SEARCH · Engineering Papers
Results for “non-convex optimization”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
COHORT: Coordination of Heterogeneous Thermostatically Controlled Loads for Demand Flexibility
Demand flexibility is increasingly important for power grids. Careful coordination of thermostatically controlled loads (TCLs) can modulate energy demand, decrease operating costs, and increase grid resiliency. We propose a novel distributed control framework for the Coordination Of HeterOgeneous Residential Thermostatically controlled loads (COHORT). COHORT is a practical, scalable, and versatile solution that coordinates a population of TCLs to jointly optimize a grid-level objective, while satisfying each TCL’s end-use requirements and operational constraints. To achieve that, we decompose the grid-scale problem into subproblems and coordi- nate their solutions to find the global optimum using the alternating direction method of multipliers (ADMM). The TCLs’ local problems are distributed to and computed in parallel at each TCL, making COHORT highly scalable and privacy-preserving. While each TCL poses combinatorial and non-convex constraints, we characterize these constraints as a convex set through relaxation, thereby making COHORT computationally viable over long planning horizons. After coordination, each TCL is responsible for its own control and tracks the agreed-upon power trajectory with its preferred strategy. In this work, we translate continuous power back to discrete on/off actuation, using pulse width modulation. COHORT is generalizable to a wide range of grid objectives, which we demonstrate through three distinct use cases: generation following, minimizing ramping, and peak load curtailment. In a notable experiment, we validated our approach through a hardware-in-the-loop simulation, including a real-world air conditioner (AC) controlled via a smart thermostat, and simulated instances of ACs modeled after real-world data traces. During the 15-day experimental period, COHORT reduced daily peak loads by an average of 12.5% and maintained comfortable temperatures.
Convergence analysis for a nonlocal gradient descent method via directional Gaussian smoothing
We analyze the convergence of a nonlocal gradient descent method for minimizing a class of high-dimensional non-convex functions, where a directional Gaussian smoothing (DGS) is proposed to define the nonlocal gradient (also referred to as the DGS gradient). The method was first proposed in [Zhang et al., Enabling long-range exploration in minimization of multimodal functions, UAI 2021], in which multiple numerical experiments showed that replacing the traditional local gradient with the DGS gradient can help the optimizers escape local minima more easily and significantly improve their performance. However, a rigorous theory for the efficiency of the method on nonconvex landscape is lacking. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by deterministic oscillating noise. We provide a convergence theory under which the iterates exponentially converge to a tightened neighborhood of the solution, whose size is characterized by the noise wavelength. Here, we also establish a correlation between the optimal values of the Gaussian smoothing radius and the noise wavelength, thus justifying the advantage of using moderate or large smoothing radii with the method. Furthermore, if the noise level decays to zero when approaching the global minimum, we prove that DGS-based optimization converges to the exact global minimum with linear rates, similarly to standard gradient-based methods in optimizing convex functions. Several numerical experiments are provided to confirm our theory and illustrate the superiority of the approach over those based on the local gradient.
Global stellarator coil optimization with quadratic constraints and objectives
Most present stellarator designs are produced by costly two-stage optimization: the first for an optimized equilibrium, and the second for a coil design reproducing its magnetic configuration. Few proxies for coil complexity and forces exist at the equilibrium stage. Rapid initial state finding for both stages is a topic of active research. Most present convex coil optimization codes use the least square winding surface method by Merkel (NESCOIL), with recent improvements in conditioning, regularization, sparsity, and physics objectives. While elegant, the method is limited to modeling the norms of linear functions in coil current. We present QUADCOIL, a global coil optimization method that targets combinations of linear and quadratic functions of the current. It can directly constrain and/or minimize a wide range of physics objectives unavailable in NESCOIL and REGCOIL, including the Lorentz force, magnetic energy, curvature, field-current alignment, and the maximum density of a dipole array. QUADCOIL requires no initial guess and runs nearly $10$ 2 x faster than filament optimization. Integrating it in the equilibrium optimization stage can potentially exclude equilibria with difficult-to-design coils, without significantly increasing the computation time per iteration. QUADCOIL finds the exact, global minimum in a large parameter space when possible, and otherwise finds a well-performing approximate global minimum. It supports most regularization techniques developed for NESCOIL and REGCOIL. We demonstrate QUADCOIL’s effectiveness in coil topology control, minimizing non-convex penalties, and predicting filament coil complexity with three numerical examples.
Multi-plane moment-of-fluid interface reconstruction in 3D
Moment-of-fluid (MOF) methods for interface reconstruction approximate the region occupied by material in each mesh element only through reference to its geometric moments. Here, we present a 3D MOF method that represents the material (POM) in each cell as the convex intersection of the cell and multiple half-spaces, each selected to minimize the least-squares error between computed moments of the approximated material and provided reference moments. This optimization problem is highly non-linear and non-convex, making the numerical result very sensitive to the initial guess. To create an effective initial guess in each cell, we construct an ellipsoid from 0th–2nd order reference moments such that its shape corresponds with that of the POM. Within this ellipsoid we inscribe a polyhedron, and initialize the minimization problem with the half-spaces defined by each of its faces. The inscribed polyhedron has minimally 4 faces, and using up to 3rd order moments permits optimization over up to 20 unknown values. We therefore define MOF methods that utilize 4, 5, or 6 half-spaces, correspondingly initialized with the faces of a single inscribed tetrahedron, triangular prism, or hexahedron. Stability of the non-linear optimization is further improved with a prepossessing step that normalizes the reference moments according to the axes of the reference ellipsoid. Using this approach, the non-linear least-squares solver reliably converges to a near-global minimum from a single initial guess. We demonstrate accuracy and robustness using single-cell and multi-cell examples over a wide spectrum of geometry. In particular, we demonstrate our ability to exactly reproduce several important and complex features defined by up to four half-spaces, such as corners, filaments, filament tips, and embedded material in the cell.
An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems
In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ 2 ) in the L∞(Ω) norm, which is optimal considering the O(δ 2 ) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Finally, based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ 2 ) convergence.
Networked Microgrid Topology Reconfiguration to Promote Fairness in Proactive Load Shedding
Increasing occurrences of natural disasters and grid emergency events consistently challenge the safe and reliable operations of power systems. During such emergency situations, system operators may proactively shed load to mitigate risks. However, uncoordinated implementation of load shedding may disrupt electricity supply and even lead to cascading failures. Meanwhile, it is crucial to address potential biases affecting different customers when executing load shedding. This paper addresses the dynamic topology reconfiguration problem for networked microgrids with distributed energy resources under emergency conditions. Specifically, we propose a novel rolling-horizon optimization model that integrates fairness-aware constraints into the networked microgrid topology reconfiguration. Unlike existing approaches that focus solely on efficiency or apply fairness considerations in static settings, our method explicitly incorporates temporal fairness constraints to restrict repeated or excessive load curtailment for load blocks. Moreover, the fairness-aware constraints are specifically developed for the context of dynamic networked microgrid topology reconfiguration, and are designed to be convex or amenable to linear reformulations, which offers a more tractable alternative to traditional models with non-convex formulations. Numerical studies on a modified IEEE 13-bus system and a larger-sized SMART-DS networked microgrid system demonstrate the performance of the proposed algorithm towards more fairness-aware networked microgrid topology reconfiguration decision-making.
Adaptive Power Flow Approximations With Second-Order Sensitivity Insights
The power flow equations are fundamental to power system planning, analysis, and control. However, the inherent non-linearity and non-convexity of these equations present formidable obstacles in problem-solving processes. To mitigate these challenges, recent research has proposed adaptive power flow linearizations that aim to achieve accuracy over wide operating ranges. The accuracy of these approximations inherently depends on the curvature of the power flow equations within these ranges, which necessitates considering second-order sensitivities. In this paper, we leverage second-order sensitivities to both analyze and improve power flow approximations. We evaluate the curvature across broad operational ranges and subsequently utilize this information to inform the computation of various sample-based power flow approximation techniques. Additionally, we leverage second-order sensitivities to guide the development of rational approximations that yield linear constraints in optimization problems. In conclusion, this approach is extended to enhance accuracy beyond the limitations of linear functions across varied operational scenarios.
Multivariate Parameter Sets for Optimal Synthesis of Compliant Mechanisms
This paper will propose the use of control maps along with discretized elements or meshes in the design parameter set for optimizing compliant mechanisms. The use of control maps will be demonstrated to encode the motion of groups of nodes or control points within a compliant mechanism design with simple mapping rules. The technique will serve as an alternative to increased mesh size or node wandering techniques that have been proposed to increase the number of alternative design shapes that may be considered. As an alternative approach, the proposed control map parameterization has the significant benefit that it minimizes the number of design parameters necessary (parameters increase linearly with the mesh size) in describing a given design making it computationally efficient. A limited number of tiles can produce a map that has a significant effect on the final shape. If the tiles are chosen appropriately, the problems such as material overlap and non-convex mesh elements are avoided automatically. This paper will describe the implementation of these control maps and provide several examples showing their implementation in the compliant mechanism topology synthesis process.
Automated Resonance Fitting for Nuclear Data Evaluation
Global and national efforts to deliver high-quality nuclear data to users have a wide-ranging impact, affecting applications in national security, reactor operations, basic science, medicine, and more. Cross section evaluation is a major part of this effort, combining theory and experimentation to produce recommended values and uncertainties for reaction probabilities. Resonance region evaluation is a specialized type of nuclear data evaluation that can require significant manual effort and months of time from expert scientists. In this article, non-convex non-linear optimization methods are combined with concepts of inferential statistics to infer a resonance model from experimental data in an automated manner that is not dependent on prior evaluation(s). This methodology aims to enhance the workflow of a resonance evaluator by minimizing time, effort, and the potential for bias from prior assumptions, while enhancing reproducibility and documentation, thereby addressing well-known challenges in the field.
A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks
To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (Biometrika 81(3):425–455, 1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform simulated and real data Monte Carlo experiments to quantify the effectiveness of our approach.
Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations
Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.
On optimal control of hybrid dynamical systems using complementarity constraints
Optimal control for switch-based dynamical systems is a challenging problem in the process control literature. In this study, we model these systems as hybrid dynamical systems with finite number of unknown switching points and reformulate them using non-smooth and non-convex complementarity constraints as a mathematical program with complementarity constraints (MPCC). We utilize a moving finite element based strategy to discretize the differential equation system to accurately locate the unknown switching points at the finite element boundary and achieve high-order accuracy at intermediate non-collocation points. We propose a globalization approach to solve the discretized MPCC problem using a mixed NLP/MILP-based strategy to converge to a non-spurious first-order optimal solution. The method is tested on three dynamic optimization examples, including a gas–liquid tank model and an optimal control problem with a sliding mode solution.
Iterative subspace algorithms for finite-temperature solution of Dyson equation
One-particle Green’s functions obtained from the self-consistent solution of the Dyson equation can be employed in the evaluation of spectroscopic and thermodynamic properties for both molecules and solids. However, typical acceleration techniques used in the traditional quantum chemistry self-consistent algorithms cannot be easily deployed for the Green’s function methods because of a non-convex grand potential functional and a non-idempotent density matrix. Moreover, the optimization problem can become more challenging due to the inclusion of correlation effects, changing chemical potential, and fluctuations of the number of particles. In this paper, we study acceleration techniques to target the self-consistent solution of the Dyson equation directly. We use the direct inversion in the iterative subspace (DIIS), the least-squared commutator in the iterative subspace (LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We observe that the definition of the residual has a significant impact on the convergence of the iterative procedure. Based on the Dyson equation, we generalize the concept of the commutator residual used in DIIS and LCIIS and compare it with the difference residual used in DIIS and KAIN. The commutator residuals outperform the difference residuals for all considered molecular and solid systems within both GW and GF2. For a number of bond-breaking problems, we found that an easily obtained high-temperature solution with effectively suppressed correlations is a very effective starting point for reaching convergence of the problematic low-temperature solutions through a sequential reduction of temperature during calculations.
Minimum Landing Error Powered-Descent Guidance for Planetary Missions
An algorithm improves the accuracy with which a lander can be delivered to the surface of Mars. The main idea behind this innovation is the use of a lossless convexification, which converts an otherwise non-convex constraint related to thruster throttling to a convex constraint, enabling convex optimization to be used. The convexification leads directly to an algorithm that guarantees finding the global optimum of the original nonconvex optimization problem with a deterministic upper bound on the number of iterations required for convergence. In this innovation, previous work in powered-descent guidance using convex optimization is extended to handle the case where the lander must get as close as possible to the target given the available fuel, but is not required to arrive exactly at the target. The new algorithm calculates the minimum-fuel trajectory to the target, if one exists, and calculates the trajectory that minimizes the distance to the target if no solution to the target exists. This approach poses the problem as two Second-Order Cone Programs, which can be solved to global optimality with deterministic bounds on the number of iterations required.