Engineering topics

Leyffer, Sven

Publications and source records attributed to Leyffer, Sven.

At least 19 records

DOE OSTI2024

Binary Quantum Control Optimization with Uncertain Hamiltonians

Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a discretized binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical simulations on various sized problem instances based on two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. In conclusion, we demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared with the controls of a deterministic model.

conditional value-at-risk (CVaR)↗

DOE OSTI2024

Remark on Algorithm 1012: Computing Projections with Large Datasets

In ACM TOMS Algorithm 1012, the DELAUNAYSPARSE software is given for performing Delaunay interpolation in medium to high dimensions. When extrapolating outside the convex hull of the training set, DELAUNAYSPARSE calls the nonnegative least squares solver DWNNLS to compute projections onto the convex hull. However, DWNNLS and many other available sum-of-squares optimization solvers were not intended for usage with many variable problems, which result from the large training sets that are typical in machine learning applications. Thus, a new PROJECT subroutine is given, based on the highly customizable quadratic program solver BQPD. This solution is shown to be as robust as DELAUNAYSPARSE for projection onto both synthetic and real-world datasets, where other available solvers frequently fail. Although it is intended as an update for DELAUNAYSPARSE, due to the difficulty and prevalence of the problem, this solution is likely to be of external interest as well.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics

Here, in this paper, we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. Cozad and Sahinidis propose a nonconvex mixed-integer nonlinear program (MINLP), in which binary variables are used to assign operators and nonlinear expressions are used to propagate data values through nonlinear operators, such as square, square root, and exponential. We extend this formulation by adding new cuts that improve the solution of this challenging MINLP. We also propose a heuristic that iteratively builds an expression tree by solving a restricted MINLP. We perform computational experiments and compare our approach with a mixed-integer program–based method and a neural network–based method from the literature.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Binary Control Pulse Optimization for Quantum Systems

Quantum control aims to manipulate quantum systems toward specific quantum states or desired operations. Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation. In this paper we focus on discrete binary quantum control problems and apply different optimization algorithms and techniques to improve computational efficiency and solution quality. Specifically, we develop a generic model and extend it in several ways. We introduce a squared L 2 -penalty function to handle additional side constraints, to model requirements such as allowing at most one control to be active. We introduce a total variation (TV) regularizer to reduce the number of switches in the control. We modify the popular gradient ascent pulse engineering (GRAPE) algorithm, develop a new alternating direction method of multipliers (ADMM) algorithm to solve the continuous relaxation of the penalized model, and then apply rounding techniques to obtain binary control solutions. We propose a modified trust-region method to further improve the solutions. Our algorithms can obtain high-quality control results, as demonstrated by numerical studies on diverse quantum control examples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

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↗

DOE OSTI2022

Sequential linear integer programming for integer optimal control with total variation regularization

We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies that the considered integer control problems admit minimizers. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Stochastic Learning Approach for Binary Optimization: Application to Bayesian Optimal Design of Experiments

Here, we present a novel stochastic approach to binary optimization suited for optimal experimental design (OED) for Bayesian inverse problems governed by mathematical models such as partial differential equations. The OED utility function, namely, the regularized optimality criterion, is cast into a stochastic objective function in the form of an expectation over a multivariate Bernoulli distribution. The probabilistic objective is then solved by using a stochastic optimization routine to find an optimal observational policy. This formulation (a) is generally applicable to binary optimization problems with soft constraints and is ideal for OED and sensor placement problems; (b) does not require differentiability of the original objective function (e.g., a utility function in OED applications) with respect to the design variable, and thus it enables direct employment of sparsity-enforcing penalty functions such as $\ell_0$, without needing to utilize a continuation procedure or apply a rounding technique; (c) exhibits much lower computational cost than traditional gradient-based relaxation approaches; and (d) can be applied to both linear and nonlinear OED problems with proper choice of the utility function. The proposed approach is analyzed from an optimization perspective with detailed convergence analysis of the optimization approach and is also analyzed from a machine learning perspective with correspondence to policy gradient reinforcement learning. The approach is demonstrated numerically by using an idealized two-dimensional Bayesian linear inverse problem and validated by extensive numerical experiments carried out for sensor placement in a parameter identification setup.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

State elimination for mixed‐integer optimal control of partial differential equations by semigroup theory

Abstract Mixed‐integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

BROOD: Bilevel and Robust Optimization and Outlier Detection for Efficient Tuning of High-Energy Physics Event Generators

The parameters in Monte Carlo (MC) event generators are tuned on experimental measurements by evaluating the goodness of fit between the data and the MC predictions. The relative importance of each measurement is adjusted manually in an often time-consuming, iterative process to meet different experimental needs. In this work, we introduce several optimization formulations and algorithms with new decision criteria for streamlining and automating this process. These algorithms are designed for two formulations: bilevel optimization and robust optimization. Both formulations are applied to the datasets used in the ATLAS A14 tune and to the dedicated hadronization datasets generated by the SHERPA generator, respectively. The corresponding tuned generator parameters are compared using three metrics. We compare the quality of our automatic tunes to the published ATLAS A14 tune. Moreover, we analyze the impact of a pre-processing step that excludes data that cannot be described by the physics models used in the MC event generators.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints

Here, we propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Compact representations of structured BFGS matrices

For general large-scale optimization problems compact representations exist in which recursive quasi-Newton update formulas are represented as compact matrix factorizations. For problems in which the objective function contains additional structure, recent structured quasi-Newton methods exploit available second-derivative information and approximate unavailable second derivatives. Here, this article develops the compact representations of two structured Broyden-Fletcher-Goldfarb-Shanno update formulas. The compact representations enable efficient limited memory and initialization strategies. Two limited memory line search algorithms are described for which extensive numerical results demonstrate the efficacy of the algorithms, including comparisons to IPOPT on large machine learning problems, and to L-BFGS on a real world large scale ptychographic imaging application.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Practical algorithms for multivariate rational approximation

We present two approaches for computing rational approximations to multivariate functions, motivated by their effectiveness as surrogate models for high-energy physics (HEP) applications. Our first approach builds on the Stieltjes process to efficiently and robustly compute the coefficients of the rational approximation. Our second approach is based on an optimization formulation that allows us to include structural constraints on the rational approximation (in particular, constraints demanding the absence of singularities), resulting in a semi-infinite optimization problem that we solve using an outer approximation approach. We present results for synthetic and real-life HEP data, and we compare the approximation quality of our approaches with that of traditional polynomial approximations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Apprentice for Event Generator Tuning

Apprentice is a tool developed for event generator tuning. It contains a range of conceptual improvements and extensions over the tuning tool Professor. Its core functionality remains the construction of a multivariate analytic surrogate model to computationally expensive Monte-Carlo event generator predictions. The surrogate model is used for numerical optimization in chi-square minimization and likelihood evaluation. Apprentice also introduces algorithms to automate the selection of observable weights to minimize the effect of mis-modeling in the event generators. We illustrate our improvements for the task of MC-generator tuning and limit setting.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

A solution framework for linear PDE-constrained mixed-integer problems

Abstract We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

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 OSTI2021

Concurrent Optimization of Capital Cost and Expected O&M

Concentrating solar power (CSP) technologies can utilize heat from concentrated sunlight from a field of tracking mirrors to generate electricity, reform fuel, provide process heat, or augment fossil plant heat sources. Electricity-generating power tower systems focus light from thousands of independent heliostats onto a thermal receiver, which uses the focused light to warm a heat transfer fluid (HTF), typically, a molten nitrate salt. The HTF is then sent to a power generation cycle or diverted into thermal energy storage (TES) for later use. Thermal storage is – in principle – a straightforward proposition. However, optimal utilization of a TES resource is complex and multi-faceted: thermal energy may be dispatched to produce electricity immediately upon first availability, or thermal energy may be reserved for next-day peak periods at risk of filling storage and dumping energy, or a portion of the thermal energy can be reserved to maintain equipment temperatures, reducing power cycle startup time, etc. Many possible dispatch permutations variously emphasize producing peak power, operating through transients, expediting daily startup, etc. The best operation strategy can change day-to-day throughout the year, depending on the weather and market pricing forecasts. The project we describe in this report develops a software package that allows users to explore design optimization, operations decisions, and performance characterization of concentrating solar power tower plants. Users interface with the tool through a scripting language, and results are reported in time series tables, plots, runtime logs, and design outputs. Users choose from a list of variables such as tower height, solar multiple, design-point irradiance, thermal storage size, etc., and specify information about the system using a list of parameters. The software can then optimize the specified variables to reduce the cost of energy produced by the system while meeting certain production requirements, accounting for uncertain weather and electricity price forecasts, and correcting for equipment failures or repair time. The software we develop is the first comprehensive design tool of its kind to incorporate all of these aspects while being deployed as open source.

14 SOLAR ENERGY↗

DOE OSTI2021

A Bilevel Approach for Identifying the Worst Contingencies for Nonconvex Alternating Current Power Systems

We address the bilevel optimization problem of identifying the most critical attacks to an alternating current (AC) power flow network. The upper-level binary maximization problem consists of choosing an attack that is treated as a parameter in the lower-level defender minimization problem. Instances of the lower-level global minimization problem by themselves are NP-hard due to the nonconvex AC power flow constraints, and bilevel solution approaches commonly apply a convex relaxation or approximation to allow for tractable bilevel reformulations at the cost of underestimating some power system vulnerabilities. Our main contribution is to provide an alternative branch-and-bound algorithm whose upper bounding mechanism (in a maximization context) is based on a reformulation that avoids relaxation of the AC power flow constraints in the lower-level defender problem. Lower bounding is provided with semidefinite programming (SDP) relaxed solutions to the lower-level problem. We establish finite termination with guarantees of either a globally optimal solution to the original bilevel problem, or a globally optimal solution to the SDP-relaxed bilevel problem which is included in a vetted list of upper-level attack solutions, at least one of which is a globally optimal solution to the bilevel problem. We demonstrate through computational experiments applied to IEEE case instances both the relevance of our contribution, and the effectiveness of our contributed algorithm for identifying power system vulnerabilities without resorting to convex relaxations of the lower-level problem. We conclude with a discussion of future extensions and improvements.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

STreCH

Symbolic regression software that formulates a symbolic regression problem as a mixed-integer nonlinear optimization problem, and solves it using a sequential tree construction heuristic

Leyffer, Sven↗