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DOE OSTI2025

Improved Guarantees for Optimal Nash Equilibrium Seeking and Bilevel Variational Inequalities

We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other problem classes, including the optimal solution selection problem and the optimal Nash equilibrium (NE) seeking problem. Our main contribution is threefold. (i) We consider bilevel VIs with monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method, named IR-EG 𝚖,𝚖 . We improve the existing iteration complexity results for addressing both bilevel VI and VI-constrained convex optimization problems. (ii) Under the strong monotonicity of the outer-level mapping, we develop a method named IR-EG 𝚜,𝚖 and derive faster guarantees than those in (i). We also study the iteration complexity of this method under a constant regularization parameter. These results appear to be new for both bilevel VIs and VI-constrained optimization. (iii) To our knowledge, complexity guarantees for computing the optimal NE in nonconvex settings do not exist. Motivated by this lacuna, we consider VI-constrained nonconvex optimization problems and devise an inexactly projected gradient method, named IPR-EG, where the projection onto the unknown set of equilibria is performed using IR-EG 𝚜,𝚖 with a prescribed termination criterion and an adaptive regularization parameter. We obtain new complexity guarantees in terms of a residual map and an infeasibility metric for computing a stationary point. Here, we validate the theoretical findings using preliminary numerical experiments for computing the best and the worst NEs.

bilevel optimization↗

DOE OSTI2023

SwitchX : Gmin-Gmax Switching for Energy-efficient and Robust Implementation of Binarized Neural Networks on ReRAM Xbars

Memristive crossbars can efficiently implement Binarized Neural Networks (BNNs) wherein the weights are stored in high-resistance states (HRS) and low-resistance states (LRS) of the synapses. We propose SwitchX mapping of BNN weights onto ReRAM crossbars such that the impact of crossbar non-idealities, that lead to degradation in computational accuracy, are minimized. Essentially, SwitchX maps the binary weights in such a manner that a crossbar instance comprises of more HRS than LRS synapses. We find BNNs mapped onto crossbars with SwitchX to exhibit better robustness against adversarial attacks than the standard crossbar mapped BNNs, the baseline. Finally, we combine SwitchX with state-aware training (that further increases the feasibility of HRS states during weight mapping) to boost the robustness of a BNN on hardware. We find that this approach yields stronger defense against adversarial attacks than adversarial training, a state-of the-art software defense. We perform experiments on a VGG16 BNN with benchmark datasets (CIFAR-10, CIFAR-100 and TinyImagenet) and use Fast Gradient Sign Method (ϵ = 0.05 to 0.3) and Projected Gradient Descent (ϵ = $\frac{2}{255}$ to $\frac{32}{255}$, α = $\frac{2}{255}$) adversarial attacks. We show that SwitchX combined with state-aware training can yield upto ~35% improvements in clean accuracy and ~6–16% in adversarial accuracies against conventional BNNs. Furthermore, an important by-product of SwitchX mapping is increased crossbar power savings, owing to an increased proportion of HRS synapses, which is furthered with state-aware training. We obtain upto ~21–22% savings in crossbar power consumption for state-aware trained BNN mapped via SwitchX on 16 × 16 and 32 × 32 crossbars using the CIFAR-10 and CIFAR-100 datasets.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

DP-TwoLevel: two-stage gradient subspace learning for differentially private federated learning

Federated learning (FL) enables collaborative model training across distributed data sources without sharing raw data, but faces fundamental challenges in communication efficiency and privacy. Differentially private (DP) training mitigates information leakage but introduces noise that degrades model performance, especially in high-dimensional settings. We propose DP-TwoLevel, a hierarchical gradient projection method that improves utility under fixed DP constraints by exploiting low-dimensional structure in model updates. Our approach learns a two-level PCA-based representation of gradients and applies DP noise in a reduced-dimensional subspace, thereby lowering the effective noise magnitude while preserving dominant signal components. We evaluate the method across three datasets (MNIST, Fashion-MNIST, CIFAR-10) and three privacy regimes (ϵ∈0.5, 1.0, 2.0). Across nine experimental settings, DP-TwoLevel consistently outperforms DP-FedAvg, achieving an average accuracy improvement of 9.44%, with larger gains observed in lower ϵ(higher-noise) regimes (up to +22.31%). We further analyze scalability across models ranging from 100K to 1.49M parameters and identify a variance-based success criterion: performance remains strong when the projection preserves more than 75% of gradient variance, degrades in a marginal regime (65–75%), and fails below this threshold. Our results demonstrate that structure-aware dimensionality reduction can significantly improve the privacy–utility tradeoff in FL without modifying formal privacy guarantees. We also provide empirical evidence of scaling limitations for global projections and motivate per-layer extensions for larger models.

Kotevska, Olivera [ORNL] (ORCID:0000000316772243)↗

NASA NTRS1980

Discrete-continuous variable structural synthesis using dual methods

Approximation concepts and dual methods are extended to solve structural synthesis problems involving a mix of discrete and continuous sizing type of design variables. Pure discrete and pure continuous variable problems can be handled as special cases. The basic mathematical programming statement of the structural synthesis problem is converted into a sequence of explicit approximate primal problems of separable form. These problems are solved by constructing continuous explicit dual functions, which are maximized subject to simple nonnegativity constraints on the dual variables. A newly devised gradient projection type of algorithm called DUAL 1, which includes special features for handling dual function gradient discontinuities that arise from the discrete primal variables, is used to find the solution of each dual problem. Computational implementation is accomplished by incorporating the DUAL 1 algorithm into the ACCESS 3 program as a new optimizer option. The power of the method set forth is demonstrated by presenting numerical results for several example problems, including a pure discrete variable treatment of a metallic swept wing and a mixed discrete-continuous variable solution for a thin delta wing with fiber composite skins.

Schmit, L. A.↗

NASA NTRS1977

A life cycle cost economics model for projects with uniformly varying operating costs

A mathematical model is developed for calculating the life cycle costs for a project where the operating costs increase or decrease in a linear manner with time. The life cycle cost is shown to be a function of the investment costs, initial operating costs, operating cost gradient, project life time, interest rate for capital and salvage value. The results show that the life cycle cost for a project can be grossly underestimated (or overestimated) if the operating costs increase (or decrease) uniformly over time rather than being constant as is often assumed in project economic evaluations. The following range of variables is examined: (1) project life from 2 to 30 years; (2) interest rate from 0 to 15 percent per year; and (3) operating cost gradient from 5 to 90 percent of the initial operating costs. A numerical example plus tables and graphs is given to help calculate project life cycle costs over a wide range of variables.

Remer, D. S.↗

DOE OSTI2025

On the Sampling-Based Computation of Nash Equilibria Under Uncertainty via the Nikaido–Isoda Function

We consider the computation of an equilibrium of a stochastic Nash equilibrium problem, where the player objectives are assumed to be L 0 -Lipschitz continuous and convex, given rival decisions with convex and closed player-specific feasibility sets. To address this problem, we consider minimizing a suitably defined value function defined using the Nikaido–Isoda function. Such an avenue does not necessitate either monotonicity properties of the concatenated gradient map or potentiality requirements on the game but does require a suitable regularity requirement under which a stationary point is a Nash equilibrium. We design and analyze a sampling-enabled projected-gradient-response method, reliant on inexact resolution of a player-level best-response subproblem. Here, by deriving suitable Lipschitzian guarantees on the value function, we derive both asymptotic guarantees for the sequence of generated iterates as well as rate and complexity guarantees for computing a stationary point by appropriate choices of the sampling rate and inexactness sequence.

Nikaido-Isoda function↗

DOE OSTI2020

MimicGAN: Robust Projection onto Image Manifolds with Corruption Mimicking

In the past few years, Generative Adversarial Networks (GANs) have dramatically advanced our ability to represent and parameterize high-dimensional, non-linear image manifolds. As a result, they have been widely adopted across a variety of applications, ranging from challenging inverse problems like image completion, to problems such as anomaly detection and adversarial defense. A recurring theme in many of these applications is the notion of projecting an image observation onto the manifold that is inferred by the generator. In this context, Projected Gradient Descent (PGD) has been the most popular approach, which essentially optimizes for a latent vector that minimizes the discrepancy between a generated image and the given observation. However, PGD is a brittle optimization technique that fails to identify the right projection (or latent vector) when the observation is corrupted, or perturbed even by a small amount. Such corruptions are common in the real world, for example images in the wild come with unknown crops, rotations, missing pixels, or other kinds of non-linear distributional shifts which break current encoding methods, rendering downstream applications unusable. To address this, we propose corruption mimicking—a new robust projection technique, that utilizes a surrogate network to approximate the unknown corruption directly at test time, without the need for additional supervision or data augmentation. The proposed method is significantly more robust than PGD and other competing methods under a wide variety of corruptions, thereby enabling a more effective use of GANs in real-world applications. Finally, more importantly, we show that our approach produces state-of-the-art performance in several GAN-based applications—anomaly detection, domain adaptation, and adversarial defense, that benefit from an accurate projection.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. Finally, this work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.

42 ENGINEERING↗

DOE OSTI2022

Learning-Based Demand Response in Grid Interactive Buildings via Gaussian Processes

This paper presents a predictive controller for a grid-interactive multi-zone building where the temperature dynamics are learned via Gaussian Process (GP) regression. We investigate the development of a learning-based predictive control with two main objectives: (i) continuously learn the temperature dynamics of the building based on data; and, (ii) use the learned dynamics to solve a multi-objective predictive control problem to guarantee occupants' comfort and energy efficiency during normal conditions and demand response events. We leverage the probabilistic non-parametric properties of GPs to estimate the (unknown) non-linear temperature dynamics of the building and to incorporate the uncertainty of those predictions in a multi-objective optimization problem. The GP-based predictive control is solved via a zero-order primal-dual projected-gradient algorithm. We evaluate numerically the performance of the proposed controller using a five-zone commercial building.

demand response↗

DOE OSTI2024

Distributed optimization for multi-commodity urban traffic control

A distributed method for concurrent traffic signal and routing control of traffic networks is proposed. The method is based on the multi-commodity store-and-forward model, in which the destinations are the commodities. The system benefits from the communication between vehicles and infrastructure, providing optimal signal timings to intersections and routes to vehicles on a link-by-link basis. Using the augmented Lagrangian to model the constraints into the objective, the baseline centralized problem is decomposed into a set of objective-coupled subproblems, one for each intersection, enabling the solution to be computed by a distributed- gradient projection algorithm. Further, the intersection agents only need to communicate and coordinate with neighboring intersections to ensure convergence to the optimal solution while tolerating suboptimal iterations that offer more flexibility, unlike other distributed approaches. Through microsimulation, we demonstrate the effectiveness of the proposed algorithm in traffic networks with time-varying demand. Computational analysis shows that the distributed problem is suitable for real-time applications. A robustness analysis show that the distributed formulation enables a graceful degradation of the system in case of failure.

Augmented Lagrangian↗

DOE OSTI2025

Single-shot in-line x-ray phase-contrast imaging of void-shockwave interactions in fusion energy materials

Recent breakthroughs in nuclear fusion, specifically the report of reactions exceeding scientific breakeven at the National Ignition Facility (NIF), highlight the potential of inertial fusion energy (IFE) as a sustainable and virtually limitless energy source. However, further progress in IFE requires characterization of defects in ablator materials and how they affect fuel capsule compression. Voids within the ablator can degrade energy yield, but their impact on the density distribution has primarily been studied through simulations, with limited high-resolution experimental validation. To address this, we used the x-ray free-electron laser (XFEL) at the matter in extreme conditions (MECs) instrument at the Linac coherent light source (LCLS) to capture 2D x-ray phase-contrast (XPC) images of a void-bearing sample with a composition similar to inertial confinement fusion (ICF) ablators. By driving a compressive shockwave through the sample using MEC's long-pulse laser system, we analyzed how voids influence shockwave propagation and density distribution during compression. To quantify this impact, we extracted phase information using two phase retrieval algorithms. First, we applied the contrast transfer function (CTF) method, paired with Tikhonov regularization and a fast optimization approach to generate an initial phase estimate. We then refined the result using a projected gradient descent (PGD) method that works directly with the sample's refractive index. Comparing these results with radiation adaptive grid Eulerian (xRAGE) radiation hydrodynamic simulations enables identification of model validation needs or improvements. By calculating phase maps in situ, it becomes possible to reconstruct areal density maps, improving understanding of laser-capsule interactions and advancing IFE research.

Hodge, D. S. [Colorado State Univ., Fort Collins, ↗

DOE OSTI2020

Optimal Energy Dispatch of Distributed PVs for the Next Generation of Distribution Management Systems

Advanced Distribution Management Systems (ADMS) are being widely adopted by electric utilities for managing and optimizing the operations of their distribution systems. Distributed photovoltaic (DPV) systems with smart inverters can be controlled to adjust active power and reactive power outputs, and they are envisioned to become a part of (centrally or distributed) controllable assets managed by the ADMS for optimizing grid operations. This paper proposes an optimal energy dispatch strategy controlling DPV systems for regulating distribution voltages and achieving conservation voltage reduction. A convex optimization model is proposed with the use of linearized power flow, and the gradient projection algorithm is used to solve the optimal active power and reactive power outputs of smart inverters. The proposed optimal energy dispatch is implemented using an open-source ADMS platform, and simulation results have demonstrated the effectiveness of the proposed approach on improving distribution grid operations.

42 ENGINEERING↗

DOE OSTI2022

Feedback Power Cost Optimization in Power Distribution Networks with Prosumers

Here, we propose a feedback control approach for solving optimal power flow (OPF) problems in power distribution networks (DNs) based on a projected gradient scheme, where the cost is given by the sum of functions of local power injections. This approach does not require detailed knowledge of the grid model and enables real-time tracking of the optimal generation configuration. Moreover, the communication burden is minimal: only a broadcast from a central entity, called the Network Supervisor (NS), to the agents in the network is needed. Hence, agents do not have to share local information that can be sensitive, e.g., their power injections or the value of the local costs. Our approach is then used to solve a particular OPF problem targeting DNs with high penetration of distributed energy resources (DERs) in which the energy cost is given by the remuneration that the DERs owners receive for the service they provide.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2023

Wildfires identification: Semantic segmentation using support vector machine classifier

This paper deals with wildfire identification in the Alaska regions as a semantic segmentation task using support vector machine classifiers. Instead of colour information represented by means of BGR channels, we proceed with a normalized reflectance over 152 days so that such time series is assigned to each pixel. We compare models associated with $\mathcal{l}1$-loss and $\mathcal{l}2$-loss functions and stopping criteria based on a projected gradient and duality gap in the presented benchmarks.

Pecha, Marek↗

DOE OSTI2020

Model-Free Primal-Dual Methods for Network Optimization with Application to Real-Time Optimal Power Flow

This paper examines the problem of real-time optimization of networked systems and develops online algorithms that steer the system towards the optimal trajectory without explicit knowledge of the system model. The problem is modeled as a dynamic optimization problem with time-varying performance objectives and engineering constraints. The design of the algorithms leverages the online zero-order primal-dual projected-gradient method. In particular, the primal step that involves the gradient of the objective function (and hence requires a networked systems model) is replaced by its zero-order approximation with two function evaluations using a deterministic perturbation signal. The evaluations are performed using the measurements of the system output, hence giving rise to a feedback interconnection, with the optimization algorithm serving as a feedback controller. The paper provides some insights on the stability and tracking properties of this interconnection. Finally, the paper applies this methodology to a real-time optimal power flow problem in power systems, and shows its efficacy on the IEEE 37-node distribution test feeder for reference power tracking and voltage regulation.

61 RADIATION PROTECTION AND DOSIMETRY↗

DOE OSTI2023

Assignment of Freight Traffic in a Large-scale Intermodal Network under Uncertainty

This paper presents a methodology for freight traffic assignment in a large-scale road-rail intermodal network under uncertainty. Network uncertainties caused by natural disasters have dramatically increased in recent years. Several of these disasters (e.g., Hurricane Sandy, Mississippi River Flooding, and Hurricane Harvey) severely disrupted the U.S. freight transportation network, and consequently, the supply chain. To account for these network uncertainties, a stochastic freight traffic assignment model is formulated. An algorithmic framework, involving the sample average approximation and gradient projection algorithm, is proposed to solve this challenging problem. The developed methodology is tested on the U.S. intermodal network with freight flow data from the Freight Analysis Framework. The experiments consider three types of natural disasters that have different risks and impacts on transportation networks: earthquakes, hurricanes, and floods. It is found that for all disaster scenarios, freight ton-miles are higher compared to the base case without uncertainty. The increase in freight ton-miles is the highest under the flooding scenario; this is because there are more states in the flood-risk areas, and they are scattered throughout the U.S.

42 ENGINEERING↗

NASA NTRS1971

A stochastic constrained optimization technique and its application to detector array processing.

A stochastic projected gradient algorithm is proposed which can be used for finding a constrained optimum point for a concave or convex objective function subject to nonlinear constraints which form a connected region even when only a noisy estimate of the objective function is available. For a constraint described by a single linear equation, convergence to the constrained optimum value is proved, and the rate of convergence of the algorithm to the constrained optimum value is determined. The algorithm is applied to the nonlinear problem of obtaining automatically an array of detectors which forms a beam in a desired direction in space in the presence of interfering noise so as to maximize the SNR subject to a constraint on the super-gain ratio.

Winkler, L. P.↗

NASA NTRS1972

Adaptive nonlinear optimization of the signal-to-noise ratio of an array subject to a constraint.

Investigation of a stochastic projected gradient algorithm which can be used to find a constrained optimum point for a concave or convex objective function subject to nonlinear constraints. When the constraint consists of only one linear equation, convergence to the constrained optimum value is proven, and bounds are obtained for the rate of convergence of the algorithm to the constrained optimum value. This algorithm is then applied to the nonlinear problem of automatically making an array of detectors form a beam in a desired direction in space when unknown interfering noise is present so as to maximize the signal-to-noise ratio subject to a constraint on the supergain ratio.

Winkler, L. P.↗