Fixed-point Constraint Research Articles

Modified inertial viscosity extrapolation method for solving quasi-monotone variational inequality and fixed point problems in real Hilbert spaces

In this paper, we introduce and study a viscous-type extrapolation algorithm for finding a solution of the variational inequality problem and a fixed point constraint of quasi-nonexpansive mappings under the scope of real Hilbert spaces when the underlying cost operator is quasi-monotone. The method involves inertial viscosity approximation and a constructed self-adjustable step size condition that depends solely on the information of the previous step. We establish a strong convergence result of the proposed method under certain mild conditions on the algorithm parameters. Finally, to demonstrate the gain of our method, some numerical examples are presented in comparison with some related methods in literature.

A novel method for hierarchical variational inequality with split common fixed point constraint

A novel method for hierarchical variational inequality with split common fixed point constraint

Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application

The class of quasimonotone mappings are known to be more general and applicable than the classes of pseudomonotone and monotone mappings. However, only very few results can be found in the literature on quasimonotone variational inequality problems and most of these results are on weak convergent algorithms. In this paper, we study the quasimonotone variational inequality problem (VIP) with constraint of fixed point problem (FPP) of quasi-pseudocontractive mappings. We introduce a new inertial Tseng’s extragradient method with self-adaptive step size for approximating the minimum-norm solutions of the aforementioned problem in the framework of Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a common (minimum-norm) solution of the quasimonotone VIP and FPP of quasi-pseudocontractive mappings without the knowledge of the Lipschitz constant of the cost operator. We provide several numerical experiments for the proposed method in comparison with existing methods in the literature. Finally, we applied our result to image restoration problem. Our result improves, extends and generalizes several of the recently announced results in this direction.

Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

This paper considers an alternated inertial-type extrapolation algorithm for solving bilevel pseudomonotone variational inequality problem in the framework of real Hilbert spaces with split variational inequality and fixed-point constraints of demimetric mapping. The algorithm which involves alternated inertial uses self-adjustment stepsize condition that depends solely on the information from previous iterative step. The bilevel problem considered in the work consists of upper-level problem with an underlying operator which is pseudomonotone, while the lower problem is associated with strongly monotone mapping and the fixed-point constraint of demimetric mapping. Our algorithm is anchored on modified projection and contraction techniques, fused with alternated inertial and relaxation. Under some suitable conditions on the algorithm control parameters, we obtain a strong convergence result of the proposed method without the prior knowledge of the operator norm or the coefficients of the underlying operators within the scope of real Hilbert spaces. Finally, some numerical examples are presented to illustrate the gain of our method in comparison to some related algorithms in the literature.

Calibration for light field cameras based on fixed point constraint of spatial plane homography.

Light field cameras can record the spatial and angular information of light rays at the same time, which facilitate many vision tasks such as 3D reconstruction and digital refocusing. In this paper, we deduce a projection model from a 3D space point to each sub-aperture image according to the transmission of light rays in the light field camera, and the projection model of traditional cameras can be considered as a special case of it. We propose a camera calibration method using this model. Firstly, the homography from a space plane to each sub-aperture image is estimated, which implies the imaging parameters of the standard light field camera, and then the principal line on each sub-aperture image is derived from the constraint of the fixed point of the plane homography. The principal point of each sub-aperture image is robustly determined from the principal lines by using RANSAC. The other parameters can be determined from the homography constrains on them. Compared with the state of art methods, the proposed method can calibrate the light field images with a varying focal length and has higher accuracy. Moreover, it provides a unified framework for the calibration of standard light field cameras and traditional cameras. Experimental results validate the proposed method.

On a system of monotone variational inclusion problems with fixed-point constraint

In this paper, we study the problem of finding the solution of the system of monotone variational inclusion problems recently introduced by Chang et al. (Optimization 70(12):2511–2525, 2020) with the constraint of a fixed-point set of quasipseudocontractive mappings. We propose a new iterative method that employs an inertial technique with self-adaptive step size for approximating the solution of the problem in Hilbert spaces and prove a strong-convergence result for the proposed method under more relaxed conditions. Moreover, we apply our results to study related optimization problems. Finally, we present some numerical experiments to demonstrate the performance of our proposed method, compare it with a related method as well as experiment on the dependency of the key parameters on the performance of our method.

Tensor completion via bilevel minimization with fixed-point constraint to estimate missing elements in noisy data

In this work, we consider the tensor completion problem of an incomplete and noisy observation. We introduce a novel completion model using bilevel minimization. Therefore, bilevel model-based denoising for the tensor completion problem is proposed. The denoising and completion tasks are fully separated. The upper-level directly addresses the completion problem with the truncated nuclear norm, while the lower-level uses the sparsity prior which is characterized by the l1-norm for the denoising task. Furthermore, we propose a simple strategy to solve our bilevel optimization problem. It formulates the lower-level as a fixed-point equation and then applies a simple but efficient iterative algorithm to get the reconstructed tensor. Numerically, the superiority of the proposal is reported via several experiments conducted on real data with an extremely small subset of observed entries.

Critical Points for Variable Length Elastica With a Fixed Point Constraint Under Displacement Control

Abstract This paper studies a variable length elastica with a fixed point constraint by an assembly method that regards the whole elastica as an assembly of two components, i.e., pinned-clamped elasticas. The pinned-clamped elastica is obtained based on the post-buckled deformed shape with one internal inflection point. Thus, multiple coexisting solutions can be located accurately, which reveals three distinct equilibrium paths for the complete load–displacement curves. Under displacement control, two critical points on two equilibrium paths are found at saddle-node bifurcations. Interestingly, a new critical point is located at the boundary point of one equilibrium path, where the shapes of two pinned-clamped elasticas are two different post-buckled deformed shapes. Changing the location of the fixed point constraint allows the position of the boundary point to be easily manipulated, and the associated snap-through phenomenon can occur on different equilibrium paths. This flexible generation of the snap-through phenomenon is useful for designing engineering systems that require controllable snap-through.

Home Position and Tool Coordinates Calibration for 6-DOF Robot Based on Fixed-point Constraint

In the application process of industrial robots, it is necessary to ensure that the home position is correct when the robot changes the end tool, so that the coordinate values of tool can be calculated more accurately and the accuracy of the robot’s trajectory can be satisfied. A comprehensive calibration method of tool coordinates and home position of serial industrial robots based on fixed-point constraints is presented in this paper. Based on Denavit-Hartenberg (D-H) kinematics theory, the error models of tool coordinates and home position errors of 6-DOF serial industrial robots are established. By constraining the tool center point (TCP) of the robot with fixed points, the joint data of calibration points with different postures at the same position are collected, and the error value of the controlled object is identified by the least square algorithm. The calibration experiment results are basically consistent with the factory calibration parameters, which proves the feasibility of the calibration algorithm.

Research on the floating performance of a novel large caliber machine gun based on the floating principle with complicated boundary conditions

Abstract In this paper, a novel large caliber machine gun was taken as the research object to analyze the floating technique based on the principle of fixed-point constraint and secondary counter-recoil. A rigid-flexible coupling multi-body dynamic model of the large caliber machine gun with muzzle brake based on floating principle was established, in which the influence of soil and human body was taken into account. The dynamic simulation was conducted and then the results were compared with the corresponding experimental data The dynamic characteristics of the machine gun with or without floating technique were analyzed to indicate the influence of floating technique upon the performance of the gun. Furthermore, the rigid-flexible coupling dynamic models with five different firing angles was constructed to study the influence caused by the angles. The results indicated that the floating mechanism could reduce the recoil effectively and improve the operational performance of this novel large caliber machine gun.

Discrete intermodal freight transportation network design with route choice behavior of intermodal operators

We consider a discrete intermodal network design problem for freight transportation, in which the network planner needs to determine whether or not to build up or expand a link to minimize the total operating cost of carriers and hub operators under a general route choice model of intermodal operators. We formulate the problem as a mixed-integer nonlinear and non-convex program that involves congestion effects, piecewise linear cost functions, and a fixed-point constraint. We develop a series of relaxed and equivalent models to reduce the hardness of the problem and provide theoretical results to show the equivalences. We present two solution methods to solve the problem with one returning heuristic solutions and the other generating a globally optimal solution. We offer two numerical experiments to test the two solution algorithms and also shed light on their performance comparisons.

The fixed-point theory of strictly causal functions

We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modeling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.

Real-time thermal error compensation method for robotic visual inspection system

The thermally induced error is a critical element in the total errors for the robotic visual inspection system during its long-term operation. This paper investigates the thermal behavior of the robotic visual inspection system and proposes a real-time thermal error compensation method. Based on the fixed-point constraint, an error calibration model is derived by associating the thermally induced parameter errors with the deviations in the measured coordinates of the fixed reference point during the robot warm up and cool down. A joint-by-joint test is performed to investigate the link parameters that change significantly with respect to temperature variations. Standard spheres are adopted as the calibration targets, and optimum projected angles for sphere center measurement by visual sensor are obtained based on theoretical analysis and experimental data. Contrary to most other works, the method needs no exact knowledge of the temperature sensors and is well suited for online dynamic thermal error compensation for the robotic visual inspection system. Verification experiment is carried out on a car-body assembly line, and results show that the max/mean residual error for the tested points has been reduced from 0.441/0.195 to 0.136/0.078 mm with the significant parameters calibrated.

Sequential Estimation of Structural Models With a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze a local condition under which the nested pseudo likelihood (NPL) algorithm converges to a consistent estimator, and derive its convergence rate. We find that the NPL algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property. To address the issue of divergence, we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the NPL algorithm does not.

Sequential Estimation of Structural Models with a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze the conditions under which the nested pseudo-likelihood (NPL) algorithm achieves convergence and derive its convergence rate. We find that the NPL algorithm may not necessarily converge when the fixed point mapping does not have a local contraction property. To address the issue of non-convergence, we propose alternative sequential estimation procedures that can achieve convergence even when the NPL algorithm does not. Upon convergence, some of our proposed estimation algorithms produce more efficient estimators than the NPL estimator.

Existence Theorems for Bilevel Problem with Applications to Mathematical Program with Equilibrium Constraint and Semi-Infinite Problem

In this paper, we establish existence theorems for bilevel problems with fixed-point constraints and bilevel problems without fixed-point constraint. The aim of this paper is to investigate under which conditions the existence of a feasible point of a bilevel problem can be assumed in advance and under which conditions there exist minimizers for this type of problems. From this, we establish existence theorems for mathematical programs with equilibrium constraints and semi-infinite problems.

Detecting Unstable Fixed Points Using Kalman Filters With Constraints

We introduce a novel application of the extended Kalman filter for detecting unstable periodic orbits of dynamical systems from experimental time series. The unknown dynamics are approximated using local polynomial functions, and the fixed-point constraint conditions are regarded as additional observation functions in the filter framework. Candidate fixed-point values are considered as parameters to be estimated. We demonstrate the application of this method to test systems and use surrogate time series data to test the statistical significance of the results

Effects of gravitational smearing on predictions of supergravity grand unification.

Limits on gravitational smearing from the dominant dimension 5 operator (with strength characterised by $cM/2M_{Planck}$) are obtained using the LEP data in supersymmetric SU(5) grand unification. Effects of c on the quasi infrared fixed point solutio ns of $m_b/m_{\tau}$ unification are also analysed. It is found that $c>0$ softens and $c<0$ stiffens the fixed point constraint. The effect of c on the upper bound of the Higgs triplet mass is analysed and the existence of a scaling induced by c is discussed. Tests for the existence of gravitational smearing are discussed.