Mixed Binary Quadratic Programs (MBQPs) are a class of NP-hard problems that arise in a wide range of applications, including finance, machine learning, and chemical and energy systems. Large-scale MBQPs are challenging to solve with exact algorithms due to the combinatorial search space and nonlinearity. Primal heuristics have been developed to quickly identify high-quality solutions to challenging combinatorial optimization problems. In this paper, we propose an extension for two well-established rounding-based primal heuristics, RENS and Undercover. Instead of using the optimal solution to a relaxation for variable rounding and search as in RENS, we use a suboptimal relaxation solution of the MBQP as the basis for rounding and guidance for searching over a restricted subproblem where a certain percentage of binary variables are free. We apply a similar idea to the Undercover heuristic that fixes a variable cover to the rounded relaxation values. Instead, we relax a subset of the cover variables based on the suboptimal relaxation and search over a larger restricted subproblem. We evaluate our proposed methods on synthetic MBQP benchmarks and real-world wind farm layout optimization problem instances. The results show that our proposed heuristics identify high-quality solutions within a small time limit and significantly reduce the primal gap and primal integral compared to RENS, Undercover, and solvers with additional primal heuristics integrated inside Branch-and-Bound.

Modeling dynamical systems is crucial for a wide range of tasks, but it remains challenging due to complex nonlinear dynamics, limited observations, or lack of prior knowledge. Recently, data-driven approaches such as Neural Ordinary Differential Equations (NODE) have shown promising results by leveraging the expressive power of neural networks to model unknown dynamics. However, these approaches often suffer from limited labeled training data, leading to poor generalization and suboptimal predictions. On the other hand, semi-supervised algorithms can utilize abundant unlabeled data and have demonstrated good performance in classification and regression tasks. We propose TS-NODE, the first semi-supervised approach to modeling dynamical systems with NODE. TS-NODE explores cheaply generated synthetic pseudo rollouts to broaden exploration in the state space and to tackle the challenges brought by lack of ground-truth system data under a teacher-student model. TS-NODE employs an unified optimization framework that corrects the teacher model based on the student's feedback while mitigating the potential false system dynamics present in pseudo rollouts. TS-NODE demonstrates significant performance improvements over a baseline Neural ODE model on multiple dynamical system modeling tasks.

We present differentiable predictive control (DPC), a method for offline learning of constrained neural control policies for nonlinear dynamical systems with performance guarantees. We show that the sensitivities of the parametric optimal control problem can be used to obtain direct policy gradients. Specifically, we employ automatic differentiation (AD) to efficiently compute the sensitivities of the model predictive control (MPC) objective function and constraints penalties. To guarantee safety upon deployment, we derive probabilistic guarantees on closed-loop stability and constraint satisfaction based on indicator functions and Hoeffding’s inequality. We empirically demonstrate that the proposed method can learn neural control policies for various parametric optimal control tasks. In particular, we show that the proposed DPC method can stabilize systems with unstable dynamics, track time-varying references, and satisfy nonlinear state and input constraints. Our DPC method has practical time savings compared to alternative approaches for fast and memory-efficient controller design. Specifically, DPC does not depend on a supervisory controller as opposed to approximate MPC based on imitation learning. We demonstrate that, without losing performance, DPC is scalable with greatly reduced demands on memory and computation compared to implicit and explicit MPC while being more sample efficient than model-free reinforcement learning (RL) algorithms.

Normalizing flows (NF) build upon invertible neural networks and have wide applications in probabilistic modeling. Currently, building a powerful yet computationally efficient flow model relies on empirical fine-tuning over a large design space. While introducing neural architecture search (NAS) to NF is desirable, the invertibility constraint of NF brings new challenges to existing NAS methods whose application is limited to unstructured neural networks. Developing efficient NAS methods specifically for NF remains an open problem. We present AutoNF, the first automated NF architectural optimization framework. First, we present a new mixture distribution formulation that allows efficient differentiable architecture search of flow models without violating the invertibility constraint. Second, under the new formulation, we convert the original NP-hard combinatorial NF architectural optimization problem to an unconstrained continuous relaxation admitting the discrete optimal architectural solution, circumventing the loss of optimality due to binarization in architectural optimization. We evaluate AutoNF with various density estimation datasets and show its superior performance-cost trade-offs over a set of existing hand-crafted baselines.

Deep Markov models (DMM) are generative models which are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we present a novel stability analysis method and provide sufficient conditions of DMM's stochastic stability. The proposed stability analysis is based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation function on the stability and overall dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we propose a few practical methods for designing constrained DMMs with guaranteed stability. We empirically substantiate our theoretical results via intuitive numerical experiments using the proposed stability constraints.