We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable performance measures such as AUC, multi-class AUC, $F$-measure and others. A common feature of the optimization model that emerges from these tasks is that it involves solving a Linear Programs (LP) during training where representations learned by upstream layers characterize the constraints or the feasible set. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of ingenious ideas proposed by Mangasarian for 1-norm SVMs -- which advocates for solving LPs with a generalized Newton method -- provides a simple and effective solution that can be run on the GPU. In particular, this strategy needs little unrolling, which makes it more efficient during backward pass. Further, even when the constraint matrix is too large to fit on the GPU memory (say large minibatch settings), we show that running the Newton method in a lower dimensional space yields accurate gradients for training, by utilizing a statistical concept called {\em sufficient} dimension reduction. While a number of specialized algorithms have been proposed for the models that we describe here, our module turns out to be applicable without any specific adjustments or relaxations. We describe each use case, study its properties and demonstrate the efficacy of the approach over alternatives which use surrogate lower bounds and often, specialized optimization schemes. Frequently, we achieve superior computational behavior and performance improvements on common datasets used in the literature.

The widespread use of big data across sectors has raised major privacy concerns, especially when sensitive information is shared or analyzed. Regulations such as GDPR and HIPAA impose strict controls on data handling, making it difficult to balance the need for insights with privacy requirements. Synthetic data offers a promising solution by creating artificial datasets that reflect real patterns without exposing sensitive information. However, traditional synthetic data methods often fail to capture complex, implicit rules that link different elements of the data and are essential in domains like healthcare. They may reproduce explicit patterns but overlook domain-specific constraints that are not directly stated yet crucial for realism and utility. For example, prescription guidelines that restrict certain medications for specific conditions or prevent harmful drug interactions may not appear explicitly in the original data. Synthetic data generated without these implicit rules can lead to medically inappropriate or unrealistic profiles. To address this gap, we propose ContextGAN, a Context-Aware Differentially Private Generative Adversarial Network that integrates domain-specific rules through a constraint matrix encoding both explicit and implicit knowledge. The constraint-aware discriminator evaluates synthetic data against these rules to ensure adherence to domain constraints, while differential privacy protects sensitive details from the original data. We validate ContextGAN across healthcare, security, and finance, showing that it produces high-quality synthetic data that respects domain rules and preserves privacy. Our results demonstrate that ContextGAN improves realism and utility by enforcing domain constraints, making it suitable for applications that require compliance with both explicit patterns and implicit rules under strict privacy guarantees.

The complexity of the traditional LP solver is still at least quadratic in the problem dimension, i.e., the

Over the past decades, Linear Programming (LP) has been widely used in different areas and considered as one of the mature technologies in numerical optimization. However, the complexity offered by state-of-the-art algorithms (i.e.

We present a Julia-based interface to the precompiled HALLaR and cuHALLaR binaries for large-scale semidefinite programs (SDPs). Both solvers are established as fast and numerically stable, and accept problem data in formats compatible with SDPA and a new enhanced data format taking advantage of Hybrid Sparse Low-Rank (HSLR) structure. The interface allows users to load custom data files, configure solver options, and execute experiments directly from Julia. A collection of example problems is included, including the SDP relaxations of the Matrix Completion and Maximum Stable Set problems.

We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable performance measures such as AUC, multi-class AUC, F -measure and others. A feature of the optimization model that emerges from these tasks is that it involves solving a Linear Programs (LP) during training where representations learned by upstream layers characterize the constraints or the feasible set. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of ingenious ideas proposed by Mangasarian for 1-norm SVMs - which advocates for solving LPs with a generalized Newton method - provides a simple and effective solution that can be run on the GPU. In particular, this strategy needs little unrolling, which makes it more efficient during the backward pass. Further, even when the constraint matrix is too large to fit on the GPU memory (say large minibatch settings), we show that running the Newton method in a lower dimensional space yields accurate gradients for training, by utilizing a statistical concept called sufficient dimension reduction. While a number of specialized algorithms have been proposed for the models that we describe here, our module turns out to be applicable without any specific adjustments or relaxations. We describe each use case, study its properties and demonstrate the efficacy of the approach over alternatives which use surrogate lower bounds and often, specialized optimization schemes. Frequently, we achieve superior computational behavior and performance improvements on common datasets used in the literature.

--This paper proposes a semidefinite relaxation for landmark-based localization with unknown data associations in planar environments. The proposed method simultaneously solves for the optimal robot states and data associations in a globally optimal fashion. Relative position measurements to a fixed set of known landmarks are used, but the data association is unknown in that the robot does not know which landmark each measurement is generated from. The relaxation is shown to be tight in a majority of cases for moderate noise levels. The proposed algorithm is compared to local Gauss-Newton baselines initialized at the dead-reckoned trajectory, and is shown to significantly improve convergence to the problem's global optimum in simulation and experiment. STIMA TING the state of a robot from noisy and incomplete sensor data is a central task associated with autonomy. In the landmark-based localization task, the robot infers its position and orientation from measurements from landmarks with known positions. State estimation methods for localization can be split into filtering methods and batch optimization methods [1].