System Of Nonlinear Inequalities Research Articles

A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities

In this study, the system of nonlinear inequalities (SNI) problem is investigated. First, a SNI is reformulated as a system of nonsmooth and nonlinear equations (SNNE). Second, a new smoothing technique for the "$\max$" function is proposed and the smoothing technique is employed for each element of the SNNE. Then, a new smoothing algorithm is developed in order to solve SNNE by combining the smoothing technique with the iterative method. The new algorithm is applied to some numerical examples to show the efficiency of our algorithm.

Optimal Design of a Parallel Manipulator for Aliquoting of Biomaterials Considering Workspace and Singularity Zones

This article presents the concept of a robotic system for aliquoting of biomaterial, consisting of a serial manipulator in combination with a parallel Delta-like robot. The paper describes a mathematical formulation for approximating the geometric constraints of the parallel robot as a set of solutions to a system of nonlinear inequalities. The analysis of the workspace is carried out, taking into account singularity zones, using a method based on the analysis of the Jacobian matrix of the mechanism and the interference of links. An optimal design procedure is proposed for the dimensional synthesis based on a criterion for maximizing the volume of the workspace, taking into account the ambiguity of the solution of the inverse kinematics. Simulation results are reported and discussed to propose a suitable design solution.

R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

Determination of Workspaces and Intersections of Robot Links in a Multi-Robotic System for Trajectory Planning

One of the problems in the development of multi-robotic systems is the safe navigation of a group of robots. To solve it, the restrictions imposed by the structural elements of its agents are determined. The article presents a multi-robotic system consisting of parallel and serial robots installed on mobile platforms. The parallel robot is made based on a tripod with the ability to rotate the robot’s base relative to the horizontal axis. The analysis of its working and technological area is carried out, taking into account singularity zones. The developed algorithms for determining the workspaces are based on deterministic methods for approximating the set of solutions to systems of nonlinear inequalities. In this case, restrictions in spaces of different coordinates are presented in the form of n-dimensional boxes. Approaches to solving two problems are proposed to determine the possible intersection of links for the collaborative performance of tasks by a multi-robotic system. The first task is to determine the intersection of the links for the given positions and the relative position of the manipulators. The second is in determining the minimum distance between the technological areas of manipulators, which consist of the workspace and all possible positions of the intermediate links.

Numerical Method for Determining the Operating Area of a Robot with Relative Manipulation Mechanisms

In this paper, we consider the application of optimization algorithms to solve the problem of determining the operating area of a robot with mechanisms of relative manipulation based on a plane 3-RPR mechanism and a tripod mechanism. A method for approximating the set of solutions to a system of nonlinear inequalities describing constraints on the geometric parameters of the robot is considered. Algorithms for constructing the operating areas of individual mechanisms were synthesized. The orientation effect of the output link on the size and shape of the operating area of each mechanism is analyzed. The operating area of the robot is obtained taking into account the constraints on the geometric parameters of the workpiece being in a moving coordinate system located in the center of the flat 3-RPR mechanism, as well as taking into account design constraints, intersections of links, and special positions.

A novel approach for coverings transformation to define a robot workspace

The article discusses the application of optimization algorithms to solve the problem of determining the workspace of robots. The development of a numerical method for approximating the set of solutions of a system of nonlinear inequalities that describe the restrictions on the geometric parameters of a robot based on the concept of non-uniform coverings is considered. An approach is proposed that allows you to reduce the number of boxes and numbers describing each of the boxes. Reducing the number of boxes is achieved by combining the boundary boxes of the covering set and the area between them along one of the axes. The dimensions along the other axes are equal for all boxes, which allows them to be described by one number. The transition to the integer space is described. Thus, converting non-uniform covering sets to a partially ordered set of integers reduces computational complexity. An algorithm for converting boxes of a covering set is presented. The approach has been tested for a 3-RPS robot.

Space-filling curves for numerical approximation and visualization of solutions to systems of nonlinear inequalities with applications in robotics

The problem of approximating and visualizing the solution set of systems of nonlinear inequalities can be frequently met in practice, in particular, when it is required to find the working space of some robots. In this paper, a method using Peano-Hilbert space-filling curves for the dimensionality reduction has been proposed for functions satisfying the Lipschitz condition. Theoretical properties of the introduced algorithm showing advantages of this reduction in the context of the present problem have been established and convergence properties of this method have been studied. A number of experiments executed on test functions and problems regarding finding workspace of robots confirm theoretical results and show a promising character of the new methodology.

A nonmonotone smoothing Newton method for system of nonlinear inequalities based on a new smoothing function

In this paper, a nonmonotone smoothing Newton method is proposed for solving system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations $$H_u(z)=0$$ . A smoothing Newton method is developed for solving the system of nonlinear inequalities by adopting a new nonmonotone line search scheme. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results are reported.

Revealed preference and the subjective state space hypothesis

This paper provides, for finite sets of choice data, revealed preference characterizations for the additive representations considered in Dekel et al. (2001). For a particular class of data sets, it is shown that the characterizing conditions can be reformulated as nonlinear systems of inequalities for which the existence of solutions can be verified using numerical methods.

Solving system of inequalities via a smoothing homotopy method

In this paper, the system of nonlinear inequalities is considered. The problem is approximated by the parameterized smooth equations which is formed by Chen-Harker-Kanzow-Smale smoothing function. A smoothing homotopy method is proposed to solve the system of smooth equations. For almost all initial points in Euclidean space, the existence and convergence of the homotopy path is proved under certain weaker condition without assumption of the monotonicity. The numerical results illustrate that the method is effective.

Approximation of the Solution Set for a System of Nonlinear Inequalities for Modelling a One-Dimensional Chaotic Process

Approximation of the Solution Set for a System of Nonlinear Inequalities for Modelling a One-Dimensional Chaotic Process

On the Existence of a Lipschitz Feedback Control in a Control Problem with State Constraints

We consider a nonlinear control system with state constraints given as a solution set for a finite system of nonlinear inequalities. The problem of constructing a feedback control that ensures the viability of trajectories of the closed system in a small neighborhood of the boundary of the state constraints is studied. Under some assumptions, the existence of a feedback control in the form of a Lipschitz function of the state of the system is proved.

Approximating a solution set of nonlinear inequalities

In this paper we propose a method for solving systems of nonlinear inequalities with predefined accuracy based on nonuniform covering concept formerly adopted for global optimization. The method generates inner and outer approximations of the solution set. We describe the general concept and three ways of numerical implementation of the method. The first one is applicable only in a few cases when a minimum and a maximum of the constraints convolution function can be found analytically. The second implementation uses a global optimization method to find extrema of the constraints convolution function numerically. The third one is based on extrema approximation with Lipschitz under- and overestimations. We obtain theoretical bounds on the complexity and the accuracy of the generated approximations as well as compare proposed approaches theoretically and experimentally.

Finding sets of solutions to systems of nonlinear inequalities

The problem of approximating the set of all solutions to a system of nonlinear inequalities is studied. A method based on the concept of nonuniform coverings is proposed. It allows one to obtain an interior and exterior approximation of this set with a prescribed accuracy. The efficiency of the method is demonstrated by determining the workspace of a parallel robot.

On the attainability problem under state constraints with piecewise smooth boundary

The paper is devoted to the problem of approximating reachable sets for a nonlinear control system with state constraints given as a solution set of a finite system of nonlinear inequalities. Each of these inequalities is given as a level set of a smooth function, but their intersection may have nonsmooth boundary. We study a procedure of eliminating the state constraints based on the introduction of an auxiliary system without constraints such that the right-hand sides of its equations depend on a small parameter. For state constraints with smooth boundary, it was shown earlier that the reachable set of the original system can be approximated in the Hausdorff metric by the reachable sets of the auxiliary control system as the small parameter tends to zero. In the present paper, these results are extended to the considered class of systems with piecewise smooth boundary of the state constraints.

Nonmonotone hybrid tabu search for inequalities and equalities: an experimental study

The main goal of this paper is to analyze the behavior of nonmonotone hybrid tabu search approaches when solving systems of nonlinear inequalities and equalities through the global optimization of an appropriate merit function. The algorithm combines global and local searches and uses a nonmonotone reduction of the merit function to choose the local search. Relaxing the condition aims to call the local search more often and reduces the overall computational eort. Two variants of a perturbed pattern search method are implemented as local search. An experimental study involving a variety of problems available in the literature is presented. Mathematics Subject Classication: 90C15; 90C56; 90C30

A smoothing and regularization predictor-corrector method for nonlinear inequalities

For a system of nonlinear inequalities, we approximate it by a family of parameterized smooth equations via a new smoothing function. We present a new smoothing and regularization predictor-corrector algorithm. The global and local superlinear convergence of the algorithm is established. In addition, the smoothing parameter μ and the regularization parameter e in our algorithm are viewed as different independent variables. Preliminary numerical results show the efficiency of the algorithm. MSC: 90C33; 90C30; 15A06

A smoothing and regularization Broyden-like method for nonlinear inequalities

In this paper, we consider the smoothing and regularization Broyden-like algorithm for the system of nonlinear inequalities. By constructing a new smoothing function \(\phi(\mu,a)=\frac{1}{2}(a+\mu(\ln2+\ln(1+\cosh\frac{a}{\mu})))\), the problem is approximated via a family of parameterized smooth equations H(μ,e,x)=0. A smoothing and regularization Broyden-like algorithm with a non-monotone linear search is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The global convergence of the algorithm is established under suitable assumptions. In addition, the smoothing parameter μ and the regularization parameter e in our algorithm are viewed as two different independent variables. Preliminary numerical results show the efficiency of the algorithm and reveal that the regularization parameter e in our algorithm plays an important role in numerical improvement, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.

Non-interior continuation algorithm for solving system of inequalities over symmetric cones

As a basic mathematical structure, the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems. In this paper, a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone. It is shown that the proposed algorithm is globally convergent and well-defined. Moreover, it can start from any point and only needs to solve one system of linear equations at most at each iteration. Under suitable assumptions, global linear and local quadratic convergence is established with Euclidean Jordan algebras. Numerical results indicate that the algorithm is efficient. The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100, ...,1 000 respectively and the problems of each size were generated randomly for 10 times. The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations. It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the second-order cones quickly. Moreover, a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones, and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.

Necessary conditions for Hadamard factorizations of Hurwitz polynomials

A property of Hurwitz polynomials is related with the Hadamard product. Garloff and Wagner proved that Hadamard products of Hurwitz polynomials are Hurwitz polynomials, and Garloff and Shrinivasan shown that there are Hurwitz polynomials of degree 4 which do not have a Hadamard factorization into two Hurwitz polynomials of the same degree 4. In this paper, we give necessary conditions for an even-degree Hurwitz polynomial to have a Hadamard factorization into two even-degree Hurwitz polynomials; such conditions are given in terms of the coefficients of the given polynomial alone. Furthermore, we show that if an odd-degree Hurwitz polynomial has a Hadamard factorization then a system of nonlinear inequalities has at least one solution.