Robotic systems routinely encounter conflicting objectives, modeling errors, and degenerate contact conditions that render quadratic programs (QPs) infeasible. Yet most optimization solvers and differentiable QP layers assume feasibility, leading to numerical failures, unstable gradients, or solver breakdown when constraints cannot be simultaneously satisfied. We present Elastic ODYN, a primal--dual non-interior-point QP solver that handles infeasibility through smooth squared-$\ell_2$ elastic relaxations. The resulting formulation remains well posed under ill-conditioning and degeneracy, supports warm starting, and converges to closest-to-feasible solutions when no feasible point exists. A lightweight refinement stage recovers physically meaningful dual variables from the elastic solution. Building on this framework, we develop Elastic OdynLayer, a differentiable QP layer with stable gradients under infeasibility, and Elastic OdynSQP, an infeasibility-aware SQP method that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint relaxation. We evaluate the framework on benchmark QPs, singular contact mechanics, differentiable parameter identification, and quadrupedal and humanoid trajectory optimization. Across all settings, Elastic ODYN consistently outperforms state-of-the-art elastic QP solvers in robustness, warm-start performance, and convergence reliability, enabling optimization, simulation, control, and learning beyond the feasibility assumptions of existing methods.

翻译：机器人系统常面临冲突目标、建模误差及退化接触条件，导致二次规划问题不可行。然而多数优化求解器与可微QP层均假设可行域存在，在约束无法同时满足时会出现数值失败、梯度不稳定或求解器崩溃。我们提出Elastic ODYN——一种通过光滑平方ℓ₂弹性松弛处理不可行性的原始-对偶非内点QP求解器。该公式在病态与退化条件下仍保持良定性，支持热启动，并在无可行点时收敛至最接近可行解。轻量级精化阶段可从弹性解中恢复物理意义明确的拉格朗日乘子。基于此框架，我们开发了Elastic OdynLayer（一种在不可行条件下具有稳定梯度的可微QP层）与Elastic OdynSQP（一种不可行感知的序列二次规划方法，通过选择性约束松弛解决不一致子问题与固有不可行的最优控制任务）。我们在基准QP测试、奇异接触力学、可微参数辨识及四足与仿人机器人轨迹优化中评估该框架。在所有场景下，Elastic ODYN在鲁棒性、热启动性能与收敛可靠性方面均持续超越现有最优弹性QP求解器，使优化、仿真、控制与学习突破现有方法对可行性的依赖假设。