A Tradeoff-Based Interactive Multi-Objective Optimization Method Driven by Evolutionary Algorithms

Multi-objective optimization problems involve two or more conflicting objectives, and they have a set of Pareto optimal solutions instead of a single optimal solution. In order to support the decision maker (DM) to find his/her most preferred solution, we propose an interactive multi-objective optimization method based on the DM’s preferences in the form of indifference tradeoffs. The method combines evolutionary algorithms with the gradient-based interactive step tradeoff (GRIST) method. An evolutionary algorithm is used to generate an approximate Pareto optimal solution at each iteration. The DM is asked to provide indifference tradeoffs whose projection onto the tangent hyperplane of the Pareto front provides a tradeoff direction. An approach for approximating the normal vector of the tangent hyperplane is proposed which is used to calculate the projection. A water quality management problem is used to demonstrate the interaction process of the interactive method. In addition, three benchmark problems are used to test the accuracy of the normal vector approximation approach and compare the proposed method with GRIST.

THE NONLINEAR STRUCTURE ON THE GENERAL QUANTUM HYPERPLANES: THE QUANTUM HYPERSURFACES AND THE NONLINEAR REALIZATIONS OF THE QUANTUM GROUPS

In this paper we prove that by the use of an extended algebra the nonlinear transformations of the general quantum hyperplanes can be obtained. The images of this transformation and its linear part, which are two quantum hyperspaces, can be interpreted as a quantum hypersurface and its quantum tangent hyperplane, respectively. The quantum group concerned is nonlinearly realized on this quantum hypersurface. The concrete results of GL q(N), as an example, are calculated.

A Differentiable Null Space Method for Constrained Dynamic Analysis

A geometric approach to the solution of the dynamic response of constrained mechanical systems is proposed. A continuous and differentiable basis of the constraint null space is automatically generated using the Gram-Schmidt process. The independent coordinates are obtained by transforming the physical velocity coordinates to the tangent hyperplane of the constraint surface. As a result the independent coordinates lie on the constraint surface and no constraint violation control is necessary.

Athwart immersions in Euclidean space

Let f and g denote immersions of the n-manifolds M and N, respectively, in Rn+1. We say that f is athwart to g if f(M) and g(N)m have no tangent hyperplane in common. In this paper necessary conditions for athwartness are obtained.

On the division of Euclidean n -space by topological ( n ─ 1)-spheres

Mazur’s theorem on the division of an n -sphere is generalized and applied to show that closed Euclidean n -space is split into two n -cells by an embedded topological ( n ─ 1)-sphere, S , if either S has a continuously varying tangent hyperplane, or S is a rectilinearly embedded complex in which the link of every simplex is a topological sphere.