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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Least H 2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms

One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by the under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. We propose a new way to update the quadratic model by minimizing the $H^{2}$ norm of the difference between neighboring quadratic models. The motivation for applying the $H^{2}$ norm is given. The theoretical properties of our new updating technique are also presented. We propose the projection in the sense of $H^{2}$ norm and the interpolation error analysis of our model function. We obtain the coefficients of the quadratic model function using the Karush–Kuhn–Tucker (KKT) conditions. Numerical results show the advantages of our model on the test set considered, and the derivative-free algorithms based on our least $H^{2}$ norm updating quadratic model functions can solve test problems with fewer function evaluations than the algorithm based on the least Frobenius norm updating model and the other compared methods.

derivative-free optimization↗
DOE OSTI2023

Chapter 6: Surrogate Model Guided Optimization Algorithms and Their Potential Use in Autonomous Experimentation

This chapter reviews the basics of derivative-free optimization methods based on surrogate models and outlines how these methods can straightforwardly be applied to autonomously steering experimentation. It summarizes general solution approaches that use surrogate models and active learning. Surrogate modeling is often combined with active learning strategies, where in each iteration of the optimization algorithm, the surrogate model is used to identify which new inputs should be evaluated next and given the new input-output pair, the surrogate model is updated. Regardless of feasibility, the surrogate models for the constraints are updated in each iteration of the optimization algorithm, while the surrogate model for the objective function is only updated when a feasible point has been found. Similarly to the case of computationally cheap constraints, the surrogate models of the constraints should be incorporated into the definition of the auxiliary optimization problem that is solved to select new sample points.

active learning↗
DOE OSTI2023

Performance and Cost Potential for Direct-Fired Supercritical CO2 Natural Gas Power Plants

Direct-fired supercritical CO2 (sCO2) power cycles are being explored as an attractive alternative to natural gas combined cycle (NGCC) plants with carbon capture and storage (CCS). Therefore, understanding their performance and cost potential is important for the commercialization of the technology. This study presents the techno-economic optimization results of natural gas-fired, utility-scale power plants based on the direct sCO2 power cycle, which are lacking in public literature. To identify the optimum plant configuration, the study considered multiple cases with varying levels of thermal integration with the plant air separation unit (ASU). Several design variables for each power cycle configuration were identified and optimized to minimize the levelized cost of electricity (LCOE) for each case. The optimization design variables include the sCO2 cooler outlet temperatures, recuperator approach temperatures, and pressure drops. High fidelity models for recuperators, coolers, and turbines were developed and used to capture the impact of design variables on plant efficiency and capital costs. The optimization was conducted using a combination of manual sensitivity analyses and automated derivative-free optimization algorithms available under NETL’s Framework for Optimization and Quantification of Uncertainty and Sensitivity platform. The optimized direct sCO2 power plants offered similar or slightly higher plant efficiencies than the reference NGCC plants based on the F-class gas turbine with CCS. The LCOE of the optimized direct sCO2 plants is 13 to 17% higher than the reference NGCC plants with CCS due to high capital costs associated with the ASU and sCO2 power block, though there is significant room for improvement due to the high uncertainty in component capital costs for these new plants. Recuperators make up over 50% of the sCO2 power block costs. Consequently, any research and development efforts to reduce the recuperator capital costs will benefit the technology’s commercialization. The study also presents preliminary results showing the impact of co-firing landfill gas and natural gas on plant efficiency, LCOE, and CO2 emissions.

Pidaparti, Sandeep↗
DOE OSTI2023

Performance and Cost Potential for Direct-Fired Supercritical CO2 Natural Gas Power Plants

Direct-fired supercritical CO2 (sCO2) power cycles are being explored as an attractive alternative to natural gas combined cycle (NGCC) plants with carbon capture and storage (CCS). Therefore, understanding their performance and cost potential is important for the commercialization of the technology. This study presents the techno-economic optimization results of natural gas-fired, utility-scale power plants based on the direct sCO2 power cycle, which are lacking in public literature. To identify the optimum plant configuration, the study considered multiple cases with varying levels of thermal integration with the plant air separation unit (ASU). Several design variables for each power cycle configuration were identified and optimized to minimize the levelized cost of electricity (LCOE) for each case. The optimization design variables include the sCO2 cooler outlet temperatures, recuperator approach temperatures, and pressure drops. High fidelity models for recuperators, coolers, and turbines were developed and used to capture the impact of design variables on plant efficiency and capital costs. The optimization was conducted using a combination of manual sensitivity analyses and automated derivative-free optimization algorithms available under NETL’s Framework for Optimization and Quantification of Uncertainty and Sensitivity platform. The optimized direct sCO2 power plants offered similar or slightly higher plant efficiencies than the reference NGCC plants based on the F-class gas turbine with CCS. The LCOE of the optimized direct sCO2 plants is 13 to 17% higher than the reference NGCC plants with CCS due to high capital costs associated with the ASU and sCO2 power block, though there is significant room for improvement due to the high uncertainty in component capital costs for these new plants. Recuperators make up over 50% of the sCO2 power block costs. Consequently, any research and development efforts to reduce the recuperator capital costs will benefit the technology’s commercialization. The study also presents preliminary results showing the impact of co-firing landfill gas and natural gas on plant efficiency, LCOE, and CO2 emissions.

Pidaparti, Sandeep↗
DOE OSTI2022

A simplified, efficient approach to hybrid wind and solar plant site optimization

Abstract. Wind plant layout optimization is a difficult, complex problem with a large number of variables and many local minima. Layout optimization only becomes more difficult with the addition of solar generation. In this paper, we propose a parameterized approach to wind and solar hybrid power plant layout optimization that greatly reduces problem dimensionality while guaranteeing that the generated layouts have a desirable regular structure. Thus far, hybrid power plant optimization research has focused on system sizing. We go beyond sizing and present a practical approach to optimizing the physical layout of a wind–solar hybrid power plant. We argue that the evolution strategy class of derivative-free optimization methods is well-suited to the parameterized hybrid layout problem, and we demonstrate how hard layout constraints (e.g., placement restrictions) can be transformed into soft constraints that are amenable to optimization using evolution strategies. Next, we present experimental results on four test sites, demonstrating the viability, reliability, and effectiveness of the parameterized evolution strategy approach for generating optimized hybrid plant layouts. Completing the tool kit for parameterized layout generation, we include a brief tutorial describing how the parameterized evolutionary approach can be inspected, understood, and debugged when applied to hybrid plant layouts.

14 SOLAR ENERGY↗
DOE OSTI2022

Isotherm Modeling and Techno-Economic Analysis of Contactor Technologies for New Tetraamine-Appended MOF for NGCC Applications

Recently, a family of tetraamine-functionalized metal-organic frameworks (MOF) has been reported as promising sorbent materials for capturing CO2 from flue gas conditions relevant to natural gas combined cycle (NGCC) applications. The main advantages of these materials are their two-step cooperative CO2 adsorption, which gives rise to unusual two step-shaped CO2 adsorption profiles and their high thermal stability. This work presents the modelling of the two-transition isotherm of the tetraamine-appended MOF, N,N'-bis(3-aminopropyl)-1,4-diaminobutane (3-4-3)-appended Mg2(dobpdc), and the techno-economic analysis (TEA) of carbon capture processes utilizing this sorbent. Due to the unusual isotherm shapes of the experimental CO2 adsorption data for tetraamine-appended Mg2(dobpdc) and the strong nonlinearity of CO2 loading with respect to temperature and pressure, we tested two different models which use logistic functions for representing the different isotherm behaviors in the different pressure ranges. The first model uses the quadratic isotherm model in the low-pressure region, the Langmuir isotherm model in the middle pressure range, and the dual site Langmuir isotherm model in the high-pressure range. To model the transition between regions we used the arctangent functions independent of temperature and the thermal effect was accounted by using the Clausius-Clapeyron relation]. The second model is an extension of the weighted dual-site Langmuir isotherm model presented by Hughes et al.. In this extended model, the dual-site Langmuir isotherm is employed in the three transition regions, using temperature-dependent logistic functions to activate or deactivate the isotherm model in the low, middle, and high-pressure ranges. Both models fit the experimental data quite well with root mean squared errors (RMSE’s) of 0.41 and 0.17 for model 1 and model 2, respectively. Since model 2 resulted in a lower RMSE, it was leveraged for the development of the gas/solid contactor models used by the TEA. Specifically, two different contactor models, an axial-flow fixed bed and moving bed contactor, were developed as part of this work. These models are dynamic, pressure-driven, and consist of mass, energy, and momentum conservation equations. A kinetic model was also developed by performing parameter estimation using experimental fixed bed breakthrough data. These models are then used to simulate CO2 capture processes from the flue gas generated from a ~600 gross MW NGCC power plant. A cost model was developed which considers the capital cost of the reactors and the significant operating costs such as steam and electricity. Using NETL’s Framework for the Optimization and Quantification of Uncertainty of Uncertainty and Surrogates tool (FOQUS), which has the capability of linking models built using numerous modelling platforms with derivative-free optimization solvers, a techno-economic optimization of the carbon capture processes was performed which minimizes the cost of capture.

Caballero, Daison↗
DOE OSTI2026

Augmenting subspace optimization methods with linear bandits

In this work, we consider the framework of methods for unconstrained minimization that are, in each iteration, restricted to a model that is only a valid approximation to the objective function on some affine subspace containing an incumbent point. These methods are of practical interest in computational settings where derivative information is either expensive or impossible to obtain. Recent attention has been paid in the literature to employing randomized matrix sketching for generating the affine subspaces within this framework. We consider a relatively straightforward, deterministic augmentation of such a generic subspace optimization method. In particular, we consider a sequential optimization framework where actions consist of one-dimensional linear subspaces and rewards consist of (approximations to) the magnitudes of directional derivatives computed in the direction of the action subspace. Reward maximization in this context is consistent with maximizing lower bounds on descent guaranteed by first-order Taylor models. This sequential optimization problem can be analysed through the lens of dynamic regret. We modify an existing linear upper confidence bound (UCB) bandit method and prove sublinear dynamic regret in the subspace optimization setting. We demonstrate the efficacy of employing this linear UCB method in a setting where forward-mode algorithmic differentiation can provide directional derivatives in arbitrary directions and in a derivative-free setting. For the derivative-free setting, we propose SS-POUNDers, an extension of the derivative-free optimization method POUNDers that employs the linear UCB mechanism to identify promising subspaces. Our numerical experiments suggest a preference, in either computational setting, for employing a linear UCB mechanism within a subspace optimization method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Manifold learning for coarse-graining atomistic simulations: Application to amorphous solids

In this work, we introduce a generalized machine learning framework to probabilistically parameterize upper-scale models in the form of nonlinear PDEs consistent with a continuum theory, based on coarse-grained atomistic simulation data of mechanical deformation and flow processes. The proposed framework utilizes a hypothesized coarse-graining methodology with manifold learning and surrogate-based optimization techniques. Coarse-grained high-dimensional data describing quantities of interest of the multiscale models are projected onto a nonlinear manifold whose geometric and topological structure is exploited for measuring behavioral discrepancies in the form of manifold distances. A surrogate model is constructed using Gaussian process regression to identify a mapping between stochastic parameters and distances. Derivative-free optimization is employed to adaptively identify a unique set of parameters of the upper-scale model capable of rapidly reproducing the system's behavior while maintaining consistency with coarse-grained atomic-level simulations. The proposed method is applied to learn the parameters of the shear transformation zone (STZ) theory of plasticity that describes plastic deformation in amorphous solids as well as coarse-graining parameters needed to translate between atomistic and continuum representations. We show that the methodology is able to successfully link coarse-grained microscale simulations to macroscale observables and achieve a high-level of parity between the models across scales.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Evaluating pulse-shaping capabilities of next-generation pulsed power architectures

This project evaluated the pulse shaping capabilities of next-generation pulsed power (NGPP) architectures. NGPP architectures share several common attributes including multiple independent pulse-generation lines, a radial water-insulated impedance transformer, and a central vacuum insulated load region. A multi-module circuit model was developed, incorporating independent pulse-generation lines and a 2-D transmission line mesh of the radial impedance transformer to assess the effects of azimuthal asymmetry in pulse-shaped experiments. Circuit model simulations demonstrated that NGPP architectures are able to produce the the desired current pulse shapes for exemplar NGPP experiments. Additionally, the project explored automated methods for experiment design, including derivative -ree optimization and machine learning. Pulse-shaped experiments require designers to determine machine parameters that reliably produce the desired current pulse at the load, a process that typically relies on expert knowledge and iterative adjustments using the Z circuit model. Given the increased complexity of NGPP systems, this manual approach may be impractical. While the evaluated methods do not eliminate the need for manual iteration, they can reduce the time required for experiment design. Derivative-free optimization automates much of the trial-and-error process, providing a close starting point for manual adjustments or making small modifications to near-final designs. Meanwhile, deep neural network methods can generate a good qualitative match to the desired current pulse in under one second without requiring circuit model simulations.

42 ENGINEERING↗
DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2020

Tuning a variational autoencoder for data accountability problem in the Mars Science Laboratory ground data system

The Mars Curiosity rover is frequently sending back engineering and science data that goes through a pipeline of systems before reaching its final destination at the mission operations center making it prone to volume loss and data corruption. A ground data system analysis (GDSA) team is charged with the monitoring of this flow of information and the detection of anomalies in that data in order to request a re-transmission when necessary. This work presents ∆-MADS, a derivative-free optimization method applied for tuning the architecture and hyperparameters of a variational autoencoder trained to detect the data with missing patches in order to assist the GDSA team in their mission.

Lakhmiri, Dounia↗
DOE OSTI2023

Structure-aware methods for expensive derivative-free nonsmooth composite optimization

We present new methods for solving a broad class of bound-constrained nonsmooth composite minimization problems. These methods are specially designed for objectives that are some known mapping of outputs from a computationally expensive function. We provide accompanying implementations of these methods: in particular, a novel manifold sampling algorithm (MS-P) with subproblems that are in a sense primal versions of the dual problems solved by previous manifold sampling methods and a method (GOOMBAH) that employs more difficult optimization subproblems. For these two methods, we provide rigorous convergence analysis and guarantees. We demonstrate extensive testing of these methods. Open-source implementations of the methods developed in this manuscript can be found at https://github.com/POptUS/ IBCDFO/.

97 MATHEMATICS AND COMPUTING↗
NASA NTRSDate not supplied

Fixed- and Free-Mode Flutter Derivatives during Aeroelastic Optimization

Aeroelastic flutter design derivatives may be approximated by ignoring the sensitivity of the structural mode shapes. This “fixed-mode” derivative is less-expensive to compute than the exact “free-mode” derivative (which accommodates the mode shape dependency), but also may provide inaccurate sensitivities for optimization. This work formulates both fixed- and free-mode flutter derivatives, and demonstrates the conditions under which they are equivalent. Two different types of fixed-mode derivatives are also developed in this work. Using a cantilevered-plate demonstration problem with both shape and sizing design parameters, the accuracy of the free-mode derivative and the two fixed-mode derivatives, are all demonstrated via gradient-based optimization.

Bret K. Stanford↗
DOE OSTI2023

Adaptive sampling quasi-Newton methods for zeroth-order stochastic optimization

Here, we consider unconstrained stochastic optimization problems with no available gradient information. Such problems arise in settings from derivative-free simulation optimization to reinforcement learning. We propose an adaptive sampling quasi-Newton method where we estimate the gradients using finite differences of stochastic function evaluations within a common random number framework. We develop modified versions of a norm test and an inner product quasi-Newton test to control the sample sizes used in the stochastic approximations and provide global convergence results to the neighborhood of a locally optimal solution. We present numerical experiments on simulation optimization problems to illustrate the performance of the proposed algorithm. When compared with classical zeroth-order stochastic gradient methods, we observe that our strategies of adapting the sample sizes significantly improve performance in terms of the number of stochastic function evaluations required.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A method for convex black-box integer global optimization

Here we study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective; such information is unavailable when the objective is a blackbox function. Rather, our underestimator uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings are shown to underestimate the objective in disconnected portions of the domain. Therefore, the union of these conditional cuts provides a nonconvex underestimator of the objective. We propose an algorithm that alternates between updating the underestimator and evaluating the objective function. We prove that the algorithm converges to a global minimum of the objective function on the feasible set. We present two approaches for representing the underestimator and compare their computational effectiveness. We also compare implementations of our algorithm with existing methods for minimizing functions on a subset of the integer lattice. We discuss the difficulty of this problem class and provide insights into why a computational proof of optimality is challenging even for moderate problem sizes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Stochastic average model methods

We consider the solution of finite-sum minimization problems, such as those appearing in nonlinear least-squares or general empirical risk minimization problems. We are motivated by problems in which the summand functions are computationally expensive and evaluating all summands on every iteration of an optimization method may be undesirable. Here we present the idea of stochastic average model (SAM) methods, inspired by stochastic average gradient methods. SAM methods sample component functions on each iteration of a trust-region method according to a discrete probability distribution on component functions; the distribution is designed to minimize an upper bound on the variance of the resulting stochastic model. We present promising numerical results concerning an implemented variant extending the derivative-free model-based trust-region solver POUNDERS, which we name SAM-POUNDERS.

97 MATHEMATICS AND COMPUTING↗