Constrained Minimization Problem Research Articles

A Deep Generative Model-Integrated Framework for 3-D Time-Difference Electrical Impedance Tomography

The time-difference image reconstruction problem of electrical impedance tomography (EIT) refers to reconstructing the conductivity change in a human body part between two time points using the boundary impedance measurements. Conventionally, the problem can be formulated as a linear inverse problem. However, due to the physical property of the forward process, the inverse problem is seriously ill-posed. As a result, traditional regularized least-squares (LSs)-based methods usually produce low-resolution images that are difficult to interpret. This work proposes a framework that uses a deep generative model (DGM) to constrain the unknown conductivity. Specifically, this framework allows the inclusion of a constraint that describes a mathematical relationship between the generative model and the unknown conductivity. The resultant constrained minimization problem is solved using an extended alternating direction method of multipliers (ADMM). The effectiveness of the framework is demonstrated by the example of 3-D time-difference chest EIT imaging. Numerical experiment shows a significant improvement of the image quality compared with total variation (TV)-regularized LSs method (peak signal-to-noise ratio (PSNR) is improved by 4.3% for 10% noise and 4.6% for 30% noise; structural similarity (SSIM) is improved by 4.8% for 10% noise and 6.0% for 30% noise). Human experiments show improved correlation between the reconstructed images and images from reference techniques.

A Method to Quantify Technical-Economic Aspects of HTS Electric Power Cables

In prevision of an expansion in the installed power and the progressive increase in electrification toward energy transition, high-efficiency electric grids are required. In this process, superconducting power cables could play an important role as they can be installed either to realize new lines or to replace existing infrastructures, potentially reducing the costs and the environmental impact compared to conventional solutions. The technical-economic convenience of the superconducting choice depends on the design parameters of the line, such as the voltage and power levels and the line length. For each condition, the configuration of the cable and its cooling system must be optimized while complying with a number of electrical and geometrical constraints. Identifying a priori the optimal cable configuration is not trivial, as the numerous variables involved are mutually dependent. This article describes a novel tool developed to quantify technical-economic aspects of superconducting cables, allowing the user to freely select the operating conditions for a generic electric line. The tool adopts a genetic algorithm to solve a constrained minimization problem for a multivariable function, returning the cable configuration characterized by the lowest costs. For this study, ac concentric cables realized with high temperature superconductors are selected, but the approach described can easily be adapted to other types of cable. By means of suitable simplifying hypotheses, the equations used to size the cable components and their corresponding costs, estimate the thermal inputs and describe the cryogen flows are introduced as parametric equations. Finally, the results of a detailed study on the impact of the relevant parameters on the cable design and costs is provided.

Pattern-Multiplicative Average of Nonnegative Matrices: When a Constrained Minimization Problem Requires Versatile Optimization Tools

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern as well. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative nature of averaging. The concept of PMA was motivated in a number of application fields, of which we consider the matrix population models and illustrate solving the PMA problem with several sets of model matrices calibrated in particular botanic case studies. The patterns of those matrices are typically nontrivial (they contain both zero and nonzero elements), the PMA problem thus having no exact solution for a fundamental reason (an overdetermined system of algebraic equations). Therefore, searching for the approximate solution reduces to a constrained minimization problem for the approximation error, the loss function in optimization terms. We consider two alternative types of the loss function and present a general algorithm of searching the optimal solution: basin-hopping global search, then local descents by the method of conjugate gradients or that of penalty functions. Theoretical disadvantages and practical limitations of both loss functions are discussed and illustrated with a number of practical examples.

The method for solving the extension of general of the split feasibility problem and fixed point problem of the cutter

This paper first introduces a new iterative method for weak and strong convergence theorem to demonstrate the estimation potential for a fixed point of the cutter and the finite general split feasibility problem. Consequently, the set of fixed points of a quasi-nonexpansive mapping and the finite general split feasibility problem, the constrained minimization problem, and the general constrained minimization problems are proved using our main results. Finally, we give two numerical examples to advocate our main results.

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

We propose a new variational model in Sobolev–Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev–Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

Feasible Newton methods for symmetric tensor Z-eigenvalue problems

Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a Karush–Kuhn–Tucker point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a symmetric tensor. Each method can generate a sequence of feasible points such that the sequence of function evaluations is decreasing. These methods can be regarded as extensions of the descent methods for unconstrained optimization problems. We pay particular attention to the Newton method. We show that under appropriate conditions, the Newton method is globally and quadratically convergent. Moreover, after finitely many iterations, the unit steplength will always be accepted. We also propose a nonlinear equations-based Newton method and establish its global and quadratic convergence. In the end, we do several numerical experiments to test the proposed Newton methods. The results show that both Newton methods are very efficient.

The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced (not necessarily minimal) objective function values. The superiorization methodology is based on interlacing the iterative steps of two separate and independent iterative processes by perturbing the iterates of one process according to the steps dictated by the other process. We include in our developed method two novel elements. The first one is the permission to restart the perturbations in the superiorized algorithm which results in a significant acceleration and increases the computational efficiency. The second element is the ability to independently superiorize subvectors. This caters to the needs of real-world applications, as demonstrated here for a problem in intensity-modulated radiation therapy treatment planning.

A dynamic mass transport method for Poisson-Nernst-Planck equations

A dynamic mass-transport method is proposed for approximately solving the Poisson–Nernst–Planck (PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally enforces solution positivity and the energy law as for the continuous PNP system. The fully discrete scheme is further formulated as a constrained minimization problem, shown to be solvable, and satisfy all three solution properties (mass conservation, positivity and energy dissipation) independent of time step size or the spatial mesh size. Numerical experiments are conducted to validate convergence of the computed solutions and verify the structure preserving property of the proposed scheme.

Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces

Abstract In this article, we present some new fixed point existence results for nonexpansive-type mappings in geodesic spaces. We also give a number of illustrative examples to settle our claims. We study the asymptotic behavior of Picard iterates generated by these class of mappings under different conditions. Finally, we approximate the solutions of the constrained minimization problem in the setting of Cartan, Alexandrov, and Toponogov (CAT(0)) spaces.

Approximating Solutions of Optimization Problems via Fixed Point Techniques in Geodesic Spaces

The principal objective of this paper is to find the solution to a constrained minimization problem and the zero of the monotone operator in geodesic spaces. The basic tool in our study is a nonexpansive mapping. Further, we employ the general Picard–Mann iterative method to approximate fixed points of nonexpansive mappings under various conditions. We obtain certain theorems concerning Δ and strong convergence.

Identification of Cavities and Inclusions in Linear Elasticity with a Phase-Field Approach

In this paper we deal with the inverse problem of determining cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement’s measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modelled as inclusions with a very small elasticity tensor.

Optimal Clipping of Structural Subgrid Stress Closures for Large-Eddy Simulation

Structural subgrid stress models for large-eddy simulation often allow for backscatter of energy from unresolved to resolved turbulent scales, but excessive model backscatter can eventually result in numerical instability. A commonly employed strategy to overcome this issue is to set predicted subgrid stresses to zero in regions of model backscatter. This clipping procedure improves the stability of structural models, however, at the cost of reduced correlation between the predicted subgrid stresses and the exact subgrid stresses. In this paper, we propose an alternative strategy that removes model backscatter from model predictions through the solution of a constrained minimization problem. This procedure, which we refer to as optimal clipping, results in a parameter-free mixed model, and it yields predicted subgrid stresses in higher correlation with the exact subgrid stresses as compared with those attained with the traditional clipping procedure. We perform a series of a priori and a posteriori tests to investigate the impact of applying the traditional and optimal clipping procedures to Clark’s gradient subgrid stress model, and we observe that optimal clipping leads to a significant improvement in model predictions as compared to the traditional clipping procedure.

Solution of the static contact problem with Coulomb friction between an elastic body and a rigid foundation

We consider a static contact problem between an elastic body and an absolutely rigid foundation with Signorini contact conditions and Coulomb friction law. The problem can be formulated as a quasi-variational inequality in which the normal stress is proportional with the friction coefficient to the friction force in the contact zone. In this case, the normal stress itself depends on the desired solution, and the existence of a solution to the problem reduces to the existence of a fixed point of a certain mapping. Consideration of Coulomb friction naturally leads to the non-differentiability of the objective functional in the auxiliary problem of the method of successive approximations and, thereby, to a significant complication of the algorithms for solving the constrained minimization problem. We propose a method of solving an auxiliary problem with given friction, which is based on the Uzawa algorithm and modified Lagrange functionals. The main advantage of the proposed method is that it allows to effectively solve auxiliary problems and to prove the theoretical convergence of the method of successive approximations to a fixed point. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.

ON THE COMPUTATION OF THE AVERAGE OF SPATIAL DISPLACEMENTS.

Many applications in biomechanics and medical imaging call for the analysis of the kinematic errors in a group of patients statistically using the average displacement and the standard deviations from the average. This paper studies the problem of computing the average displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It has been shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the average for each quaternion component separately and independently. While the resulting algorithm is simple, the change of the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacement. This kinematic measure is then used to formulate a constrained least squares minimization problem. It has been shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of given translation vectors and the latter can be obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. It is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.

Hessian Informed Mirror Descent

Inspired by the recent paper (L. Ying, Journal of Scientific Computing, 84, 1–14 (2020), we explore the relationship between the mirror descent and the variable metric method. When the metric in the mirror decent is induced by a convex function, whose Hessian is close to the Hessian of the objective function, this method enjoys both robustness from the mirror descent and superlinear convergence for Newton type methods. When applied to a linearly constrained minimization problem, we prove the global and local convergence, both in the continuous and discrete settings. As applications, we compute the Wasserstein gradient flows and Cahn-Hillard equation with degenerate mobility. When formulating these problems using a minimizing movement scheme with respect to a variable metric, our mirror descent algorithm offers a fast convergence speed for the underlying optimization problem while maintaining the total mass and bounds of the solution.

MEC enabled cooperative sensing and resource allocation for industrial IoT systems

In industrial Internet of Things systems, state estimation plays an important role in multisensor cooperative sensing. However, the state information received by remote control center experiences random delay, which inevitably affects the state estimation performance. Moreover, the computation and storage burden of remote control center is very huge, due to the large amount of state information from all sensors. To address this issue, we propose a layered network architecture and design the mobile edge computing (MEC) enabled cooperative sensing scheme. In particular, we first characterize the impact of random delay on the error of state estimation. Based on this, the cooperative sensing and resource allocation are optimized to minimize the state estimation error. The formulated constrained minimization problem is a mixed integer programming problem, which is effectively solved with problem decomposition based on the information content of delivered data packets. The improved marine predators algorithm (MPA) is designed to choose the best edge estimator for each sensor to pretreat the sensory information. Finally, the simulation results show the advantage and effectiveness of proposed scheme in terms of estimation accuracy.

Nonexistence and existence of solutions with prescribed norm for semilinear nonlocal elliptic equations

In this paper, using Pohozaev's identity and discussing minimizing sequences for the related constrained minimization problem, we study the nonexistence and existence of solutions with prescribed -norm for the following nonlocal equation: where , and .

Inversion using adaptive physics‐based neural network: Application to magnetotelluric inversion

ABSTRACTA new trend to solve geophysical problems aims to combine the advantages of deterministic inversion with neural network inversion. The neural networks applied to geophysical inversion have had limited success due to the need for extensive training of datasets and the lack of generalizability to out‐of‐sample scenarios. Deterministic regularized inversion often requires a good starting model to avoid possible local minima in highly nonlinear problems. We have developed a physics‐based neural network procedure that combines the advantages of deterministic and neural network inversions in a coupled inversion scheme. The new inversion algorithm is formulated as a constrained minimization problem with a composite objective function composed of data misfit, neural network training function and a coupling model objective function that links the two inversion schemes through a reference model. We investigate two strategies to update the reference model in the coupling model objective function using either a fully trained network or an adaptively trained network that keeps evolving and learning during the inversion. First, we propose a procedure to generate the training dataset required for the fully trained network to improve the network generalization. Second, we extend the analysis of the adaptively trained network inversion without preparing a training dataset, so it is suitable for most exploration scenarios. Our approach enables the network to learn only the relevant model distribution. Predictions from the network are used to constrain the deterministic inversion using a new predicted reference model. We demonstrate the convergence of our procedure in recovering high‐resolution resistivity models when applied to synthetic magnetotellurics data. We compare the recovered resistivity model from a deterministic inversion, a neural network inversion and physics‐based neural network inversions using fully and adaptively trained networks, respectively. The results indicate that the adaptive physics‐based neural network is superior as it does not require preparing a training dataset but can still recover reliable resistivity models.

An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.

A marker-less monocular vision point positioning method for industrial manual operation environments

Vision-assisted technologies in industrial manual operation such as augmented reality (AR) are increasingly popular. They require high positioning accuracy and robustness to operate normally. However, narrow spaces, moving hands, or tools may occlude or obscure local visual features of operation environments and thus affect the positioning accuracy and robustness of operating position. The resultant misguidance may even cause misoperation of operators. This paper proposes a marker-less monocular vision point positioning method for vision-assisted manual operation in industrial environments. The proposed method can accurately and robustly locate the target point of operation using constraint minimization method even if the target area has no corresponding visual features in the case of occlusion and improper illumination. The proposed method has three phases: intersection generation, intersection optimization, and target point solving. In the intersection generation phase, a certain number of intersections of epipolar lines are generated as candidate target points using fundamental matrices. Here, the solving constraint is converted from point-to-line to point-to-points. In the intersection optimization phase, the intersections are optimized to two different sets through the iterative linear fitting and geometric mean absolute error methods. Here, the solving constraint is further converted from point-to-points to point-to-point sets. In the target point solving phase, the target point is solved as a constrained minimization problem based on the distribution constraint of the two intersection sets. Here, the solving constraint is finally converted from point-to-point sets to point-to-point, and the unique optimal solution is obtained as the target point. The experimental results show that this method has a better accuracy and robustness than the traditional homography matrix method for the practical industrial operation scenes.