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Results for “mathematical programming formulations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A mixed integer linear programming approach for the design of chemical process families

Tackling climate change goals requires widespread deployment of process technology variants across many decentralized sites with different geographical, environmental, and operational requirements. Conventional engineering approaches focus on unique designs for each installation (process variant), while missing opportunities for manufacturing standardization. Here, instead we seek to optimize a process platform of common unit designs while simultaneously designing an entire family of process variants that make use of that platform. This reduces engineering effort, deployment timelines, and manufacturing costs. We propose a nonlinear generalized disjunctive programming formulation and convert this to an efficient mixed-integer linear programming (MILP) formulation through discretization of the design space. We formulate our optimization in Pyomo with costing from IDAES, and we demonstrate the computational performance and solution quality on a water treatment desalination system from the PARETO framework and a carbon capture system built in Aspen Plus as part of CCSI2.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Optimal decision trees for categorical data via integer programming

Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if allowed to grow large, they lose interpretability. In this paper, we present a mixed integer programming formulation to construct optimal decision trees of a prespecified size. We take the special structure of categorical features into account and allow combinatorial decisions (based on subsets of values of features) at each node. Our approach can also handle numerical features via thresholding. Here we show that very good accuracy can be achieved with small trees using moderately-sized training sets. The optimization problems we solve are tractable with modern solvers.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2011

Systems, methods and apparatus for modeling, specifying and deploying policies in autonomous and autonomic systems using agent-oriented software engineering

Systems, methods and apparatus are provided through which in some embodiments, an agent-oriented specification modeled with MaCMAS, is analyzed, flaws in the agent-oriented specification modeled with MaCMAS are corrected, and an implementation is derived from the corrected agent-oriented specification. Described herein are systems, method and apparatus that produce fully (mathematically) tractable development of agent-oriented specification(s) modeled with methodology fragment for analyzing complex multiagent systems (MaCMAS) and policies for autonomic systems from requirements through to code generation. The systems, method and apparatus described herein are illustrated through an example showing how user formulated policies can be translated into a formal mode which can then be converted to code. The requirements-based programming systems, method and apparatus described herein may provide faster, higher quality development and maintenance of autonomic systems based on user formulation of policies.

Hinchey, Michael G.↗
DOE OSTI2021

Disjunctive linear separation conditions and mixed-integer formulations for aircraft conflict resolution

In this paper, we address the aircraft conflict resolution problem in air traffic control. We introduce new mixed-integer programming formulations for aircraft conflict resolution with speed, heading and altitude control which are based on disjunctive linear separation conditions. We first examine the two-dimensional aircraft conflict resolution problem with speed and heading control represented as continuous decision variables. We show that the proposed disjunctive linear separation conditions are equivalent to the classical nonlinear conditions for aircraft separation. Further, we characterise conflict-free trajectories based on aircraft velocity bounds and propose a simple pre-processing algorithm to identify aircraft pairs which are either always conflict-free, or which cannot be separated using speed and heading control only. We then incorporate altitude control and propose a lexicographic optimisation formulation that aims to minimise the number of flight level changes before resolving outstanding conflicts via two-dimensional velocity control. The proposed mixed-integer programming formulations are nonconvex, and we propose convex relaxations, decomposition methods and constraint generation algorithms to solve the two-dimensional and lexicographic optimisation formulations to guaranteed optimality. Numerical experiments on four types of conflict resolution benchmarking instances are conducted to test the performance of the proposed mixed-integer formulations. Further, the proposed method is compared against two benchmarks based on state-of-the-art approaches for the aircraft conflict resolution problem. Our numerical results show that the proposed method largely outperforms both benchmarks in terms of runtime and is able to solve significantly more instances to global optimality.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Solving equilibrium problems using extended mathematical programming

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP.

Equilibrium programming↗
NASA NTRS1995

A Variational Formulation for the Finite Element Analysis of Sound Wave Propagation in a Spherical Shell

Development of design tools to furnish optimal acoustic environments for lightweight aircraft demands the ability to simulate the acoustic system on a workstation. In order to form an effective mathematical model of the phenomena at hand, we have begun by studying the propagation of acoustic waves inside closed spherical shells. Using a fully-coupled fluid-structure interaction model based upon variational principles, we have written a finite element analysis program and are in the process of examining several test cases. Future investigations are planned to increase model accuracy by incorporating non-linear and viscous effects.

Lebiedzik, Catherine↗
DOE OSTI2016

Features in simulation of crystal growth using the hyperbolic PFC equation and the dependence of the numerical solution on the parameters of the computational grid

We investigate the three-dimensional mathematical model of crystal growth called PFC (Phase Field Crystal) in a hyperbolic modification. This model is also called the modified model PFC (originally PFC model is formulated in parabolic form) and allows to describe both slow and rapid crystallization processes on atomic length scales and on diffusive time scales. Modified PFC model is described by the differential equation in partial derivatives of the sixth order in space and second order in time. The solution of this equation is possible only by numerical methods. Previously, authors created the software package for the solution of the Phase Field Crystal problem, based on the method of isogeometric analysis (IGA) and PetIGA program library. During further investigation it was found that the quality of the solution can strongly depends on the discretization parameters of a numerical method. In this report, we show the features that should be taken into account during constructing the computational grid for the numerical simulation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2016

Strengthening the SDP Relaxation of AC Power Flows with Convex Envelopes, Bound Tightening, and Valid Inequalities

Here this work revisits the Semidefine Programming (SDP) relaxation of the AC power flow equations in light of recent results illustrating the benefits of bounds propagation, valid inequalities, and the Convex Quadratic (QC) relaxation. By integrating all of these results into the SDP model a new hybrid relaxation is proposed, which combines the benefits from all of these recent works. This strengthened SDP formulation is evaluated on 71 AC Optimal Power Flow test cases from the NESTA archive and is shown to have an optimality gap of less than 1% on 63 cases. This new hybrid relaxation closes 50% of the open cases considered, leaving only 8 for future investigation.

42 ENGINEERING↗
NASA NTRS1986

The mathematical modeling of rapid solidification processing

The detailed formulation of and the results obtained from a continuum mechanics-based mathematical model of the planar flow melt spinning (PFMS) rapid solidification system are presented and discussed. The numerical algorithm proposed is capable of computing the cooling and freezing rates as well as the fluid flow and capillary phenomena which take place inside the molten puddle formed in the PFMS process. The FORTRAN listings of some of the most useful computer programs and a collection of appendices describing the basic equations used for the modeling are included.

Gutierrez-Miravete, E.↗
DOE OSTI2021

FLASSH 1.0: Full Law Analysis Scattering System Hub

Nuclear technologies, such as thermal reactors and critical systems, are driven by reaction rates which are directly dependent on thermal neutrons. The ability to calculate reaction rates is directly linked to the capability to predict and understand the cross sections. For atoms bound within a material structure, the available modes of energy exchange are captured in the Thermal Scattering Law (TSL, i.e. S(α,β)). The TSL is fundamentally a material property and can be defined explicitly to account for the excitation and deexcitation of the lattice energy states. This work describes the development of the Full Law Analysis Scattering System Hub (FLASSH) which calculates the TSL and cross sections. FLASSH is a modern code which has been designed to produce accurate TSL evaluations, remove historical approximations, and improve formulations while making advance physics accessible and simple to use.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Convex Programs for Minimal-Area Problems

The minimal-area problem that defines string diagrams in closed string field theory asks for the metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. Here, we consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Nonlinear and Digital Man-machine Control Systems Modeling

An adaptive modeling technique is examined by which controllers can be synthesized to provide corrective dynamics to a human operator's mathematical model in closed loop control systems. The technique utilizes a class of Liapunov functions formulated for this purpose, Liapunov's stability criterion and a model-reference system configuration. The Liapunov function is formulated to posses variable characteristics to take into consideration the identification dynamics. The time derivative of the Liapunov function generate the identification and control laws for the mathematical model system. These laws permit the realization of a controller which updates the human operator's mathematical model parameters so that model and human operator produce the same response when subjected to the same stimulus. A very useful feature is the development of a digital computer program which is easily implemented and modified concurrent with experimentation. The program permits the modeling process to interact with the experimentation process in a mutually beneficial way.

Mekel, R.↗
NASA NTRS2013

Educational NASA Computational and Scientific Studies (enCOMPASS)

Educational NASA Computational and Scientific Studies (enCOMPASS) is an educational project of NASA Goddard Space Flight Center aimed at bridging the gap between computational objectives and needs of NASA's scientific research, missions, and projects, and academia's latest advances in applied mathematics and computer science. enCOMPASS achieves this goal via bidirectional collaboration and communication between NASA and academia. Using developed NASA Computational Case Studies in university computer science/engineering and applied mathematics classes is a way of addressing NASA's goals of contributing to the Science, Technology, Education, and Math (STEM) National Objective. The enCOMPASS Web site at http://encompass.gsfc.nasa.gov provides additional information. There are currently nine enCOMPASS case studies developed in areas of earth sciences, planetary sciences, and astrophysics. Some of these case studies have been published in AIP and IEEE's Computing in Science and Engineering magazines. A few university professors have used enCOMPASS case studies in their computational classes and contributed their findings to NASA scientists. In these case studies, after introducing the science area, the specific problem, and related NASA missions, students are first asked to solve a known problem using NASA data and past approaches used and often published in a scientific/research paper. Then, after learning about the NASA application and related computational tools and approaches for solving the proposed problem, students are given a harder problem as a challenge for them to research and develop solutions for. This project provides a model for NASA scientists and engineers on one side, and university students, faculty, and researchers in computer science and applied mathematics on the other side, to learn from each other's areas of work, computational needs and solutions, and the latest advances in research and development. This innovation takes NASA science and engineering applications to computer science and applied mathematics university classes, and makes NASA objectives part of the university curricula. There is great potential for growth and return on investment of this program to the point where every major university in the U.S. would use at least one of these case studies in one of their computational courses, and where every NASA scientist and engineer facing a computational challenge (without having resources or expertise to solve it) would use enCOMPASS to formulate the problem as a case study, provide it to a university, and get back their solutions and ideas.

Memarsadeghi, Nargess↗
NASA NTRS2017

Mind the Gap: Exploring the Underground of the NASA Space Cancer Risk Model

The REID quantifies the lifetime risk of death from radiation-induced cancer in an exposed astronaut. The NASA Space Cancer Risk (NSCR) 2012 mode incorporates elements from physics, biology, epidemiology, and statistics to generate the REID distribution. The current model quantifies the space radiation environment, radiation quality, and dose-rate effects to estimate a NASA-weighted dose. This weighted dose is mapped to the excess risk of radiation-induced cancer mortality from acute exposures to gamma rays and then transferred to an astronaut population. Finally, the REID is determined by integrating this risk over the individual's lifetime. The calculated upper 95% confidence limit of the REID is used to restrict an astronaut's permissible mission duration (PMD) for a proposed mission. As a statistical quantity characterized by broad, subjective uncertainties, REID estimates for space missions result in wide distributions. Currently, the upper 95% confidence level is over 350% larger than the mean REID value, which can severely limit an astronaut's PMD. The model incorporates inputs from multiple scientific disciplines in the risk estimation process. Physics and particle transport models calculate how radiation moves through space, penetrates spacecraft, and makes its way to the human beings onboard. Epidemiological studies of exposures from atomic bombings, medical treatments, and power plants are used to quantify health risks from acute and chronic low linear energy transfer (LET) ionizing radiation. Biological studies in cellular and animal models using radiation at various LETs and energies inform quality metrics for ions present in space radiation. Statistical methodologies unite these elements, controlling for mathematical and scientific uncertainty and variability. Despite current progress, these research platforms contain knowledge gaps contributing to the large uncertainties still present in the model. The NASA Space Radiation Program Element (SRPE) defines the knowledge gaps that impact our understanding of the cancer risks. These gaps are outlined in NASA's Human Research Roadmap [4], which identifies the research questions and actions recommended for reducing the uncertainty in the current NSCR model and for formulation of future models. The greatest contributors to uncertainty in the current model include radiation quality, dose rate effects, and the transfer of exposure-based risk from other populations to an astronaut population. Future formulations of the risk model may benefit from including other potential sources of uncertainty such as space dosimetry, errors in human epidemiology data, and the impact of microgravity and other spaceflight stressors. Here, we discuss the current capabilities of the NSCR-2012 model and several immediate research needs, highlighting areas expected to have an operational impact on the current model schema. The following subway-style route map outlines the NSCR-2012 model (Green Line), emphasizing the research gaps in the Human Research Roadmap for risk of radiation-induced carcinogenesis (Stops on Dashed Lines). The map diagrams how these research gaps feed specific portions of the model.

Chappell, L. J.↗
DOE OSTI2020

Analytic Insights into the Neutronic Characteristics of Neutron Moderators from MCNP Calculations

In the course of re-processing the ENDF/B VIII.0 thermal scattering law data, formulas and data checking insights were developed from MCNP calculations into the neutronic characteristics of moderators. Infinite medium test calculations were carried out in MCNP as part of the verification effort for the ACE formatted thermal scattering files. Edit results from MCNP were combined with derived results to formulate several equations which give analytic insights into the neutronic behavior of moderators.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

graphenv: a Python library for reinforcement learning on graph search spaces

Many important and challenging problems in combinatorial optimization (CO) can be expressed as graph search problems, in which graph vertices represent full or partial solutions and edges represent decisions that connect them. Graph structure not only introduces strong relational inductive biases for learning (Battaglia et al., 2018) - in this context, by providing a way to explicitly model the value of transitioning (along edges) between one search state (vertex) and the next - but lends itself to problems both with and without clearly defined algebraic structure. For example, classic CO problems on graphs such as the Traveling Salesman Problem (TSP) can be expressed as either pure graph search or integer programs. Other problems, however, such as molecular optimization, do no have concise algebraic formulations and yet are readily implemented as a graph search (V. et al., 2022; Zhou et al., 2019). Such "model-free" problems constitute a large fraction of modern reinforcement learning (RL) research owing to the fact that it is often much easier to write a forward simulation that expresses all of the state transitions and rewards, than to write down the precise mathematical expression of the full optimization problem. In the case of molecular optimization, for example, one can use domain knowledge alongside existing software libraries to model the effect of adding a single bond or atom to an existing but incomplete molecule, and let the RL algorithm build a model of how good a given decision is by "experiencing" the simulated environment many times through. In contrast, a model-based mathematical formulation that fully expresses all the chemical and physical constraints is intractable. In recent years, RL has emerged as an effective paradigm for optimizing searches over graphs and led to state-of-the-art heuristics for games like Go and chess, as well as for classical CO problems such as the TSP. This combination of graph search and RL, while powerful, requires non-trivial software to execute, especially when combining advanced state representations such as Graph Neural Networks (GNN) with scalable RL algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Basic Research Needs for Inverse Methods for Complex Systems under Uncertainty

Inverse problems, which aim to infer unknown properties of a system using experimental and observational data, are central to addressing many of the U.S. Department of Energy’s (DOE) most critical scientific and engineering challenges. Accurate, computationally efficient, and data-efficient solutions to inverse problems are essential for advancing DOE mission-critical science drivers, including analyzing data from large-scale experimental facilities, optimizing fusion reactor performance, accelerating materials discovery, enhancing geophysical imaging, improving wildfire predictions, and enabling autonomous systems and digital twins. However, these problems are becoming increasingly complex, often involving nonlinear, highdimensional, and interconnected systems and models that span multiple physics and scales, while relying on data with varying quantity, quality, and information content. Compounding these challenges is the uncertainty inherent in DOE-relevant systems, where errors in inputs, noise in data, incompleteness of data, and discrepancies between models and reality constrain the accuracy and precision of solutions. At the same time, the convergence of recent scientific computing trends—scientific machine learning, artificial intelligence, and computing advances such as exascale computing—is creating unprecedented opportunities for tackling these challenges. The cross-cutting nature of inverse problems, combined with their growing complexity and rapidly evolving data and algorithmic demands, strongly motivates the formulation of a prioritized research agenda to maximize their capabilities and impact. In response to this need, DOE’s Advanced Scientific Computing Research (ASCR) program in the Office of Science convened the Workshop on Basic Research Needs for Inverse Problems for Complex Systems Under Uncertainty in June 2025. This workshop brought together experts across disciplines to identify grand challenges and major opportunities in the field. Through collaborative discussions, the workshop defined transformative research directions aimed at addressing the mathematical, statistical, and computational challenges posed by inverse problems under uncertainty. As a result of these efforts, four priority research directions (PRDs) were identified to guide future research and development in this area. These PRDs, summarized below, represent a roadmap for advancing the foundational science and mathematics of inverse problems, enabling robust, scalable, and uncertainty-aware solutions that are critical for DOE applications.

97 MATHEMATICS AND COMPUTING↗