When two worlds collide : a continuous mathematician's view of challenges in MINLP and combinatorial optimization.
No abstract prepared.
99 GENERAL AND MISCELLANEOUS↗
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Results for “Nonlinear programming.”
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Second Symposium on Nonlinear Reduced Order Models (NROM 2018) Program.
Abstract not provided.
Brink, Adam Ray↗
Modeling the AC Power Flow Equations with Optimally Compact Neural Networks: Application to Unit Commitment
Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using neural networks. These neural networks can be exactly transformed into mixed integer linear programs and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the neural network. Accordingly, this paper develops a technique for training an "optimally compact'' neural network, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We demonstrate the use of this neural network as an approximator of the nonlinear power flow equations by embedding it in the AC unit commitment problem, transforming the problem from a mixed integer nonlinear program into a more manageable mixed integer linear program. We use the 14-, 57-, and 89-bus networks as test cases and compare the AC-feasibility of commitment decisions resulting from the neural network, DC, and linearized power flow approximations. Our results show that the neural network model outperforms both the DC and linearized power flow approximations when embedded in the unit commitment problem. The neural network formulation most often selects a feasible unit commitment schedule, and furthermore, it only s
AC power flow↗
Mixed-Integer Linear Programming Formulation with Embedded Machine Learning Surrogates for the Design of Chemical Process Families
In previous work, we introduced process family design. The main idea is to design a platform of common elements, and, allowing us to capture additional cost savings, simultaneously design a family of processes, and reducing both engineering and deployment timelines. We formulate this as an optimization problem, specifically a nonlinear generalized disjunctive program (GDP). We have proposed two approaches for reformulating and solving this problem: one based on full-discretization of the design space and one that uses Machine Learning (ML) surrogates to replace the nonlinear process models. Using ML surrogates to predict required system costs and performance indicators allows us to reformulate the nonlinearities in the GDP generate an efficient MILP formulation. In this work, we apply the ML surrogate approach to two case studies. One case study involves designing a family of carbon capture systems to cover a set of different flue gas flow rates and inlet CO 2 concentrations, where we consider the absorber and stripper as common unit module types. The second case study focuses on a water-desalination process, where we design a family of these processes for a variety of salt concentrations and flow rates. In both of these case studies, we demonstrate a scalable optimization approach that enables the design of multiple processes simultaneously, reducing the time-to-market and overall costs by maximizing the cost savings due to both economies of scale and economies of numbers.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Automatic Identification of Closed-Loop Wind Turbine Dynamics via Genetic Programming
Wind turbines are nonlinear systems that operate in turbulent environments. As such, their behavior is difficult to characterize accurately across a wide range of operating conditions by physically meaningful models. Customarily, data-based models of wind turbines are defined in 'black box' format, lacking in both conciseness and physical intelligibility. To address this deficiency, we identify models of a modern horizontal-axis wind turbine in symbolic form using a recently developed symbolic regression method. The method used relies on evolutionary multi-objective optimization to produce succinct dynamic models from operational data without 'a priori' knowledge of the system. We compare the produced models with models derived by other methods for their estimation capacity and evaluate the tradeoff between model intelligibility and accuracy. Several succinct models are found that predict wind turbine behavior as well as or better than more complex alternatives derived by other methods.
17 WIND ENERGY↗
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↗
MEASURING THE EVOLUTIONARY RATE OF COOLING OF ZZ Ceti
We have finally measured the evolutionary rate of cooling of the pulsating hydrogen atmosphere (DA) white dwarf ZZ Ceti (Ross 548), as reflected by the drift rate of the 213.13260694 s period. Using 41 yr of time-series photometry from 1970 November to 2012 January, we determine the rate of change of this period with time to be dP/dt = (5.2 {+-} 1.4) Multiplication-Sign 10{sup -15} s s{sup -1} employing the O - C method and (5.45 {+-} 0.79) Multiplication-Sign 10{sup -15} s s{sup -1} using a direct nonlinear least squares fit to the entire lightcurve. We adopt the dP/dt obtained from the nonlinear least squares program as our final determination, but augment the corresponding uncertainty to a more realistic value, ultimately arriving at the measurement of dP/dt = (5.5 {+-} 1.0) Multiplication-Sign 10{sup -15} s s{sup -1}. After correcting for proper motion, the evolutionary rate of cooling of ZZ Ceti is computed to be (3.3 {+-} 1.1) Multiplication-Sign 10{sup -15} s s{sup -1}. This value is consistent within uncertainties with the measurement of (4.19 {+-} 0.73) Multiplication-Sign 10{sup -15} s s{sup -1} for another similar pulsating DA white dwarf, G 117-B15A. Measuring the cooling rate of ZZ Ceti helps us refine our stellar structure and evolutionary models, as cooling depends mainly on the core composition and stellar mass. Calibrating white dwarf cooling curves with this measurement will reduce the theoretical uncertainties involved in white dwarf cosmochronometry. Should the 213.13 s period be trapped in the hydrogen envelope, then our determination of its drift rate compared to the expected evolutionary rate suggests an additional source of stellar cooling. Attributing the excess cooling to the emission of axions imposes a constraint on the mass of the hypothetical axion particle.
79 ASTRONOMY AND ASTROPHYSICS↗
Minimax Program Terminal Control in Two-Level Hierarchic Nonlinear Discrete Dynamical System
We consider a discrete dynamical system consisting of a number of controllable objects whose dynamics is governed by vector nonlinear discrete recurrent relations with control parameters and perturbations (noises). In such system, two levels of making management decisions are distinguished: the dominating (first) level and the subordinate (second) level, which have different criteria of functioning. These levels are integrated by certain information and control relations. We consider the problem of two-level minimax program terminal control in a discrete dynamical system and propose its mathematical statement and a general scheme of its solution.
97 MATHEMATICS AND COMPUTING↗
WE-AB-207B-11: Optimizing Tumor Control Probability in Radiation Therapy Treatment - Application to HDR Cervical Cancer
Purpose: The ultimate goal of radiotherapy treatment planning is to find a treatment that will yield a high tumor-control-probability(TCP) with an acceptable normal-tissue-complication probability(NTCP). Yet most treatment planning today is not based upon optimization of TCPs and NTCPs, but rather upon meeting physical dose and volume constraints defined by the planner. We design treatment plans that optimize TCP directly and contrast them with the clinical dose-based plans. PET image is incorporated to evaluate gain in TCP for dose escalation. Methods: We build a nonlinear mixed integer programming optimization model that maximizes TCP directly while satisfying the dose requirements on the targeted organ and healthy tissues. The solution strategy first fits the TCP function with a piecewise-linear approximation, then solves the problem that maximizes the piecewise linear approximation of TCP, and finally performs a local neighborhood search to improve the TCP value. To gauge the feasibility, characteristics, and potential benefit of PET-image guided dose escalation, initial validation consists of fifteen cervical cancer HDR patient cases. These patients have all received prior 45Gy of external radiation dose. For both escalated strategies, we consider 35Gy PTV-dose, and two variations (37Gy-boost to BTV vs 40Gy-boost) to PET-image-pockets. Results: TCP for standard clinical plans range from 59.4% - 63.6%. TCP for dose-based PET-guided escalated-dose-plan ranges from 63.8%–98.6% for all patients; whereas TCP-optimized plans achieves over 91% for all patients. There is marginal difference in TCP among those with 37Gy-boosted vs 40Gy-boosted. There is no increase in rectum and bladder dose among all plans. Conclusion: Optimizing TCP directly results in highly conformed treatment plans. The TCP-optimized plan is individualized based on the biological PET-image of the patients. The TCP-optimization framework is generalizable and has been applied successfully to other external-beam delivery modalities. A clinical trial is on-going to gauge the clinical significance. Partially supported by the National Science Foundation.
60 APPLIED LIFE SCIENCES↗
Energy-efficient resin wafer electrodeionization for impaired water reclamation
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water desalination, electrodeionization, imparied ↗
Deep Learning Explicit Differentiable Predictive Control Laws for Buildings
We present a differentiable predictive control (DPC) methodology for learning constrained control laws for unknown nonlinear systems. DPC poses an approximate solution to multiparametric programming problems emerging from explicit nonlinear model predictive control (MPC). Contrary to approximate MPC, DPC does not require supervision by an expert controller. Instead, a system dynamics model is learned from a small dataset of recorded observations of the perturbed system's dynamics and the control law is optimized offline by interaction with the learned system model. The DPC method is based on two sequential steps, i) system identification using a constrained neural state-space model, and ii) optimization of an explicit control law parametrized by another neural network in closed-loop simulation with the identified neural state-space model. The combination of a differentiable closed-loop system and penalty methods for constraint handling of system outputs and inputs allows us to optimize the control law's parameters directly by backpropagating economic MPC loss through the learned system model. By incorporating domain knowledge and leveraging established techniques from optimal control, our method leverages deep neural networks as nonlinear function approximators for system identification and control while avoiding concomitant costs of intractably large datasets, and computationally expensive over-parametrized models. The scalability, data efficiency, and constrained optimal control capability of the proposed DPC method are demonstrated in simulation using a multi-zone building emulator.
Drgona, Jan↗
Modeling design and control problems involving neural network surrogates
Here, we consider nonlinear optimization problems that involve surrogate models represented by neural networks. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific case of feedforward neural networks with ReLU activation: as a mixed-integer optimization problem and as a mathematical program with complementarity constraints. For the latter formulation we prove that stationarity at a point for this problem corresponds to stationarity of the embedded formulation. Each of these formulations may be solved with state-of-the-art optimization methods, and we show how to obtain good initial feasible solutions for these methods. We compare our formulations on three practical applications arising in the design and control of combustion engines, in the generation of adversarial attacks on classifier networks, and in the determination of optimal flows in an oil well network.
97 MATHEMATICS AND COMPUTING↗
Alternative mixed integer linear programming optimization for joint job scheduling and data allocation in grid computing
This paper presents a novel approach to the joint optimization of job scheduling and data allocation in grid computing environments. We formulate this joint optimization problem as a mixed integer quadratically constrained program. To tackle the nonlinearity in the constraint, we alternatively fix a subset of decision variables and optimize the remaining ones via Mixed Integer Linear Programming (MILP). We solve the MILP problem at each iteration via an off-the-shelf MILP solver. Our experimental results show that our method significantly outperforms existing heuristic methods, employing either independent optimization or joint optimization strategies. We have also verified the generalization ability of our method over grid environments with various sizes and its high robustness to the algorithm setting.
97 MATHEMATICS AND COMPUTING↗
LOGISTIC FUNCTION PROFILE FIT: A least-squares program for fitting interface profiles to an extended logistic function
The extended logistic function provides a physically reasonable description of interfaces such as depth profiles or line scans of surface topological or compositional features. It describes these interfaces with the minimum number of parameters, namely, position, width, and asymmetry. Logistic Function Profile Fit (LFPF) is a robust, least-squares fitting program in which the nonlinear extended logistic function is linearized by a Taylor series expansion (equivalent to a Newton-Raphson approach) with no apparent introduction of bias in the analysis. The program provides reliable confidence limits for the parameters when systematic errors are minimal and provides a display of the residuals from the fit for the detection of systematic errors. The program will aid researchers in applying ASTM E1636-10, 'Standard practice for analytically describing sputter-depth-profile and linescan-profile data by an extended logistic function,' and may also prove useful in applying ISO 18516: 2006, 'Surface chemical analysis-Auger electron spectroscopy and x-ray photoelectron spectroscopy-determination of lateral resolution.' Examples are given of LFPF fits to a secondary ion mass spectrometry depth profile, an Auger surface line scan, and synthetic data generated to exhibit known systematic errors for examining the significance of such errors to the extrapolation of partial profiles.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Multispectrum measurements of spectral line parameters including temperature dependences of N2- and self-broadened half-width coefficients in the region of the v9 band of 12C2H6
Ethane is a prominent contributor to the spectrum of Titan, particularly in the region of the v9 band at 12μm. A multispectrum nonlinear least squares fitting program was applied to laboratory spectra of ethane to measure accurate positions, absolute intensities, N2- and selfbroadened half- width coefficients and their temperature dependences for a large number transitions. These measurements include several pQ and rQ sub-bands (and other sub-bands such as pP, rR) in the v9 fundamental band of 12C2H6 centered near 822 cm-1. Positions were measured for 2958 transitions and intensities for 3771 transitions. N2- and self-broadened half-width coefficients were determined for over 1700 transitions while temperature dependence exponents were retrieved for over 1350 of those transitions. Of these, many measurements (mostly line positions and intensities) belong to the v9+v4-v4 hot band, v9+2v4-2v4 hot band, 13C12CH6 v9 band and unidentified transitions. Forty-three high resolution (0.0016-0.005 cm-1) infrared laboratory absorption spectra recorded at temperatures between 148 and 298 K were fitted simultaneously to retrieve these parameters. Forty-one of these spectra were obtained in the temperature range of 211-298 K using the Bruker IFS 120HR interferometer located at the Pacific Northwest National Laboratory (PNNL) in Richland, Washington. Two additional spectra at 148 K were recorded using a new temperature stabilized cryogenic cell designed to work inside the sample compartment of the high resolution Bruker IFS 125HR interferometer of the Jet Propulsion Laboratory (JPL) in Pasadena California. The specialized cooling cell developed at Connecticut College and capable of achieving gas sample temperatures down to 70 K with a temperature stability and uniformity of better than ±0.05 K was employed to record the 148 K spectra. Constraints to intensity ratios, doublet separations, half-width coefficients and their temperature dependence exponents were required to determine these parameters for each of the two torsional split components. Similar to N2- and self-broadened half-width coefficients, their temperature dependence exponents were also found to follow distinctively different patterns. The variations of the observed half-width coefficients and their temperature dependences with respect to J, K quantum numbers are discussed. Because of the high density of torsionally split spectral lines, hot-band ransitions as well as blends, it was not possible to retrieve any information on the small pressure-induced shift coefficients. Present results are compared to other available measurements.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Discarding Information from Previous Iterations in an Optimal Way To Solve the Coupled Cluster Amplitude Equations
The direct inversion of the iterative subspace (DIIS) convergence acceleration algorithm is used in most electronic structure programs to solve the nonlinear coupled cluster amplitude equations. When the DIIS algorithm is used, the storage of previous trial vectors may become a bottleneck and the discarding of trial vectors may lead to a degradation of the convergence or even divergence. Here, we discuss an alternative way of storing information from trial vectors where only the last three trial vectors are needed to maintain the convergence of the full set of previous trial vectors, and which requires only minor modifications of an existing DIIS code
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
CICART Center For Integrated Computation And Analysis Of Reconnection And Turbulence
CICART is a partnership between the University of New Hampshire (UNH) and Dartmouth College. CICART addresses two important science needs of the DoE: the basic understanding of magnetic reconnection and turbulence that strongly impacts the performance of fusion plasmas, and the development of new mathematical and computational tools that enable the modeling and control of these phenomena. The principal participants of CICART constitute an interdisciplinary group, drawn from the communities of applied mathematics, astrophysics, computational physics, fluid dynamics, and fusion physics. It is a main premise of CICART that fundamental aspects of magnetic reconnection and turbulence in fusion devices, smaller-scale laboratory experiments, and space and astrophysical plasmas can be viewed from a common perspective, and that progress in understanding in any of these interconnected fields is likely to lead to progress in others. The establishment of CICART has strongly impacted the education and research mission of a new Program in Integrated Applied Mathematics in the College of Engineering and Applied Sciences at UNH by enabling the recruitment of a tenure-track faculty member, supported equally by UNH and CICART, and the establishment of an IBM-UNH Computing Alliance. The proposed areas of research in magnetic reconnection and turbulence in astrophysical, space, and laboratory plasmas include the following topics: (A) Reconnection and secondary instabilities in large high-Lundquist-number plasmas, (B) Particle acceleration in the presence of multiple magnetic islands, (C) Gyrokinetic reconnection: comparison with fluid and particle-in-cell models, (D) Imbalanced turbulence, (E) Ion heating, and (F) Turbulence in laboratory (including fusion-relevant) experiments. These theoretical studies make active use of three high-performance computer simulation codes: (1) The Magnetic Reconnection Code, based on extended two-fluid (or Hall MHD) equations, in an Adaptive Mesh Refinement (AMR) framework, (2) the Particle Simulation Code, a fully electromagnetic 3D Particle-In-Cell (PIC) code that includes a collision operator, and (3) GS2, an Eulerian, electromagnetic, kinetic code that is widely used in the fusion program, and simulates the nonlinear gyrokinetic equations, together with a self-consistent set of Maxwell’s equations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Computation of the Seismic Stability of Earth Retaining Structures Final Report CRADA No. TSB-954-94
The purpose of this CRADA project was to evaluate the seismic stability of block retaining wall systems. Retaining wall systems are used extensively in private and commercial developments. This study was designed to develop and demonstrate a computer modeling technology to be used to predict the seismic stability of any block wall system design. The nonlinear finite element computer programs developed at LLNL and employed in the Computational Earthquake Initiative were utilized in this small business CRADA to analyze the seismic stability of the block retaining walls. The unique capability of the LLNL programs to rigorously model frictional contact in a dynamic analysis problem were used in a computer simulation of the dynamic interaction the block waiVsoil systems under seismic excitation.
58 GEOSCIENCES↗