Oscar Dowson

Oscar Dowson, Robert B Parker, Russel Bent

We present \texttt{MathOptAI.jl}, an open-source Julia library for embedding trained machine learning predictors into a JuMP model. \texttt{MathOptAI.jl} can embed a wide variety of neural networks, decision trees, and Gaussian Processes into a larger mathematical optimization model. In addition to interfacing a range of Julia-based machine learning libraries such as \texttt{Lux.jl} and \texttt{Flux.jl}, \texttt{MathOptAI.jl} uses Julia's Python interface to provide support for PyTorch models. When the PyTorch support is combined with \texttt{MathOptAI.jl}'s gray-box formulation, the function, Jacobian, and Hessian evaluations associated with the PyTorch model are offloaded to the GPU in Python, while the rest of the nonlinear oracles are evaluated on the CPU in Julia. \MathOptAI is available at https://github.com/lanl-ansi/MathOptAI.jl under a BSD-3 license.

Robert Parker, Oscar Dowson, Nicole LoGiudice et al.

We propose a reduced-space formulation for optimizing over trained neural networks where the network's outputs and derivatives are evaluated on a GPU. To do this, we treat the neural network as a "gray box" where intermediate variables and constraints are not exposed to the optimization solver. Compared to the full-space formulation, in which intermediate variables and constraints are exposed to the optimization solver, the reduced-space formulation leads to faster solves and fewer iterations in an interior point method. We demonstrate the benefits of this method on two optimization problems: Adversarial generation for a classifier trained on MNIST images and security-constrained optimal power flow with transient feasibility enforced using a neural network surrogate.

David P. Morton, Oscar Dowson, Bernardo K. Pagnoncelli

We study a class of multi-stage stochastic programs, which incorporate modeling features from Markov decision processes (MDPs). This class includes structured MDPs with continuous state and action spaces. We extend policy graphs to include decision-dependent uncertainty for one-step transition probabilities as well as a limited form of statistical learning. We focus on the expressiveness of our modeling approach, illustrating ideas with a series of examples of increasing complexity. As a solution method, we develop new variants of stochastic dual dynamic programming, including approximations to handle non-convexities.

Robert B. Parker, Oscar Dowson, Nicole LoGiudice et al.

We compare full-space, reduced-space, and gray-box formulations for representing trained neural networks in nonlinear constrained optimization problems. We test these formulations on a transient stability-constrained, security-constrained alternating current optimal power flow (SCOPF) problem where the transient stability criteria are represented by a trained neural network surrogate. Optimization problems are implemented in JuMP and trained neural networks are embedded using a new Julia package: MathOptAI.jl. To study the bottlenecks of the three formulations, we use neural networks with up to 590 million trained parameters. The full-space formulation is bottlenecked by the linear solver used by the optimization algorithm, while the reduced-space formulation is bottlenecked by the algebraic modeling environment and derivative computations. The gray-box formulation is the most scalable and is capable of solving with the largest neural networks tested. It is bottlenecked by evaluation of the neural network's outputs and their derivatives, which may be accelerated with a graphics processing unit (GPU). Leveraging the gray-box formulation and GPU acceleration, we solve our test problem with our largest neural network surrogate in 2.5$\times$ the time required for a simpler SCOPF problem without the stability constraint.