arXiv subjects

Robert Parker

Publications and source records attributed to Robert Parker.

8 recordsLinked to original sources

arXiv2026

Exploiting block triangular submatrices in KKT systems

We propose a method for solving Karush-Kuhn-Tucker (KKT) systems that exploits block triangular submatrices by first using a Schur complement decomposition to isolate the block triangular submatrices then performing a block backsolve where only diagonal blocks of the block triangular form need to be factorized. We show that factorizing reducible symmetric-indefinite matrices with standard 1$\times$1 or 2$\times$2 pivots yields fill-in outside the diagonal blocks of the block triangular form, in contrast to our proposed method. While exploiting a block triangular submatrix has limited fill-in, unsymmetric matrix factorization methods do not reveal inertia, which is required by interior point methods for nonconvex optimization. We show that our target matrix has inertia that is known \textit{a priori}, letting us compute inertia of the KKT matrix by Sylvester's law. Finally, we demonstrate the computational advantage of this method on KKT systems from optimization problems with neural network surrogates in their constraints. Our method achieves up to 15$\times$ speedups over state-of-the-art symmetric indefinite matrix factorization methods MA57 and MA86 in a constant-hardware comparison.

math.OC↗

arXiv2026

Generating adversarial inputs for a graph neural network model of AC power flow

This work formulates and solves optimization problems to generate input points that yield high errors between a neural network's predicted AC power flow solution and solutions to the AC power flow equations. We demonstrate this capability on an instance of the CANOS-PF graph neural network model, as implemented by the PF$\Delta$ benchmark library, operating on a 14-bus test grid. Generated adversarial points yield errors as large as 3.7 per-unit in reactive power and 0.08 per-unit in voltage magnitude. When minimizing the perturbation from a training point necessary to satisfy adversarial constraints, we find that the constraints can be met with as little as an 0.04 per-unit perturbation in voltage magnitude on a single bus. This work motivates the development of rigorous verification and robust training methods for neural network surrogate models of AC power flow.

cs.LG↗

arXiv2025

Maximal Load Shedding Verification for Neural Network Models of AC Line Switching

Solving for globally optimal line switching decisions in AC transmission grids can be intractability slow. Machine learning (ML) models, meanwhile, can be trained to predict near-optimal decisions at a fraction of the speed. Verifying the performance and impact of these ML models on network operation, however, is a critically important step prior to their actual deployment. In this paper, we train a Neural Network (NN) to solve the optimal power shutoff line switching problem. To assess the worst-case load shedding induced by this model, we propose a bilevel attacker-defender verification approach that finds the NN line switching decisions that cause the highest quantity of network load shedding. Solving this problem to global optimality is challenging (due to AC power flow and NN nonconvexities), so our approach exploits a convex relaxation of the AC physics, combined with a local NN search, to find a guaranteed lower bound on worst--case load shedding. These under-approximation bounds are solved via MathOptAI.jl. We benchmark against a random sampling approach, and we find that our optimization-based approach always finds larger load shedding. Test results are collected on multiple PGLib test cases and on trained NN models which contain more than 10 million model parameters.

eess.SY↗

arXiv2025

Nonlinear Optimization with GPU-Accelerated Neural Network Constraints

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.

cs.LG↗

arXiv2025

Variable aggregation for nonlinear optimization problems

Variable aggregation has been largely studied as an important pre-solve algorithm for optimization of linear and mixed-integer programs. Although some nonlinear solvers and algebraic modeling languages implement variable aggregation as a pre-solve, the impact it can have on constrained nonlinear programs is unexplored. In this work, we formalize variable aggregation as a pre-solve algorithm to develop reduced-space formulations of nonlinear programs. A novel approximate maximum variable aggregation strategy is developed to aggregate as many variables as possible. Furthermore, aggregation strategies that preserve the problem structure are compared against approximate maximum aggregation. Our results show that variable aggregation can generally help to improve the convergence reliability of nonlinear programs. It can also help in reducing total solve time. However, Hessian evaluation can become a bottleneck if aggregation significantly increases the number of variables appearing nonlinearly in many constraints.

math.OC↗

arXiv2025

Transient Stability-Constrained OPF: Neural Network Surrogate Models and Pricing Stability

A Transient Stability-Constrained Optimal Power Flow (TSC-OPF) problem is proposed that enforces frequency stability constraints using Neural Network (NN) surrogate models. NNs are trained using a novel model-driven active sampling algorithm that iteratively generates NN training data located near the stability boundary and contained within the feasible set of the Alternating Current Optimal Power Flow (AC-OPF) problem. In the context of wholesale electricity markets, pricing structures are analyzed along with their dependencies on the selected input features to the NN surrogate model. An important insight identifies a trade-off between the accuracy of the NN surrogate model and sensible locational pricing structures. NN surrogate models for frequency stability are validated by ensuring the resulting TSC-OPF solution is stable over randomly generated load samples using a small Hawaii test case. The proposed TSC-OPF problem is shown to significantly enhance frequency stability at low computational cost and low financial cost to the system. For certain selections of NN inputs, the TSC-OPF problem is able to stabilize all load scenarios for which the solution to the AC-OPF problem resulted in instability.

math.OC↗

arXiv2024

Managing power balance and reserve feasibility in the AC unit commitment problem

Incorporating the AC power flow equations into unit commitment models has the potential to avoid costly corrective actions required by less accurate power flow approximations. However, research on unit commitment with AC power flow constraints has been limited to a few relatively small test networks. This work investigates large-scale AC unit commitment problems for the day-ahead market and develops decomposition algorithms capable of obtaining high-quality solutions at industry-relevant scales. The results illustrate that a simple algorithm that only seeks to satisfy unit commitment, reserve, and AC power balance constraints can obtain surprisingly high-quality solutions to this AC unit commitment problem. However, a naive strategy that prioritizes reserve feasibility leads to AC infeasibility, motivating the need to design heuristics that can effectively balance reserve and AC feasibility. Finally, this work explores a parallel decomposition strategy that allows the proposed algorithm to obtain feasible solutions on large cases within the two hour time limit required by typical day-ahead market operations.

eess.SY↗

arXiv2024

Towards Perturbation-Induced Static Pivoting on GPU-Based Linear Solvers

Linear system solving is a key tool for computational power system studies, e.g., optimal power flow, transmission switching, or unit commitment. CPU-based linear system solver speeds, however, have saturated in recent years. Emerging research shows that GPU-based linear system solvers are beginning to achieve notable speedup over CPU-based alternatives in some applications. Due to the architecture of GPU memory access, numerical pivoting represents the new bottleneck which prevents GPU-based solvers from running even faster. Accordingly, this paper proposes a matrix perturbation-based method to induce static pivoting. Using this approach, a series of perturbed, well-conditioned, pivot-free linear systems are solved in parallel on GPUs. Matrix expansion routines are then used to linearly combine the results, and the true solution is recovered to an arbitrarily high degree of theoretical accuracy. We showcase the validity of our approach on distributed-slack AC power flow solve iterations associated with the PGLib 300-bus test case.

eess.SY↗