness analysis under noise and convergence rate characterization in real-time or high-dimensional control applications.

Barrier Sequential Quadratic Programming thus represents an overview of classical SQP and modern interior-point/barrier function methodology, with extensions toward large-scale, stochastic, and quantum-accelerated optimization. The contributions documented in (Guo et al., 2012, Ek et al., 2020, Dehaghani et al., 20 Oct 2025), and related works anchor much of this methodology and set a foundation for further technical advances.