Nonlinear Cost Function Research Articles

DOA Estimation in Low SNR Environment through Coprime Antenna Arrays: An Innovative Approach by Applying Flower Pollination Algorithm

The design of the modern computing paradigm of heuristics is an innovative development for parameter estimation of direction of arrival (DOA) using sparse antenna arrays. In this study, the optimization strength of the flower pollination algorithm (FPA) is exploited for the DOA estimation in a low signal to noise ratio (SNR) regime by applying coprime sensor arrays (CSA). The enhanced degree of freedom (DOF) is achieved with FPA by investigating the global minima of highly nonlinear cost function with multiple local minimas. The sparse structure of CSA demonstrates that the DOF up to O(MN) is achieved by employing M+N CSA elements, where M and N are the numbers of antenna elements used to construct the CSA. Performance analysis is conducted for estimation accuracy, robustness against noise, robustness against snapshots, frequency distribution of root mean square error (RMSE), variability analysis of RMSE, cumulative distribution function (CDF) of RMSE over Monte Carlo runs and the comparative studies of particle swarm optimization (PSO). These reveal the worth of the proposed methodology for estimating DOA.

Information-Theoretic Registration with Explicit Reorientation of Diffusion-Weighted Images

We present an information-theoretic approach to the registration of images with directional information, especially for diffusion-weighted images (DWIs), with explicit optimization over the directional scale. We call it locally orderless registration with directions (LORDs). We focus on normalized mutual information as a robust information-theoretic similarity measure for DWI. The framework is an extension of the LOR-DWI density-based hierarchical scale-space model that varies and optimizes the integration, spatial, directional and intensity scales. As affine transformations are insufficient for inter-subject registration, we extend the model to nonrigid deformations. We illustrate that the proposed model deforms orientation distribution functions (ODFs) correctly and is capable of handling the classic complex challenges in DWI registrations, such as the registration of fiber crossings along with kissing, fanning, and interleaving fibers. Our experimental results clearly illustrate a novel promising regularizing effect, which comes from the nonlinear orientation-based cost function. We show the properties of the different image scales, and we show that including orientational information in our model makes the model better at retrieving deformations in contrast to standard scalar-based registration.

Modulation of control authority in adaptive haptic shared control paradigms

This paper presents an adaptive haptic shared control framework wherein a driver and an automation system are physically connected through a motorized steering wheel. An adaptive impedance controller for the automation system is introduced to enable the automation system to dynamically and smoothly exchange the control authority with the human partner. To determine an optimal modulation policy, a cost function is considered, and its terms are defined to minimize the performance error and reduce the disagreement between the human and automation system. We employed a nonlinear stochastic model predictive approach to solve the cost function subjected to probabilistic uncertainties in human’s biomechanics. The polynomial chaos expansions are employed to obtain a computationally tractable form of the cost function, and the continuation generalized minimum residual method is utilized to solve the tractable nonlinear cost function. To demonstrate the effectiveness of the proposed approach, we consider a scenario where the human and the automation system both detect an obstacle and negotiate on controlling the steering wheel so that the obstacle can be avoided safely. For this scenario, four interaction modes are defined based on the cooperation status (cooperative and uncooperative) and the control transfer’s desired direction (human to automation or automation to human). The numerical results demonstrate that when the human’s control command is sufficient for avoiding the obstacle, by modulating and adopting smaller values for the impedance controller, the disagreement between the human and automation systems can be reduced. On the other hand, when the human’s control command is insufficient, by modulating and adopting larger values for the impedance controller parameters, the automation system gains the control authority and ensures the safety of the obstacle avoidance task. Additionally, we performed a set of tests with processors in the loop (PIL) to show the proposed predictive controller can compute the optimal modulation policy in real-time. Using two low-cost micro-controllers, the PIL results show high computational speed and numerical accuracy for the proposed method.

Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function

In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.

Simultaneous compensation of systematic errors of a low-cost MEMS triaxial accelerometer and its temperature dependency without accurate laboratory equipment

PurposeLaboratory calibration methods are time-consuming and require accurate devices to find the error coefficients of the low-cost microelectromechanical system (MEMS) accelerometer. Besides, low-cost MEMS sensors highly depend on temperature because of their silicon property and the effect of temperature on error coefficients should also be considered for compensation. This paper aims to present a field calibration method in which the accelerometer is placed in different positions without any accurate equipment in a few minutes and its temperature is changed by a simple device like a hairdryer.Design/methodology/approachIn this paper, a non-linear cost function is defined based on this rule that the magnitude of the acceleration measured by the accelerometer in static mode is equal to the gravity plus error factors. Also, the dependency of error coefficients of the accelerometer is presented as a second-order polynomial in this cost function. By minimizing the cost function, the accelerometer error coefficients include bias, scale factor and non-orthogonality and their temperature dependency are obtained simultaneously.FindingsSimulation results in MATLAB and empirical results of a MPU6050 accelerometer verify the good performance of the proposed calibration method.Originality/valueFinding a fast and simple field calibration method to calibrate a low-cost MEMS accelerometer and compensate for the temperature dependency without using accurate laboratory equipment can help a wide range of industries that use advanced and expensive sensors or use expensive laboratory equipment to calibrate their sensors, to decrease their costs.

Resource allocation in rooted trees subject to sum constraints and nonlinear cost functions

We study resource allocation problems in rooted trees in which demand values are given in the leaves. Single-type resources (weights) are to be assigned in the tree nodes such that the total weight in the rooted path from each leaf to the root equals its demand. The goal is to minimize the total costs of the allocated resources. It is known that the linear cost case, i.e., when the cost of a resource is proportional to its weight, is solvable in linear time. In this paper we show that when costs are monotone nondecreasing functions, which reflect, e.g., (dis)economies of scale, the problem becomes intractable, and design for it a fully polynomial time approximation scheme by formulating it as a dynamic program and using the technique of K-approximation sets and functions.

A performance measure analysis for traffic networks with random data and general monotone cost functions

We consider a congested traffic network where users behave according to the Wardrop equilibrium principle, but the data are uncertain and only known through their probability distributions. Within this framework, we propose a stochastic equilibrium model to analyse the network performance, which allows for nonlinear cost functions. The effectiveness of our approach is shown through numerical experiments on medium-size networks.

Stochastic Dynamic Cutting Plane for Multistage Stochastic Convex Programs

We introduce Stochastic Dynamic Cutting Plane (StoDCuP), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors. Finally, numerical experiments are presented on nondifferentiable multistage stochastic programs where Inexact StoDCuP computes a good approximate policy quicker than StoDCuP while SDDP and the previous inexact variant of SDDP combined with Mosek library to solve subproblems were not able to solve the differentiable reformulation of the problem.

Complex Dynamics in a Mixed Duopoly Game Based on Relative Profit Maximization

In this paper, the complex dynamic behavior of a mixed duopoly game model is studied. Based on the principle of relative profit maximization and bounded rational expectation, the corresponding discrete dynamic systems are constructed in the case of nonlinear cost function. In theory, the conditions for the local stability of Nash equilibrium are given. In terms of numerical experiments, bifurcation diagrams are used to depict the effects of product differences, adjustment speed, and other parameters on the stability of Nash equilibrium.

Multiscale design of nonlinear materials using a Eulerian shape optimization scheme

AbstractMotivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale‐level design of particulate composites for an optimal macroscale‐level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS‐based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three‐parameter isotropic damage law is used to model microstructure‐level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress‐strain response.

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

For two matroids $$M_1$$ and $$M_2$$ with the same ground set V and two cost functions $$w_1$$ and $$w_2$$ on $$2^V$$ , we consider the problem of finding bases $$X_1$$ of $$M_1$$ and $$X_2$$ of $$M_2$$ minimizing $$w_1(X_1)+w_2(X_2)$$ subject to a certain cardinality constraint on their intersection $$X_1 \cap X_2$$ . For this problem, Lendl et al. (Matroid bases with cardinality constraints on the intersection, arXiv:1907.04741v2 , 2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $$|X_1 \cap X_2|\le k$$ or $$|X_1 \cap X_2|\ge k$$ ; and designed a new primal-dual algorithm for the case where the constraint is $$|X_1 \cap X_2|=k$$ . The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $$\mathrm {M}$$ -convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.

Sequencing situations and games with non-linear cost functions under optimal order consistency

This paper considers sequencing situations with non-linear cost functions under optimal order consistency. Specifically, we study sequencing situations with discounting cost functions and logarithmic cost functions of the completion time. In both settings, we show that the neighbor switching gains are non-negative and non-decreasing for every misplaced pair of players. We derive new conditions on the time-dependent neighbor switching gains in a sequencing situation under optimal order consistency to guarantee convexity of the associated sequencing game. Furthermore, we define two types of gain splitting rules for the class of sequencing situations under optimal order consistency. Each one of them is based on a procedure that specifies a path from the initial order to an optimal order, dividing the neighbor switching gains in every step among the two involved players. We prove that these allocations are stable under the same conditions that are required for convexity. These requirements are fulfilled for discounting and logarithmic sequencing situations, as well as in other settings, such as in sequencing situations with exponential cost functions.

Simplification of calibration of low-cost MEMS accelerometer and its temperature compensation without accurate laboratory equipment

A nonlinear cost function is defined for field calibration of the accelerometer, using the rule that the norm of the measured vector in a static state is equal to the magnitude of the gravity vector. To solve this cost function, various optimization methods like Newton and Levenberg–Marquardt have been presented in different references. However, these methods are complicated, time-consuming, and require an initial value. This study presents a method that simplifies the cost function and obtains the error coefficients, including bias, scale factor, and non-orthogonality using the linear least-squares method which is simpler and faster than other optimization methods and does not need initial values. Also, the output of the low-cost MEMS accelerometer depends on temperature due to its silicon property. Thus, by finding the dependency of the error coefficients on temperature, they can be compensated. This paper models dependency of error coefficients on temperature using cubic spline interpolation and minimizes the temperature effect. Simulation results of MATLAB and the proposed field calibration method and temperature compensation on the low-cost MPU6050 sensor show its good performance.

A Fast Covariance Union Algorithm for Inconsistent Sensor Data Fusion

We consider a challenging scenario in this research, where the sensors may receive spurious sensor data, potentially causing inconsistent state estimates. Covariance union (CU) is a fault-tolerant algorithm that can deal with inconsistent state estimation fusion. However, existing CU algorithms suffer from high computational costs due to optimizing nonlinear cost functions when generating fusion weights. To overcome this deficiency, an efficient CU algorithm named fast covariance union (FCU) is developed. We have proved that the fusion weight of FCU can be optimally generated by a closed-form algorithm without optimizing any nonlinear cost function, leading to better fusion efficiency. In addition, the FCU algorithm ensures the fused estimate be consistent as long as one of the estimates is consistent. Finally, the Monte Carlo simulation results show that the FCU algorithm has higher computational efficiency than the existing CU algorithms and handles the spurious sensor data fusion effectively.

A Novel Adaptive Filter for Cooperative Localization Under Time-Varying Delay and Non-Gaussian Noise

In this study, to cope with time-varying delay in acoustic communication under non-Gaussian noise caused by outliers, a novel robust delay filtering algorithm is proposed for cooperative localization of autonomous underwater vehicles (AUVs). First, the modified measurement equations of the nonlinear cooperative location system with time-varying delay are derived, and the delay caused by the information processing and propagation of the underwater acoustic modem is converted into measurement bias. Second, to improve the robustness of the system with outliers caused by abnormal measurement values of underwater acoustic communication and Doppler velocity log (DVL), the statistical similarity measure (SSM) is introduced to construct the cost function. This is accomplished by quantifying the similarity between the state vector and the predicted state vector, similarity between the delay measurement, and modified predicted delay measurement. Finally, the modified measurement noise variance is obtained by maximizing the cost function, and two algorithms for the approximate solution of the nonlinear cost function are presented. The proposed robust delayed algorithm alleviates the impacts of delayed measurement with outliers on localization accuracy. The effectiveness and potential of the proposed filter are verified by lake trials.

A consistently fast and accurate algorithm for estimating camera pose from point correspondences

This paper presents an accurate and simultaneously efficient algorithm for the Perspective-n-Point (PnP) problem that estimates the absolute pose of a fully-calibrated camera from given 3D-to-2D point correspondences. Previous works typically use the depth of each 3D point to formulate the PnP problem, which will bring extra variables. In contrast, the presented algorithm does not involve any depth factor. By introducing the Cayley–Gibbs–Rodriguez (CGR) parameterization, the modified formulation is first compressed to a nonlinear least-square cost function that only depends on three unknown rotation parameters and a known symmetric coefficient matrix. The Gröbner basis method is then adopted to find a set of solutions for the rotation parameters by solving a third-order polynomial system arising from the first-order optimality conditions of the cost function. Lastly, the rotation and translation are efficiently computed by back-substitution. Furthermore, a novel approach is developed for handling singularities of the CGR parameterization. It is improved by applying fixed pre-rotations, as opposed to randomly generated rotations in previous works, to 3D points. This improvement will facilitate the calculation of the coefficient matrix involved in a cost function when re-solving the PnP problem. Extensive experiments on both simulated and real data demonstrate that the presented algorithm can achieve the state-of-the-art accuracy with reduced computational requirements.

Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem

This paper investigates the solution for an inverse of a parametric nonlinear transportation problem, in which, for a certain values of the parameters, the cost of the unit transportation in the basic problem are adapted as little as possible so that the specific feasible alternative become an optimal solution. In addition, a solution stability set of these parameters was investigated to keep the new optimal solution (feasible one) is unchanged. The idea of this study based on using a tuning parameters λ∈Rm in the function of the objective and input parameters υ∈Rl in the set of constraint. The inverse parametric nonlinear cost transportation problem P(λ,υ), where the tuning parameters λ∈Rm in the objective function are tuned (adapted) as less as possible so that the specific feasible solution x∘ has been became the optimal ones for a certain values of υ∈Rl, then, a solution stability set of the parameters was investigated to keep the new optimal solution x∘ unchanged. The proposed method consists of three phases. Firstly, based on the optimality conditions, the parameter λ∈Rm are tuned as less as possible so that the initial feasible solution x∘ has been became new optimal solution. Secondly, using input parameters υ∈Rl resulting problem is reformulated in parametric form P(υ). Finally, based on the stability notions, the availability domain of the input parameters was detected to keep its optimal solution unchanged. Finally, to clarify the effectiveness of the proposed algorithm not only for the inverse transportation problems but also, for the nonlinear programming problems; numerical examples treating the inverse nonlinear programming problem and the inverse transportation problem of minimizing the nonlinear cost functions are presented.

Generalized Lagrange–Jacobi–Gauss–Radau collocation method for solving a nonlinear optimal control problem with the classical diffusion equation

In this paper, a well-known nonlinear optimal control problem (OCP) arisen in a variety of biological, chemical, and physical applications is considered. The quadratic form of the nonlinear cost function is endowed with the state and control functions. In this problem, the dynamic constraint of the system is given by a classical diffusion equation. This article is concerned with a generalization of Lagrange functions. Besides, a generalized Lagrange–Jacobi–Gauss–Radau (GLJGR) collocation method is introduced and applied to solve the aforementioned OCP. Based on initial and boundary conditions, the time and space variables, t and x, are clustered with Jacobi–Gauss–Radau points. Then, to solve the OCP, Lagrange multipliers are used and the optimal control problem is reduced to a parameter optimization problem. Numerical results demonstrate its accuracy, efficiency, and versatility of the presented method.

A Phaseless Inverse Source Method (PISM) Based on Near-Field Scanning for Radiation Diagnosis and Prediction of PCBs

In this article, we present an efficient phaseless inverse source method (PISM) to reconstruct equivalent sources on a printed circuit board (PCB) using magnitude-only single-plane near-field scanning. The equivalent source distribution reconstructed by the conventional source reconstruction method requires accurate near-field magnitude and phase information, but phase measurement is difficult in many cases due to inaccuracy and complexity. In order to describe the inverse problem between phaseless fields and radiation sources, a nonlinear cost function based on surface integral equations is established. The cost function of the PISM is rapidly minimized by an iterative optimization and regularization technique for nonlinear ill-posed systems. The proposed method gives the characterization of PCBs both for diagnostic tasks of radiation sources and the near-field radiation behavior prediction when the phase information is missing. It also shows excellent robustness in tackling with various near-field data and noises. Both numerical examples and laboratory experiments are given to verify the effectiveness of the proposed method.

Approximate Calculation of Equilibria in the Nonlinear Stackelberg Oligopoly Model: A Linearization Based Approach

The game-theoretic problem of choosing optimal strategies for oligopoly market agents with linear demand functions and nonlinear cost functions is considered. Necessary conditions for the existence of a solution of a system of nonlinear equations with power functions are established. The system of equations for the optimal responses of agents is linearized by expanding the power functions in Taylor series. As a result, the linearized system depends on the vector of linearization parameters, and the calculation of game equilibria is reduced finding fixed points of nonlinear mappings. The deviations of the approximate equilibrium from the exact solution are investigated. Analytical formulas for calculating equilibria in the game of oligopolists under an arbitrary level of Stackelberg leadership are derived. Analysis of duopoly and tripoly demonstrates that the game equilibrium is determined by two factors as follows. First, the concavity of the agent’s cost function (the positive scale effect) leads to an increase in his payoff compared to the agents with convex cost functions (the negative scale effect). Second, the agent’s payoff increases if he is a Stackelberg leader; however, the advantage of his environment by the type of cost function reduces the effect of the second factor.