A General and Streamlined Differentiable Optimization Framework

Differentiating through constrained optimization problems is increasingly central to learning, control, and large-scale decision-making systems, yet practical integration remains challenging due to solver specialization and interface mismatches. This paper presents a general and streamlined framework-an updated DiffOpt.jl-that unifies modeling and differentiation within the Julia optimization stack. The framework computes forward - and reverse-mode solution and objective sensitivities for smooth, potentially nonconvex programs by differentiating the KKT system under standard regularity assumptions. A first-class, JuMP-native parameter-centric API allows users to declare named parameters and obtain derivatives directly with respect to them - even when a parameter appears in multiple constraints and objectives - eliminating brittle bookkeeping from coefficient-level interfaces. We illustrate these capabilities on convex and nonconvex models, including economic dispatch, mean-variance portfolio selection with conic risk constraints, and nonlinear robot inverse kinematics. Two companion studies further demonstrate impact at scale: gradient-based iterative methods for strategic bidding in energy markets and Sobolev-style training of end-to-end optimization proxies using solver-accurate sensitivities. Together, these results demonstrate that differentiable optimization can be deployed as a routine tool for experimentation, learning, calibration, and design-without deviating from standard JuMP modeling practices and while retaining access to a broad ecosystem of solvers.

Sobolev Training of End-to-End Optimization Proxies

Optimization proxies - machine learning models trained to approximate the solution mapping of parametric optimization problems in a single forward pass - offer dramatic reductions in inference time compared to traditional iterative solvers. This work investigates the integration of solver sensitivities into such end to end proxies via a Sobolev training paradigm and does so in two distinct settings: (i) fully supervised proxies, where exact solver outputs and sensitivities are available, and (ii) self supervised proxies that rely only on the objective and constraint structure of the underlying optimization problem. By augmenting the standard training loss with directional derivative information extracted from the solver, the proxy aligns both its predicted solutions and local derivatives with those of the optimizer. Under Lipschitz continuity assumptions on the true solution mapping, matching first order sensitivities is shown to yield uniform approximation error proportional to the training set covering radius. Empirically, different impacts are observed in each studied setting. On three large Alternating Current Optimal Power Flow benchmarks, supervised Sobolev training cuts mean squared error by up to 56 percent and the median worst case constraint violation by up to 400 percent while keeping the optimality gap below 0.22 percent. For a mean variance portfolio task trained without labeled solutions, self supervised Sobolev training halves the average optimality gap in the medium risk region (standard deviation above 10 percent of budget) and matches the baseline elsewhere. Together, these results highlight Sobolev training whether supervised or self supervised as a path to fast reliable surrogates for safety critical large scale optimization workloads.