Calculation of multivariate Chebyshev-type inequalities

AbstractA general multivariate Chebyshev inequality has been obtained by Whittle and Olkin and Pratt. Application of their inequality is made difficult because of the presence of a certain intractable matrix equation. Dharmadhikari and Joag-Dev have obtained a bivariate Gauss inequality. In this paper, we extend the bivariat Gauss inequality to the general multivariate case, encountering the same intractable matrix equation. We develop a general method for the solution of this equation, applying recent results in the solution of systems of nonlinear equations

Deciding Not to Decide Using Resource-Bounded Sensing

We view the problem of sensor-based decision-making in terms of two components: a sensor fusion component that isolates a set of models consistent with observed data, and an evaluation component that uses this information and task-related information to make model-based decisions. In previous work we have described a procedure for computing the solution set of parametric equations describing a sensor-object imaging relationship, and also discussed the use of task-specific information to support set-based decision-making methods. In this paper, we investigate the implications of allowing one of the decision-making options to be no decision, whereupon a human might be called to aid or interact with the system. In particular, this type of capability supports the construction of supervised or partially autonomous systems. We discuss how such situations might arise and give concrete examples of how a system might reach such a decision using our techniques

Homotopies for solving polynomial systems within a bounded domain

AbstractThe problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouche's theorem by homotopy continuation concepts yields a new class of homotopy methods, the so-called regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. As the dominant part has a sparser structure, it is easier to solve. It will be used as start system in the regional homotopy. The paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain. Applications illustrate the usefulness of the approach

A weighted biobjective transformation technique for locating multiple optimal solutions of nonlinear equation systems

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Due to the fact that a nonlinear equation system may contain multiple optimal solutions, solving nonlinear equation systems is one of the most important challenges in numerical computation. When applying evolutionary algorithms to solve nonlinear equation systems, two issues should be considered: i) how to transform a nonlinear equation system into a kind of optimization problem, and ii) how to develop an optimization algorithm to solve the transformed optimization problem. In this paper, we tackle the first issue by transforming a nonlinear equation system into a weighted biobjective optimization problem. By the above transformation, not only do all the optimal solutions of an original nonlinear equation system become the Pareto optimal solutions of the transformed biobjective optimization problem, but also their images are different points on a linear Pareto front in the objective space. In addition, we suggest an adaptive multiobjective differential evolution, the goal of which is to effectively locate the Pareto optimal solutions of the transformed biobjective optimization problem. Once these solutions are found, the optimal solutions of the original nonlinear equation system can also be obtained correspondingly. By combining the weighted biobjective transformation technique with the adaptive multiobjective differential evolution, we propose a generic framework for the simultaneous locating of multiple optimal solutions of nonlinear equation systems. Comprehensive experiments on 38 nonlinear equation systems with various features have demonstrated that our framework provides very competitive overall performance compared with several state-of-the-art methods

Biased random-key genetic algorithm for bound-constrained global optimization

Global optimization seeks a minimum or maximum of a multimodal function over a discrete orcontinuous domain. In this paper, we propose a biased random-key genetic algorithm for findingapproximate solutions for continuous global optimization problems subject to box constraints. Experimentalresults illustrate its effectiveness on the robot kinematics problem, a challenging problemaccording to [7]