Van Geemen's families of lines on special quintic threefolds

We study the families of lines on the quintic threefolds of the pencilx ₀ ⁵ +x ₁ ⁵ +x ₂ ⁵ +x ₃ ⁵ +x ₄ ⁵ -5tx ₀ x ₁ x ₂ x ₃ x ₄=0. We show that on the generic threefold of the pencil there exists a 1-dimensional family of lines that is not a cone. Partially supported by funds M.P.I.     

Crystal and molecular structure of phenylbenzyl-1-(2,2,3-trimethylcyclopropyl)phosphine oxide

The title compound crystallizes in the orthorhombic space groupPna2₁, witha=15.576(2),b=19.134(3),c=5.693(1) Å andz=4. The structure was solved by direct methods, and refined to a finalR value of 0.059. This study confirms both the structure and stereochemistry of the title compound.     

Global optima results for the Kauffman NK model

The Kauffman NK model has been used in theoretical biology, physics and business organizations to model complex systems with interacting components. Recent NK model results have focused on local optima. This paper analyzes global optima of the NK model. The resulting global optimization problem is transformed into a stochastic network model that is closely related to two well-studied problems in operations research. This leads to applicable strategies for explicit computation of bounds on the global optima particularly with K either small or close to N. A general lower bound, which is sharp for K = 0, is obtained for the expected value of the global optimum of the NK model. A detailed analysis is provided for the expectation and variance of the global optimum when K = N−1. The lower and upper bounds on the expectation obtained for this case show that there is a wide gap between the values of the local and the global optima. They also indicate that the complexity catastrophe that occurs with the local optima does not arise for the global optima.     

Analyzing the Complexity of Finding Good Neighborhood Functions for Local Search Algorithms

A drawback to using local search algorithms to address NP-hard discrete optimization problems is that many neighborhood functions have an exponential number of local optima that are not global optima (termed L-locals). A neighborhood function η is said to be stable if the number of L-locals is polynomial. A stable neighborhood function ensures that the number of L-locals does not grow too large as the instance size increases and results in improved performance for many local search algorithms. This paper studies the complexity of stable neighborhood functions for NP-hard discrete optimization problems by introducing neighborhood transformations. Neighborhood transformations between discrete optimization problems consist of a transformation of problem instances and a corresponding transformation of solutions that preserves the ordering imposed by the objective function values. In this paper, MAX Weighted Boolean SAT (MWBS), MAX Clause Weighted SAT (MCWS), and Zero-One Integer Programming (ZOIP) are shown to be NPO-complete with respect to neighborhood transformations. Therefore, if MWBS, MCWS, or ZOIP has a stable neighborhood function, then every problem in NPO has a stable neighborhood function. These results demonstrate the difficulty of finding effective neighborhood functions for NP-hard discrete optimization problems.This research is supported in part by the Air Force Office of Scientific Research (F49620-01-1-0007, FA9550-04-1-0110).     

Analyzing the performance of simultaneous generalized hill climbing algorithms

Simultaneous generalized hill climbing (SGHC) algorithms provide a framework for using heuristics to simultaneously address sets of intractable discrete optimization problems where information is shared between the problems during the algorithm execution. Many well-known heuristics can be embedded within the SGHC algorithm framework. This paper shows that the solutions generated by an SGHC algorithm are a stochastic process that satisfies the Markov property. This allows problem probability mass functions to be formulated for particular sets of problems based on the long-term behavior of the algorithm. Such results can be used to determine the proportion of iterations that an SGHC algorithm will spend optimizing over each discrete optimization problem. Sufficient conditions that guarantee that the algorithm spends an equal number of iterations in each discrete optimization problem are provided. SGHC algorithms can also be formulated such that the overall performance of the algorithm is independent of the initial discrete optimization problem chosen. Sufficient conditions are obtained guaranteeing that an SGHC algorithm will visit the globally optimal solution for each discrete optimization problem. Lastly, rates of convergence for SGHC algorithms are reported that show that given a rate of convergence for the embedded GHC algorithm, the SGHC algorithm can be designed to preserve this rate.     

Metaheuristics for a Flexible Assembly System Design Problem

Flexibility has become an important priority in the formulation and implementation of manufacturing strategies. This in turn has opened up a new class of design problems for such systems. Flexible assembly systems (FAS), consisting of a variety of processors and operations, provide the opportunity for improving product manufacturing flexibility, hence gaining competitive advantages. This paper considers a particular design decision problem for FAS. A matrix-based, polynomial-time lower bound algorithm is presented. Simulated annealing and tabu search metaheuristics are formulated to address the problems. Computational experience with these metaheuristics is reported.