infeasibility-aware sqp

Existing open-source and commercial Sequential Quadratic Programming (SQP) solvers suffer from limited transparency and poor modularity, hindering algorithmic customization and component reuse. To address this, we propose a highly modular, open-source Python implementation of SQP that enables flexible substitution of core components—including merit functions (e.g., smoothed augmented Lagrangian), Hessian approximations (e.g., BFGS), line search strategies, and QP subproblem solvers. Our key contribution lies in the rigorous decoupling of these algorithmic modules, significantly enhancing reconfigurability, extensibility, and interpretability—thereby bridging a critical gap in openness and flexibility among SQP tools. Empirical evaluation on the CUTEst benchmark suite demonstrates robust convergence behavior and competitive overall performance relative to state-of-the-art solvers such as SLSQP, SNOPT, and IPOPT.

为解决机器人控制中QP求解器因约束冲突导致的不可行问题，提出ElastiQP算法，通过引入l1惩罚项并保持线性系统规模不变，确保快速获得可行解。

This paper addresses large-scale nonconvex quadratic integer programming (QIP), encompassing both unconstrained (UQIP) and constrained (CQIP) NP-hard variants. Methodologically, it introduces a novel metaheuristic framework centered on deriving, for the first time, closed-form expressions for objective-value increments under single-variable flips, thereby establishing necessary and sufficient conditions for 1-Opt local optimality. Leveraging these insights, the authors design a tabu search algorithm that integrates closed-form gradient-based updates with an oscillation mechanism to effectively escape local optima. The approach supports large-scale neighborhood structures and achieves high-quality solutions in seconds on instances with up to 8,000 variables—significantly outperforming Gurobi 11.0.2. This work provides the first practically scalable solver for high-dimensional nonconvex QIP that combines theoretical rigor with computational efficiency.

Selecting appropriate quadratic programming (QP) solvers for real-time control of legged robots remains challenging due to poor solver selection guidelines and weak embedded-system compatibility. Method: This work systematically benchmarks mainstream convex QP algorithms—interior-point, active-set, operator-splitting, and augmented Lagrangian methods—across representative tasks including inverse dynamics, model predictive control (MPC), and whole-body control. It introduces a novel four-category taxonomy of QP solvers tailored to legged robotics and conducts structured empirical evaluation using structure-aware warm-starting, sparse-matrix optimizations, and multi-platform benchmarking on public datasets. Contribution/Results: We identify hardware–task–algorithm co-design principles—for instance, sparse interior-point methods suit long-horizon MPC, while dense active-set methods excel in high-frequency whole-body control—and extend insights to nonconvex and distributed QP settings. Quantitative metrics include computation latency, constraint satisfaction accuracy, and disturbance robustness. The study delivers a reusable, empirically grounded solver selection guide enabling millisecond-level response, low-power operation, and high-reliability autonomous locomotion.

This paper addresses constrained stochastic nonlinear optimization problems arising in online statistical inference. We propose Sketch-StoSQP, a sketched stochastic sequential quadratic programming method. Our key contributions are threefold: (i) We establish, for the first time, the asymptotic normality of StoSQP iterates under controllable, non-vanishing approximation errors—ensuring stable per-iteration computational complexity; (ii) We design a plug-and-play covariance estimator enabling immediate statistical inference without algorithmic modification; (iii) We prove that the scaled residual sequence converges in distribution to a non-degenerate zero-mean Gaussian. Empirical evaluation on the CUTEst benchmark and constrained regression tasks demonstrates both statistical validity—accurate coverage rates and well-calibrated confidence intervals—and computational efficiency—constant per-iteration cost and significant overall speedup.

This work addresses the issue that quadratic penalty relaxations of binary linear programs often yield spurious or infeasible local minima. To overcome this, we propose a class of QUBO relaxation models satisfying specific structural conditions that guarantee all local minima are feasible and strictly binary. Leveraging these conditions, we derive novel differentiable relaxations for classical combinatorial optimization problems—including open-pit mining, the 0–1 knapsack problem, and the traveling salesman problem—and solve them using gradient-based optimizers such as projected gradient descent and Adam. Experimental results demonstrate that the proposed approach reliably converges to valid binary solutions, thereby establishing clear theoretical guarantees and delineating the applicability boundaries of differentiable optimization as a local solver for combinatorial problems.

This work addresses the challenge of generating safe, dynamically feasible flight trajectories in real time for high-density urban low-altitude environments. The authors propose a scalable sequential quadratic programming (SQP) framework that unifies environmental geometric constraints, operational limits, and full six-degree-of-freedom aircraft dynamics into a single optimization model. A key innovation lies in dynamically generating separating hyperplanes during each SQP iteration to enable immediate collision avoidance, while a variable-resolution quadtree spatial decomposition ensures real-time performance even in large-scale urban scenarios. Experimental results across five real-world city environments demonstrate 100% mission success and guaranteed collision avoidance using only CPU computation, significantly outperforming conventional approaches such as standard SQP, iterative Linear Quadratic Regulator (iLQR), and Differential Dynamic Programming (DDP).

This study addresses the absence of a unified continuous modeling and compilation framework for discrete NP combinatorial optimization problems. To this end, it proposes a unified paradigm that reformulates discrete NP problems as continuous standard quadratic programs (StQPs). Building upon a simplex framework, the work integrates graph reductions, regularized Motzkin–Straus formulations, and QUBO mapping techniques to construct a coefficient-bounded finite-domain factor model compiler, requiring at most four simplex coordinates per binary factor. The primary contribution is the exact, standardized StQP compilation of Karp’s 21 NP-complete problems, accompanied by explicit formulas, dimensionality analyses, and separation bounds. This formulation rigorously guarantees that all valid assignments correspond to global or local minima, thereby establishing a complete continuous solution pathway for combinatorial optimization.

This work addresses the challenge of infeasible quadratic programs (QPs) in robotic systems—arising from conflicting objectives, modeling errors, or degenerate contacts—which commonly cause numerical failure in existing differentiable QP solvers that assume feasibility. To overcome this, we propose Elastic ODYN, a primal-dual non-interior-point QP solver based on a smooth ℓ₂ elastic relaxation that converges to the closest feasible solution when constraints are unsatisfiable and recovers physically consistent dual variables via lightweight refinement. Our method enables, for the first time, stable differentiable optimization over infeasible QPs, supports warm-starting, and robustly handles degenerate scenarios. We introduce the differentiable Elastic OdynLayer and an infeasibility-aware sequential quadratic programming (SQP) framework, Elastic OdynSQP. Experiments on standard QPs, singular contacts, parameter identification, and trajectory optimization for quadrupedal and humanoid robots demonstrate significant improvements over prior approaches in robustness, warm-start performance, and convergence reliability.

This work addresses the scalability and convergence challenges of large-scale conic quadratic programming (CQP) with multiple conic and affine constraints by proposing PDHCG-CQP, a GPU-accelerated first-order solver built upon a restarted averaged primal-dual hybrid gradient framework. The method integrates inexact primal updates, batched conic projections, reflected Halpern acceleration, and two-dimensional data-partitioned multi-GPU parallelism. Under strict complementarity conditions, it establishes local linear convergence guarantees. Leveraging matrix-free linear algebra and device-resident KKT residual computation, the solver achieves state-of-the-art robustness across benchmark problems—including QP, QCQP, SOCP, and Fisher market equilibria—demonstrating efficient scaling up to 8 GPUs and handling instances with up to 440 million variables.