A minimax search plan for constrained optimization problems

A minimax search strategy is described for locating the boundary point of a region on a line joining a feasible point to an infeasible point. Asymptotic strategies, useful when the number of experiments to be used in the search is not predetermined, are also given. These strategies are useful subroutines for many multidimensional optimization algorithms.The authors thank G. V. Reklaitis for initial discussions concerning this problem. John H. Beamer was an NSF Graduate Fellow at the time when this research was conducted.     

A new decision support framework for managing foot-and-mouth disease epidemics

Animal disease epidemics such as the foot-and-mouth disease (FMD) pose recurrent threat to countries with intensive livestock production. Efficient FMD control is crucial in limiting the damage of FMD epidemics and securing food production. Decision making in FMD control involves a hierarchy of decisions made at strategic, tactical, and operational levels. These decisions are interdependent and have to be made under uncertainty about future development of the epidemic. Addressing this decision problem, this paper presents a new decision-support framework based on multi-level hierarchic Markov processes (MLHMP). The MLHMP model simultaneously optimizes decisions at strategic, tactical, and operational levels, using Bayesian forecasting methods to model uncertainty and learning about the epidemic. As illustrated by the example, the framework is especially useful in contingency planning for future FMD epidemics.     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

A maximum problem for matrices

The following maximum problem is considered: To find among all contractions T on an n-dimensional Hilbert space whose spectral radius does not exceed a given number p< 1, the operator T for which |Tⁿ| is maximum. A matrix T of Toeplitz type is constructed for which this maximum is attained.     

A transportation problem formulation for the MAC Airlift Planning problem

The Military Airlift Command (MAC) is responsible for planning the allocation of airlift resources for the movement of cargo and passengers. A heuristic algorithm, the Airlift Planning Algorithm (APA), has recently been developed under subcontract to the Oak Ridge National Laboratory to assist MAC in scheduling airlift resources. In this paper, we present a transportation problem formulation which can be used as a preprocessor to the APA or as an estimator for the APA. This paper examines the robustness and sensitivity of the transportation problem formulation. In particular, the performance of the APA improves by approximately 10% when the transportation problem is used as a preprocessor for two hypothetical problems and improves by up to 50% for derived airlift constrained problems.     

A framework for managing uncertain inputs: An axiomization of rewarding

The success postulate in belief revision ensures that new evidence (input) is always trusted. However, admitting uncertain input has been questioned by many researchers. Darwiche and Pearl argued that strengths of evidence should be introduced to determine the outcome of belief change, and provided a preliminary definition towards this thought. In this paper, we start with Darwiche and Pearl’s idea aiming to develop a framework that can capture the influence of the strengths of inputs with some rational assumptions. To achieve this, we first define epistemic states to represent beliefs attached with strength, and then present a set of postulates to describe the change process on epistemic states that is determined by the strengths of input and establish representation theorems to characterize these postulates. As a result, we obtain a unique rewarding operator which is proved to be a merging operator that is in line with many other works. We also investigate existing postulates on belief merging and compare them with our postulates. In addition, we show that from an epistemic state, a corresponding ordinal conditional function by Spohn can be derived and the result of combining two epistemic states is thus reduced to the result of combining two corresponding ordinal conditional functions proposed by Laverny and Lang. Furthermore, when reduced to the belief revision situation, we prove that our results induce all the Darwiche and Pearl’s postulates as well as the Recalcitrance postulate and the Independence postulate.     

The master bay plan problem: a solution method based on its connection to the three-dimensional bin packing problem

This paper addresses the problem of determining stowage plansfor containers in a ship, referred to as the Master Bay PlanProblem (MBPP). MBPP is NP-complete. We present a heuristic method for solvingMBPP based on its relation with the three-dimensional bin packingproblem (3D-BPP), where items are containers and bins are differentportions of the ship. Our aim is to find stowage plans, takinginto account structural and operational constraints relatedto both the containers and the ship, that minimize the timerequired for loading all containers on board. A validation of the proposed approach with some test casesis given. The results of real instances of the problem involvingmore than 1400 containers show the effectiveness of the proposedapproach for large scale applications.     

A stability problem for holonomic systems

Sunto Si studia il problema della stabilità di una posizione di equilibrio di un sistema meccanico olonomo soggetto ad una sollecitazione derivante da un potenziale del tipo U(q, t)=f(t) U*(q), conf funzione periodica, e ad una sollecitazione dissipativa. I risultati ottenuti estendono a sistemi con un numero finito qualsiasi di gradi di libertà risultati noti per sistemi ad un solo grado di libertà.

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Work performed under the auspices of the Italian Council of Research (C.N.R.): National Group of Mathematical Physics (G.N.F.M.), and Italian Ministry of Public Education (M.P.I.).

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The author wishes to thank professors S.Busenberg and L.Salvadori for many useful discussions.     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

A variational problem for affine connections

We give a variational problem for affine connections which characterizes the Riemannian connection of an Einstein metric of negative scalar curvature. Received: 10 May 2005; revised: 25 July 2005     

A variational problem for pullback metrics

Let (M, g) and (N, h) be Riemannian manifolds without boundary and let f : M → N be a smooth map. Let ||f^*h||{\|f^*h\|} denote the norm of the pullback metric of h by f. In this paper, we consider the functional F(f) = ò_M ||f^*h||² dv_g{{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}. We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f ₁ and f ₂ are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.     

A Transmission problem for bi-isotropic media

In this work we study the solvability of the transmission problem describing the scattering of a plane time-harmonic electromagnetic wave by a penetrable bi-isotropic obstacle. Existence and uniqueness of the solution to this problem is established by a weak solutions approach, with the use of the Dirichlet-to-Neumann operator. Moreover, in the low frequency case, we reduce the problem to a sequence of potential problems which can be solved iteratively.     

A two-point problem for first-order systems

Let f be a continuous function from [a, b] ×\mathbbRⁿ [a, b] \times \mathbb{R}^n into \mathbbRⁿ \mathbb{R}^n . In this paper we prove that the problem¶¶ { llu^￠ = f(t,u)+ lu(a)=u(b)=0  \left \{ \begin{array}{ll}u^{\prime}= f(t,u)+ \lambda \\[3pt]u(a)=u(b)=0\end{array}\right.\ ¶¶ has a (classical) solution for a wide class of functions f. Next we point out a particular case.