A Penalty Approach for Differentiation Through Black-Box Quadratic Programming Solvers

A new framework called dXPP proposes to differentiate through quadratic program solutions without relying on explicit Karush–Kuhn–Tucker (KKT) conditions, a shift its authors say improves computational efficiency and numerical robustness at scale [1][2]. The work, posted to the arXiv preprint server by researcher Yuxuan Linghu, introduces a penalty-based method that decouples the solving step from the differentiation step [1][2]. In the forward pass, dXPP can use any black-box quadratic programming (QP) solver. The backward pass maps the solution onto a smooth approximate penalty problem and implicitly differentiates through it, requiring only a smaller linear system in the primal variables [2]. This design bypasses the numerical difficulties that can arise when differentiating through the KKT system directly [2]. Quadratic programming underpins many machine-learning pipelines, from support vector machines to control and portfolio optimization. Differentiable optimization—where gradients flow through the solution of an optimization subproblem—has become a building block in end-to-end learnable systems. The dominant approach has been to differentiate through the KKT optimality conditions, but that route can become expensive and fragile as problem size grows [2]. The authors evaluated dXPP on randomly generated QPs, large-scale sparse projection problems, and a multi-period portfolio optimization task [2]. They report that the method is competitive with KKT-based differentiation and achieves substantial speedups on large-scale instances [2]. The implementation has been released as open source [2]. The preprint appeared on arXiv, an open-access repository that hosts e-prints across physics, mathematics, computer science, and related fields [8]. arXiv, which began in 1991, now receives roughly 24,000 submissions per month and has surpassed two million articles [8]. The dXPP manuscript was first submitted on 15 February 2026, with revisions following on 3 March 2026 and 14 June 2026 [1]. Differentiable optimization intersects with broader trends in automated machine learning, where techniques such as Bayesian optimization are used to tune hyperparameters of expensive black-box functions [3]. Multi-objective optimization, which handles trade-offs among conflicting goals, represents another domain where efficient QP differentiation could prove useful [4]. The dXPP framework does not directly address multi-objective settings, but its solver-agnostic design may lower the barrier for integrating differentiable QP layers into larger learning architectures [2].