We develop a theoretically-grounded learning method for the Generalized Linear Programming Value Function (GVF), which models the optimal value of a linear programming (LP) problem as its objective and constraint bounds vary. This function plays a fundamental role in algorithmic techniques for large-scale optimization, particularly in decomposition for two-stage mixed-integer linear programs (MILPs). This paper establishes a structural characterization of the GVF that enables it to be modeled as a particular neural network architecture, which we then use to learn the GVF in a way that benefits from three notable properties. First, our method produces a true under-approximation of the value function with respect to the constraint bounds. Second, the model is input-convex in the constraint bounds, which not only matches the structure of the GVF but also enables the trained model to be efficiently optimized over using LP. Finally, our learning method is unsupervised, meaning that training data generation does not require computing LP optimal values, which can be prohibitively expensive at large scales. We numerically show that our method can approximate the GVF well, even when compared to supervised methods that collect training data by solving an LP for each data point. Furthermore, as an application of our framework, we develop a fast heuristic method for large-scale two-stage MILPs with continuous second-stage variables, via a compact reformulation that can be solved faster than the full model linear relaxation at large scales and orders of magnitude faster than the original model.

We present a new approach for modeling avoidance constraints in 2D environments, in which waypoints are assigned to obstacle-free polyhedral regions. Constraints of this form are often formulated as mixed-integer programming (MIP) problems employing big-M techniques-however, these are generally not the strongest formulations possible with respect to the MIP's convex relaxation (so called ideal formulations), potentially resulting in larger computational burden. We instead model obstacle avoidance as combinatorial disjunctive constraints and leverage the independent branching scheme to construct small, ideal formulations. As our approach requires a biclique cover for an associated graph, we exploit the structure of this class of graphs to develop a fast subroutine for obtaining biclique covers in polynomial time. We also contribute an open-source Julia library named ClutteredEnvPathOpt to facilitate computational experiments of MIP formulations for obstacle avoidance. Experiments have shown our formulation is more compact and remains competitive on a number of instances compared with standard big-M techniques, for which solvers possess highly optimized procedures.

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in- and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.

We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is significantly stronger than the commonly used univariate-input relaxation which has been proposed as a natural convex relaxation barrier for verification. While our description of the relaxation may require an exponential number of inequalities, we show that they can be separated in linear time and hence can be efficiently incorporated into optimization algorithms on an as-needed basis. Based on this novel relaxation, we design two polynomial-time algorithms for neural network verification: a linear-programming-based algorithm that leverages the full power of our relaxation, and a fast propagation algorithm that generalizes existing approaches. In both cases, we show that for a modest increase in computational effort, our strengthened relaxation enables us to verify a significantly larger number of instances compared to similar algorithms.

We present an auto-tuning system for optimizing I/O performance of HDF5 applications and demonstrate its value across platforms, applications, and at scale. The system uses a genetic algorithm to search a large space of tunable parameters and to identify effective settings at all layers of the parallel I/O stack. The parameter settings are applied transparently by the auto-tuning system via dynamically intercepted HDF5 calls.