Instance generators and test suites for the multiobjective quadratic assignment problem

We describe, and make publicly available, two problem instance generators for a multiobjective version of the well-known quadratic assignment problem (QAP). The generators allow a number of instance parameters to be set, including those controlling epistasis and inter-objective correlations. Based on these generators, several initial test suites are provided and described. For each test instance we measure some global properties and, for the smallest ones, make some initial observations of the Pareto optimal sets/fronts. Our purpose in providing these tools is to facilitate the ongoing study of problem structure in multiobjective (combinatorial) optimization, and its effects on search landscape and algorithm performance.

Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft

This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

Optimal kinematic control of an autonomous underwater vehicle

This note investigates the motion control of an autonomous underwater vehicle (AUV). The AUV is modeled as a nonholonomic system as any lateral motion of a conventional, slender AUV is quickly damped out. The problem is formulated as an optimal kinematic control problem on the Euclidean Group of Motions SE(3), where the cost function to be minimized is equal to the integral of a quadratic function of the velocity components. An application of the Maximum Principle to this optimal control problem yields the appropriate Hamiltonian and the corresponding vector fields give the necessary conditions for optimality. For a special case of the cost function, the necessary conditions for optimality can be characterized more easily and we proceed to investigate its solutions. Finally, it is shown that a particular set of optimal motions trace helical paths. Throughout this note we highlight a particular case where the quadratic cost function is weighted in such a way that it equates to the Lagrangian (kinetic energy) of the AUV. For this case, the regular extremal curves are constrained to equate to the AUV's components of momentum and the resulting vector fields are the d'Alembert-Lagrange equations in Hamiltonian form.

Complexity of Monte Carlo algorithms for a class of integral equations

In this work we study the computational complexity of a class of grid Monte Carlo algorithms for integral equations. The idea of the algorithms consists in an approximation of the integral equation by a system of algebraic equations. Then the Markov chain iterative Monte Carlo is used to solve the system. The assumption here is that the corresponding Neumann series for the iterative matrix does not necessarily converge or converges slowly. We use a special technique to accelerate the convergence. An estimate of the computational complexity of Monte Carlo algorithm using the considered approach is obtained. The estimate of the complexity is compared with the corresponding quantity for the complexity of the grid-free Monte Carlo algorithm. The conditions under which the class of grid Monte Carlo algorithms is more efficient are given.

Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

On two rational difference equations

In this paper, we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations [GRAPHICS] and [GRAPHICS] where a is a nonnegative constant. (c) 2005 Elsevier Inc. All rights reserved.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.