Nonlinear Minimization Problem Research Articles

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We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the W-1,p′(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p'}(\\Omega )$$\\end{document} norm with exponents p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p> 2$$\\end{document}. The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents p≫2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\gg 2$$\\end{document}. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.

Two new limited-memory preconditioned conjugate gradient algorithms for nonlinear optimization problems

The conjugate gradient (CG) techniques are a class of unconstrained optimization algorithms with strong local and global convergence qualities and minimal memory needs. While the quasi-Newton methods are reliable and efficient on a wide range of problems and these methods are converge faster than the conjugate gradient methods and require fewer function evaluations, however, they are request substantially more storage, and if the problem is ill-conditioned, they may require several iterations. There is another class, termed preconditioned conjugate gradient method, it is a technique that combines two methods conjugate gradient with quasi-Newton. In this work, we proposed a new two limited memory preconditioned conjugate gradient methods (New1 and New2), to solve nonlinear unconstrained minimization problems, by using new modified symmetric rank one (NMSR1) and new modified Davidon, Fletcher, Powell (NMDFP), and also using projected vectors. We proved that these modifications fulfill some conditions. Also, the descent condition of the new technique has been proved. The numerical results showed the efficiency of the proposed new algorithms compared with some standard nonlinear, unconstrained problems.

Calculating Global Minimum Points to Binary Polynomial Optimization Problem: Optimizing the Optimal PMU Localization Problem as a Case-Study

State estimation (SE) is an algorithmic function of an energy management system (EMS). SE provides an actual-time monitoring and control of modern electrical power grids. State Estimation can be worked with sufficiency using Phasor Measurement Units optimally placed within a power grid. This paper concerns the implementation of proper algorithms embedded in optimization solvers to the optimal PMU localization problem solving globally. The optimization model is formulated as a 0 - 1 nonlinear minimization problem. The problem is transformed to a polyhedron using linearization methods and B&B tree. In this model, we use a linear cost function under polynomial constraints and binary restrictions on the design variables in a symbolic format. This mathematical model is programmed in the YALMIP environment which is fully compatible with MATLAB. The 0 - 1 Nonlinear Programming (NLP) model is suitable for getting concisely global optimal solutions. The optimal solution is given by a wrapped optimization engine including a local optimizer routine performing together with a mixed-Integer-Linear Programming routine. The solution is achieved within a zero-gap precisely encountered during the iterative process. This tolerance criterion is a necessity for a successful implementation of the B&B tree because it ensures global optimality with an acceptance relative gap. The minimization model is implemented in a YALMIP code fully compatible with MATLAB in two stages. Initially, an objective function with one term is minimized to discover a number of sensors for wide-area monitoring, control and state estimator applications. Then, an extra product is considered in the objective to suffice maximum reliability for observing the network buses. The numerical minimization models are applied to standard power networks in the direction to be solved globally.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of 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, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.

User grouping and resource management in efficient NOMA‐OMA transmission scheme for infotainment message dissemination in VANETs

AbstractThe data distribution for infotainment services at vehicular ad‐hoc networks (VANETs) has become a research hotspot in recent years. Non‐orthogonal multiple access (NOMA) is accepted as an enabling technology for 5G and beyond due to its improved spectral efficiency. In this work, we consider a VANET communication scenario where users have common data demands such as high‐density maps, local entertainment/shopping options or other popular media content. For such a scenario, we propose an efficient clustering scheme together with a hybrid NOMA‐OMA scheme for improved spectral and power efficiency. For this setup, we also develop an optimal bandwidth allocation strategy to improve the power efficiency further. We formulate a one dimensional nonlinear power minimization problem such that the bandwidth allocation for the system can be optimized. The complexity and optimality analysis of optimization algorithms used for power minimization are performed. The improved performance of the proposed scheme in terms of power efficiency, spectral efficiency and latency performance is showcased in comparison to conventional NOMA and OMA approaches.

Non-convex activated zeroing neural network model for solving time-varying nonlinear minimization problems with finite-time convergence

Zeroing neural network (ZNN) model is a powerful tool for solving time-varying nonlinear minimization problems. This study presents some limitations of existing ZNN models, mainly related to convex and unsaturation constraints. To this end, a projection method is introduced for constructing non-convex activation functions, whereby a finite-time convergent non-convex zeroing neural network (FT-NCZNN) model is proposed based on this method. The model has a faster convergence rate than the original zeroing neural network (OZNN) model. Rigorous theoretical analyses are provided to verify the convergence of the model as well as the upper boundary on convergence time. Subsequently, through a series of simulations, the superior performance of the FT-NCZNN model under different noise conditions is demonstrated. Finally, an engineering application on motion generation is introduced for a double-linked manipulator to further illustrate the validity and feasibility of the FT-NCZNN model.

Event-triggered nonfragile predictive control for networked control system with redundant channels

This paper is concerned with the nonfragile predictive control problem for the networked control system (NCS) with input delay and network-induced time delays (NITDs). First, the transmission strategy governed by redundant channels is introduced to reduce the occurrence of data packet dropout between the sensor and the observer. Moreover, the event-triggered mechanism (ETM) is employed to save the network resources, where the maximum triggered interval is given. Subsequently, a nonfragile networked predictive control scheme (NPCS) is designed to address the input delay and the NITDs. Accordingly, the sufficient criterion ensuring mean-square exponential stability (MSES) is derived by employing the Lyapunov–Krasovskii stability theorem. Both the observer gain matrix (OGM) and the controller gain matrix (CGM) are gained through solving the nonlinear minimization problem with convex constraints. Finally, a simulation example is utilized to demonstrate the usefulness of the proposed main results.

Estimating and Calibrating DER Model Parameters Using Levenberg–Marquardt Algorithm in Renewable Rich Power Grid

The proliferation of inverter-based distributed energy resources (IBDERs) has increased the number of control variables and dynamic interactions, leading to new grid control challenges. For stability analysis and designing appropriate protection controls, it is important that IBDER models are accurate. This paper focuses on the accurate estimation and parameter calibration of DER_A, a recently proposed aggregated IBDER model. In particular, we focus on the parameters of the reactive power–voltage regulation module. We formulate the problem of parameter tuning as a non-linear least square minimization problem and solve it using the Levenberg–Marquardt (LM) method. The LM method is primarily chosen due to its flexibility in adaptively selecting between the steepest descent and Gauss–Newton methods through a damping parameter. The LM approach is used to minimize the error between the actual measurements and the estimated response of the model. Further, the computational challenges posed by the numerical calculation of the Jacobian are tackled using a quasi-Newton root-finding approach. The proposed method is validated on a real feeder model in the northeastern part of the United States. The feeder is modeled in OpenDSS and the measurements thus obtained are fed to the DER_A model for calibration. The simulation results indicate that our approach is able to successfully calibrate the relevant model parameters quickly and with high accuracy, with a total sum of square error of 3.57×10−7.

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Identification of Linear Time-Invariant Systems: A Least Squares of Orthogonal Distances Approach

This work describes the parameter identification of servo systems using the least squares of orthogonal distances method. The parameter identification problem was reconsidered as data fitting to a plane, which in turn corresponds to a nonlinear minimization problem. Three models of a servo system, having one, two, and three parameters, were experimentally identified using both the classic least squares and the least squares of orthogonal distances. The models with two and three parameters were identified through numerical routines. The servo system model with a single parameter only considered the input gain. In this particular case, the analytical conditions for finding the critical points and for determining the existence of a minimum were presented, and the estimate of the input gain was obtained by solving a simple quadratic equation whose coefficients depended on measured data. The results showed that as opposed to the least squares method, the least squares of orthogonal distances method experimentally produced consistent estimates without regard for the classic persistency-of-excitation condition. Moreover, the parameter estimates of the least squares of orthogonal distances method produced the best tracking performance when they were used to compute a trajectory-tracking controller.

Approximating the span of principal components via iterative least-squares

In the course of the last century, Principal Component Analysis (PCA) have become one of the pillars of modern scientific methods. Although PCA is normally addressed as a statistical tool aiming at finding orthogonal directions on which the variance is maximized, its first introduction by Pearson at 1901 was done through defining a non-linear least-squares minimization problem of fitting a plane to scattered data points. Thus, it seems natural that PCA and linear least-squares regression are somewhat related, as they both aim at fitting planes to data points. In this paper, we present a connection between the two approaches. Specifically, we present an iterated linear least-squares approach, yielding a sequence of subspaces, which converges to the space spanned by the leading principal components (i.e., principal space).

A New Spectral Compression Method based on the Minimization of the Color Difference and Root Mean Square Error

This paper describes two new interim connection spaces (ICSs), P-2OC and P-3OC, for spectral image compression and reconstruction. For this type of ICS, the weighting table or compression matrix H is modelled as a variable. The associated reconstruction matrix N is chosen as the Weiner estimation matrix. The objective function f(H) is the combination of the averages of the CIEDE2000 color difference (ΔE00) and the root mean square error (RMSE) between the original and reconstructed reflectance values. Hence, the compression matrix H is determined by solving the nonlinear minimization problem based on the Munsell training dataset. The proposed ICSs, P-2OC and P-3OC, were tested and compared, respectively, with ICS-2SI and ICS-3SI developed by Zhang et al. (JOSAA, Vol. 29, pp. 1027–1034) in 2012 using the NCS dataset, and 2 spectral images. Performance tests showed that the proposed P-2OC and P-3OC ICSs are better than the ICS-2SI and ICS-3SI ICSs, respectively, in terms of RMSE, goodness of fit coefficient (GFC), and ΔE00 under CIE illuminants D65, A, C and F11. Therefore, it is expected that the P-2OC and P-3OC ICSs can find applications in spectral image compression and cross-media reproduction.

Determination of time-dependent coefficient in time fractional heat equation

The aim of this work is to determine the time-dependent heat coefficient in a type of inverse problem for one dimensional time-fractional heat equations defined by the Caputo operator for (0<α<1) by applying a method based on the finite difference scheme and Tikhonov’s regularization. First, a stable implicit finite difference scheme is applied to find a numerical solution to the (forward) direct problem. While the inverse problem was reformulated as a nonlinear least-square minimization problem with a simple physical bound on the unknown coefficient and solved efficiently by MATLAB routine lsqnonlin from the optimization toolbox. But the latter problem will remain ill-posed because the presence of any error in the input data will lead to a large error in the output data. So, Tikhonov’s technique is applied to obtain stable results. In the end, two numerical test examples show that the proposed method is stable, accurate, and works well with different noise levels

Fast yet predictable braking manoeuvers for real-time robot control

This paper proposes a framework for generating fast, smooth and predictable braking manoeuvers for a controlled robot. The proposed framework integrates two approaches to obtain feasible modal limits for designing braking trajectories. The first approach is realtime capable but conservative considering the usage of the available feasible actuator control region, resulting in longer braking times. In contrast, the second approach maximizes the used braking control inputs at the cost of requiring more time to evaluate larger, feasible modal limits via optimization. Both approaches allow for predicting the robot's stopping trajectory online. In addition, we also formulated and solved a constrained, nonlinear final-time minimization problem to find optimal torque inputs. The optimal solutions were used as a benchmark to evaluate the performance of the proposed predictable braking framework. A comparative study was compiled in simulation versus a classical optimal controller on a 7-DoF robot arm with only three moving joints. The results verified the effectiveness of our proposed framework and its integrated approaches in achieving fast robot braking manoeuvers with accurate online predictions of the stopping trajectories and distances under various braking settings.

Impact of inflation and discount on optimum ordering and preservation technology expenses scheme of perishable inventory model

This study coins an inventory model for deteriorating articles. The work devises an optimal approach of expenditure on preservation technology to attenuate the degradation process. The demand in the inventory model is both price and stock level sensitive. Demand in the form of price dependence is again a convex combination of linear and nonlinear prices. The investigation proliferates state-of-the-art strategy by presenting the concept of discount if payment is done in advance. The suggested model scrutinises the impact of inflation on optimality plan as well. The object of this exercise is to find optimal cycle length and preservation technology expenses to minimise the total cost per unit time. Stock is backlogged partially. Solution methodology of nonlinear minimisation problem embodies derivation of theorems and propositions. The final results are validated with the help of three examples, convexity of graphical schemes and sensitivity analysis. The paper also outlines the managerial point of view.

A Nullspace-Based Predictive Control Allocation for the Control of a Quadcopter Manipulating an Object Attached to the Ground

In this paper, we propose a novel optimal predictive control allocation for the control of a quadcopter manipulating an object attached to the ground. This controller sprucely takes advantage of the nullspace of the quadcopter attitude mapped to the effective force to compute the optimal sequence of controls and attitude references. In particular, at each time instant, the controller solves a nonlinear minimization problem, where a specialized term is added to the cost function to penalize the deviation of the desired attitude from its nullspace, which is the attitude subspace for which any thrust generates zero effective force on the system. The weight associated to this specialized term uses a Gaussian bell-curve function to make the tuning of the attitude-reference generation more flexible and enhance the performance of the controller. To demonstrate the effectiveness of the proposed controller, numerical simulations of two different cases are provided.

Equiareal Parameterization of Triangular Bézier Surfaces

Parameterization is the key property of a parametric surface and significantly affects many kinds of applications. To improve the quality of parameterization, equiareal parameterization minimizes the equiareal energy, which is presented as a measure to describe the uniformity of iso-parametric curves. With the help of the binary Möbius transformation, the equiareal parameterization is extended to the triangular Bézier surface on the triangular domain for the first time. The solution of the corresponding nonlinear minimization problem can be equivalently converted into solving a system of bivariate polynomial equations with an order of three. All the exact solutions of the equations can be obtained, and one of them is chosen as the global optimal solution of the minimization problem. Particularly, the coefficients in the system of equations can be explicitly formulated from the control points. Equiareal parameterization keeps the degree, control points, and shape of the triangular Bézier surface unchanged. It improves the distribution of iso-parametric curves only. The iso-parametric curves from the new expression are more uniform than the original one, which is displayed by numerical examples.

A New Modified Conjugate Gradient for Nonlinear Minimization Problems

The conjugate gradient is a highly effective technique to solve the unconstrained nonlinear minimization problems and it is one of the most well-known methods. It has a lot of applications. For large-scale, unconstrained minimization problems, conjugate gradient techniques are widely applied. In this paper, we will suggest a new parameter of conjugate gradient to solve the nonlinear unconstrained minimization problems, based on parameter of Dai and Liao. We will study the property of the descent ,the property of the sufficient descent and property of the global convergence a new method. We introduce some numerical data to prove the efficacy of the our method.

An accelerated Sobolev gradient method for unconstrained optimization problems based on variable inner products

We propose an accelerated version of the classical gradient method for unconstrained optimization problems defined on a Sobolev space H with Hilbert structure. Motivated by empirical results available in the literature demonstrating improved convergence of Sobolev gradient methods with suitably chosen weights, we develop a rigorous and constructive approach allowing one to identify the optimal gradient gk=g(λk) among gradients g(λ) parameterized by a weight function λ belonging to a finite-dimensional space of weights, which defines the inner product 〈⋅,⋅〉λ in the space H. At the kth iteration of the method, where an approximation uk∈H to the minimizer is given, an optimal weight λk is found as a solution of a nonlinear minimization problem in the space of weights R+N. The weight λk defines the optimal gradient gk equal to the projection of the Newton step hk onto a certain finite-dimensional subspace Tk, in the sense that Pk(σgk−hk)=0, where Pk is the projection operator onto Tk and σ a fixed step size. This property ensures that thus constructed gradient method attains a quadratic convergence in a certain sense for error components in Tk, in addition to the linear convergence typical of the classical gradient method. A numerical implementation of the new approach is also proposed. Computational results based on two model problems confirm the theoretically established convergence properties, demonstrating that the proposed approach outperforms the standard steepest-descent method based on Sobolev gradients and compares favorably to the Newton–Krylov method.

An inverse problem of fourth-order partial differential equation with nonlocal integral condition

In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and finding stable and accurate solutions, we employ the Quintic B-spline (QBS) collocation and Tikhonov regularization methods, respectively. The following nonlinear minimization problem is solved using MATLAB. The collected findings demonstrate that accurate and stable solutions can be found.