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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Pseudospectral convex optimization for on-ramp merging control of connected vehicles

It can be a daunting task for human drivers to merge into highways because of the intricate vehicle negotiations and potential risk within limited time and space. Connected vehicle (CV) technologies could be a solution to this problem and offer many benefits to the road safety, traffic mobility, and energy efficiency. However, real-time optimal control of CVs is still an open challenge, due to the nonlinear vehicle dynamics, non-convex fuel consumption model, and highly dynamic uncertain inter-vehicle interactions. To tackle these issues, a novel real-time optimal control approach that balances the computational efficiency and solution optimality is proposed for the purpose of onboard application. To this end, the pseudospectral collocation method is integrated with a sequential convex programming approach to develop two new optimization algorithms, which are implemented within a model predictive control (MPC) framework to allow for real-time generation of optimal merging speed profiles. One algorithm leverages the line search technique to improve convergence, and the other benefits from the trust region method for better computational efficiency. The optimality and convergence process of both proposed algorithms are investigated by comparing their solutions with a popular non-linear solver. Furthermore, simulation results show that the proposed methods outperform the benchmark in terms of computational cost, fuel consumption, and traffic efficiency. In particular, the proposed fuel-economy merging rule can save 57.1% fuel consumption on average on four different traffic volumes. Meanwhile, the proposed optimal control algorithms can reduce 2.2% travel time on average comparing to the “first-in-first-out” merging rule.

33 ADVANCED PROPULSION SYSTEMS↗
NASA NTRSDate not supplied

A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions

Sliced Normal (SN) distributions are a generalization of Gaussian distributions where the quadratic argument of the exponential is replaced with a sum of squares polynomial. SNs may be used to represent the distribution of a diverse set of random variables including multi-modal, non-symmetric, and skewed distributions. Unfortunately, the likelihood function of a SN includes a normalization constant and the inclusion of this normalization constant makes the likelihood a non-convex function of the hyperparameters which define the SN. In previous work, suboptimal fitting of the hyperparameters was performed by transforming the given data into a higher dimensional monomial basis and selecting the optimal hyperparameters of a Gaussian fit in this space. However, this approach did not account for the effect of lifting on the normalization constant. Indeed, it was observed that as the number of monomials is increased the likelihood of the Sliced Normal can decrease. In this paper, we increase the likelihood of Sliced Normals found using the previous method by developing a convex formulation which scales the covariance matrix of the Gaussian fit such that the likelihood of the Sliced Normal is maximized. The result is significant improvements of the log likelihood of fitted SN distributions, including a significant increase, especially for problems with 500+ monomials.

Convex optimization approach to improving suboptim↗
DOE OSTI2022

Real-Time On-Ramp Merging Control of Connected and Automated Vehicles using Pseudospectral Convex Optimization

Highway on-ramp merging can be a challenging task for human drivers due to the complex vehicle negotiations and interactions in limited time and space. Connected and automated vehicles (CAVs) have great potential to address the problem and offer many benefits in terms of safety, traffic efficiency, and fuel economy. However, real-time optimal control of CAVs still faces many challenges, including nonlinear dynamics, complex inter-vehicle interactions, and a highly dynamic and uncertain traffic environment. To address these challenges, we develop a novel control approach that balances the solution optimality and computational efficiency to determine optimal merging speed profiles in real time. Specifically, by employing a pseudospectral method and a sequential convex programming approach, two algorithms are proposed and implemented within the model predictive control (MPC) framework to enable real-time generation of optimal solutions for potential on-vehicle applications. The convergence and optimality of the proposed algorithms are validated by comparing with a general-purpose solver under different traffic scenarios.

Shi, Yang↗
DOE OSTI2023

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, in this work, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2021

Convex Optimization Guidance for Precision Landing on Titan

Precision landing is an anticipated technology for future interplanetary missions. Autonomous spacecraft Entry, Descent and Landing (EDL) on the surface of a planetary body with a degree of precision in the order of meters is highly challenging. In this paper, a successive convexification guidance algorithm is utilized to simulate autonomous precision landing sequences on Saturn’s moon Titan. Due to its unique geophysical features, studying the science of matter within Titan’s atmosphere and beneath its surface is one of NASA’s most important planetary science objectives. As part of the Space Exploration Technology Directorate, a parafoil is proposed for landing on Titan due to its cost effectiveness, ease of deployment, low mass compared to the prospective payload and capabilities of precise autonomous delivery. This paper focuses on path optimization and guidance law development for high-fidelity dynamics parafoil tuning in the dense and adverse wind atmosphere of Titan, defined as a nonlinear and nonconvex optimal control problem. The powerful successive convexification method is used to solve the problem accordingly. The algorithm is designed such that the converged solution adheres to the nonlinear dynamics and kinematics in accordance with the original formulation, while respecting the state and control constraints. The six-degree-of-freedom (6DoF) simulations results show that this robust method is suitable for autonomous interplanetary applications.

Mooij, Edwin↗
DOE OSTI2023

Quantum Optimization: Potential, Challenges, and the Path Forward

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of algorithmic approaches, often with little linkage. This is further complicated by the fragmented nature of the field of mathematical optimization, where major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions, have devoted communities. With these aspects in mind, this work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory — highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains – finance and sustainability – as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Huang, Bowen↗
DOE OSTI2024

Precision Computations in Strongly Coupled Conformal Field Theories (Final Technical Report)

Conformal Field Theories (CFTs) are quantum field theories that are invariant under the conformal symmetry group (which includes translations and rotations, but also local rescalings of spacetime). They are building blocks of general quantum field theories, and appear in many areas of physics, including statistical physics, condensed matter physics, particle physics, and quantum gravity. Because of their extra symmetries, the mathematical structure of CFTs is tightly constrained, and this leads to the idea of the ``conformal bootstrap," which is to use these mathematical structures to constrain, and in some cases determine, CFT observables. A new numerical implementation of the conformal bootstrap idea appeared in 2008 with the work of Rattazzi, Rychkov, Tonni, and Vichi. Their observation was that certain bootstrap constraints (conformal symmetry and unitarity) could be combined to yield a convex optimization problem that constraints CFT data. By solving this convex optimization problem on a computer, one could obtain bounds on observables like critical exponents and operator product expansion (OPE) coefficients. Over the course of this award, the PI has improved numerical bootstrap techniques by optimizing known algorithms and finding new ones for performing the required convex optimization computations. The PI has applied these techniques to compute high-precision observables in several important strongly-coupled systems. The PI has also explored both analytical and numerical bootstrap methods for constraining the space of low energy effective field theories of quantum gravity, and developed new analytical techniques for CFT and QFT more broadly.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS2011

Safe Onboard Guidance and Control Under Probabilistic Uncertainty

An algorithm was developed that determines the fuel-optimal spacecraft guidance trajectory that takes into account uncertainty, in order to guarantee that mission safety constraints are satisfied with the required probability. The algorithm uses convex optimization to solve for the optimal trajectory. Convex optimization is amenable to onboard solution due to its excellent convergence properties. The algorithm is novel because, unlike prior approaches, it does not require time-consuming evaluation of multivariate probability densities. Instead, it uses a new mathematical bounding approach to ensure that probability constraints are satisfied, and it is shown that the resulting optimization is convex. Empirical results show that the approach is many orders of magnitude less conservative than existing set conversion techniques, for a small penalty in computation time.

Blackmore, Lars James↗
DOE OSTI2022

A Non-Cooperative Game-Based Distributed Beam Scheduling Framework for 5G Millimeter-Wave Cellular Networks

Here, this paper studies the problem of distributed beam scheduling for 5G millimeter-Wave (mm-Wave) cellular networks where base stations (BSs) belonging to different operators share the same spectrum without centralized coordination among them. Our goal is to design efficient distributed scheduling algorithms to maximize the network utility, which is a function of the achieved throughput by the user equipment (UEs), subject to the average and instantaneous power consumption constraints of the BSs. We propose a Media Access Control (MAC) and a power allocation/adaptation mechanism utilizing the Lyapunov stochastic optimization framework and non-cooperative games. In particular, we first decompose the original utility maximization problem into two sub-optimization problems for each time frame, which are a convex optimization problem and a non-convex optimization problem, respectively. By formulating the distributed scheduling problem as a non-cooperative game where each BS is a player attempting to optimize its own utility, we provide a distributed solution to the non-convex sub-optimization problem via finding the Nash Equilibrium (NE) of the game whose weights are determined optimally by the Lyapunov optimization framework. Finally, we conduct simulation under various network settings to show the effectiveness of the proposed game-based beam scheduling algorithm in comparison to that of several reference schemes.

42 ENGINEERING↗
NASA NTRS2012

Active Mirror Predictive and Requirements Verification Software (AMP-ReVS)

This software is designed to predict large active mirror performance at various stages in the fabrication lifecycle of the mirror. It was developed for 1-meter class powered mirrors for astronomical purposes, but is extensible to other geometries. The package accepts finite element model (FEM) inputs and laboratory measured data for large optical-quality mirrors with active figure control. It computes phenomenological contributions to the surface figure error using several built-in optimization techniques. These phenomena include stresses induced in the mirror by the manufacturing process and the support structure, the test procedure, high spatial frequency errors introduced by the polishing process, and other process-dependent deleterious effects due to light-weighting of the mirror. Then, depending on the maturity of the mirror, it either predicts the best surface figure error that the mirror will attain, or it verifies that the requirements for the error sources have been met once the best surface figure error has been measured. The unique feature of this software is that it ties together physical phenomenology with wavefront sensing and control techniques and various optimization methods including convex optimization, Kalman filtering, and quadratic programming to both generate predictive models and to do requirements verification. This software combines three distinct disciplines: wavefront control, predictive models based on FEM, and requirements verification using measured data in a robust, reusable code that is applicable to any large optics for ground and space telescopes. The software also includes state-of-the-art wavefront control algorithms that allow closed-loop performance to be computed. It allows for quantitative trade studies to be performed for optical systems engineering, including computing the best surface figure error under various testing and operating conditions. After the mirror manufacturing process and testing have been completed, the software package can be used to verify that the underlying requirements have been met.

Basinger, Scott A.↗
DOE OSTI2021

An infeasible-start framework for convex quadratic optimization, with application to constraint-reduced interior-point and other methods

A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating rule. Under mild assumptions, the framework is proved to converge on CQPs with both inequality and equality constraints and, at a negligible additional cost per iteration, produces an infeasibility certificate, together with a feasible point for an (approximately) ℓ 1 -least relaxed feasible problem, when the given problem does not have a feasible solution. The framework is applied to a feasible-start constraint-reduced interior-point algorithm previously proved to be highly performant on problems with many more inequality constraints than variables (“imbalanced”). Numerical comparison with popular codes (OSQP, qpOASES, MOSEK) is reported on both randomly generated problems and support-vector machine classifier training problems. The results show that the former typically outperforms the latter on imbalanced problems. Finally, application of the proposed infeasible-start framework to other feasible-start algorithms is briefly considered, and is tested on a simplex iteration.

97 MATHEMATICS AND COMPUTING↗