New global optima results for the Kauffman NK model: handling dependency

The Kauffman NK model has been used in theoretical biology, physics and business organizations to model complex systems with interacting components. This paper presents new global optima results for the NK model by developing tools for handling dependency in the cases where K grows with N; this generalizes the previous work that focused on the analysis of the (independent) case K=N−1. A dependency graph is defined and studied to handle dependencies among underlying random variables in the NK model. Order statistics (with dependencies) and the expected value of the global optima, E _{N, K} , are bounded using equitable coloring of the dependency graph. These bounds convert the problem of bounding order statistics of dependent random variables into that of independent random variables while incorporating quantitative information about the mutual dependencies between the underlying random variables. An alternative upper bound on E _{N, K} using direct arguments is also proposed. A detailed analysis of E _{N, K} for K close to N (K=N−α and K=β N, α ∈ Z ⁺,β ∈ (0,1)) is given for underlying uniform and normal distributions. Finally, for bounded underlying distributions, the global optima is shown to be concentrated around its mean E _{N, K} .     

A Comparison of Global Optimization Methods for the Design of a High-speed Civil Transport

The conceptual design of aircraft often entails a large number of nonlinear constraints that result in a nonconvex feasible design space and multiple local optima. The design of the high-speed civil transport (HSCT) is used as an example of a highly complex conceptual design with 26 design variables and 68 constraints. This paper compares three global optimization techniques on the HSCT problem and two test problems containing thousands of local optima and noise: multistart local optimizations using either sequential quadratic programming (SQP) as implemented in the design optimization tools (DOT) program or Snyman's dynamic search method, and a modified form of Jones' DIRECT global optimization algorithm. SQP is a local optimizer, while Snyman's algorithm is capable of moving through shallow local minima. The modified DIRECT algorithm is a global search method based on Lipschitzian optimization that locates small promising regions of design space and then uses a local optimizer to converge to the optimum. DOT and the dynamic search algorithms proved to be superior for finding a single optimum masked by noise of trigonometric form. The modified DIRECT algorithm was found to be better for locating the global optimum of functions with many widely separated true local optima.     

Bounding Ramsey numbers through large deviation inequalities

We develop a new approach for proving lower bounds for various Ramsey numbers, based on using large deviation inequalities. This approach enables us to obtain the bounds for the off-diagonal Ramsey numbers R(K^r, K^k), r ≤ k, that match the best known bounds, obtained through the local lemma. We discuss also the bounds for a related Ramsey-type problem and show, for example, that there exists a K⁴-free graph G on n vertices in which every cn^3/5 log^1/2 n vertices span a copy of K³. © 1995 John Wiley & Sons, Inc.     

The expectation and variance of the joint linear complexity of random periodic multisequences

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. By using the generalized discrete Fourier transform for multisequences, Meidl and Niederreiter determined the expectation of the joint linear complexity of random N-periodic multisequences explicitly. In this paper, we study the expectation and variance of the joint linear complexity of random periodic multisequences. Several new lower bounds on the expectation of the joint linear complexity of random periodic multisequences are given. These new lower bounds improve on the previously known lower bounds on the expectation of the joint linear complexity of random periodic multisequences. By further developing the method of Meidl and Niederreiter, we derive a general formula and a general upper bound for the variance of the joint linear complexity of random N-periodic multisequences. These results generalize the formula and upper bound of Dai and Yang for the variance of the linear complexity of random periodic sequences. Moreover, we determine the variance of the joint linear complexity of random periodic multisequences with certain periods.     

Global holomorphic 2-forms and pluricanonical systems on threefolds

The change of zero locus of a global holomorphic 2-form on a threefold under birational transformations is investigated. It is proved that existence of 2-forms with certain conditions on their zero loci on a threefold of nonnegative Kodaira dimension limits types of terminal singularities appearing on its minimal models. As a result of the restriction on the types of terminal singularities and Reid's Riemann-Roch formula, a universal bound N is found such that the linear system NK defines a birational map from a threefold of general type admitting those 2-forms, where K is the canonical bundle of the threefold. Received March 10, 2000 / Published online October 11, 2000     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

Sharp Estimates for the Global Attractor of Scalar Reaction–Diffusion Equations with a Wentzell Boundary Condition

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor A_W\mathcal{A}_{\mathrm{W}} for scalar reaction–diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds on the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor A_K\mathcal{A}_{K}, K∈{D,N,P}, of reaction–diffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor A_W\mathcal {A}_{\mathrm{W}} is of different order than the dimension of A_K\mathcal{A}_{K}, for each K∈{D,N,P}, in all space dimensions that are greater than or equal to three.     

The variable neighborhood search for the two machine flow shop problem with a passive prefetch

We consider the two machine flow shop scheduling problem with passive loading of the buffer on the second machine. To compute lower bounds for the global optimum, we present four integer linear programming formulations of the problem. Three local search methods with variable neighborhoods are developed for obtaining upper bounds. Some new large neighborhood is designed. Our methods use this neighborhood along with some other well-known neighborhoods. For computational experiments, we present a new class of test instances with known global optima. Computational results indicate a high efficiency of the proposed approach for the new class of instances as well as for other classes of instances of the problem.     

Cache-Oblivious Range Reporting with Optimal Queries Requires Superlinear Space

We consider a number of range reporting problems in two and three dimensions and prove lower bounds on the amount of space used by any cache-oblivious data structure for these problems that achieves the optimal query bound of O(log  _B N+K/B) block transfers, where K is the size of the query output.     

Approximate computation of the permeability tensor of a periodic porous medium

The permeability tensor K of an infinite periodic porous medium, obtained using the homogenization theory, is considered. The solutions of an optimal control problem for the Dirichlet or Neumann equation are used to obtain optimal upper bounds for K. The test functions used for the estimations are simpler than those obtained by other authors. Some possibilities are given to obtain also lower bounds.     

Global optimization for molecular conformation problems

A primal-relaxed dual global optimization algorithm is presented along with an extensive review for finding the global minimum energy configurations of microclusters composed by particles interacting with any type of two-body central forces. First, the original nonconvex expression for the total potential energy is transformed to the difference of two convex functions (DC transformation) via an eigenvalue analysis performed for each pair potential that constitutes the total potential energy function. Then, a decomposition strategy based on the GOP algorithm [1–4] is designed to provide tight upper and lower bounds on the global minimum through the solutions of a sequence of relaxed dual subproblems. A number of theoretical results are included which expedite the computational effort by exploiting the special mathematical structure of the problem. The proposed approach attains-convergence to the global minimum in a finite number of iterations. Based on this procedure global optimum solutions are generated for small Lennard-Jones and Morse (a=3) microclustersn7. For larger clusters (8N24 for Lennard-Jones and 8N30 for Morse), tight lower and upper bounds on the global solution are provided which serve as excellent initial points for local optimization approaches.     

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

We prove explicit upper and lower bounds for the L ¹-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P ^m in ambient Riemannian spaces N ⁿ . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N ⁿ themselves.     

A Population-based Approach for Hard Global Optimization Problems based on Dissimilarity Measures

When dealing with extremely hard global optimization problems, i.e. problems with a large number of variables and a huge number of local optima, heuristic procedures are the only possible choice. In this situation, lacking any possibility of guaranteeing global optimality for most problem instances, it is quite difficult to establish rules for discriminating among different algorithms. We think that in order to judge the quality of new global optimization methods, different criteria might be adopted like, e.g.:

Noyaux de Tate et capitulation

Following Kahn, and Assim and Movahhedi, we look for bounds for the order of the capitulation kernels of higher K-groups of S-integers into abelian S-ramified p-extensions. The basic strategy is to change twists inside some Galois-cohomology groups, which is done via the comparison of Tate Kernels of higher order. We investigate two ways: a global one, valid for twists close to 0 (in a certain sense), and a local one, valid for twists close to 1 in cyclic extensions. The global method produces lower bounds for abelian p-extensions which are S-ramified, but not Zp-embeddable. The local method is close to that of [J. Assim, A. Movahhedi, Bounds for étale capitulation kernels, K-Theory 33 (2004) 199-213], but is improved to take into consideration what happens when S consists of only the p-places. In contrast to the first one, one can expect this second method to produce nontrivial lower bounds in certain Zp-extensions. For example, we construct Zp-extensions in which the capitulation kernel is as big as we want (when letting the twist vary). We also include a complete solution to the problem of comparing Tate Kernels.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Concentration bounds for the probabilities $\mathbb{P}(N\ge M+r)$ and $\mathbb{P}(N\le M?r)$ are proved, where $M$ is a median or the expectation of a subgraph count $N$ associated with a random geometric graph built over a Poisson process. The lower tail bounds have a Gaussian decay and the upper tail inequalities satisfy an optimality condition. A remarkable feature is that the underlying Poisson process can have a.s. infinitely many points.The estimates for subgraph counts follow from tail inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques involving the convex distance for Poisson processes.     

On the geometry of metric measure spaces. II

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.     

Steiner symmetrals and their distance from a ball

It is known that given any convex bodyK ⊂ ℝ ⁿ there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant. In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.     

Expected hitting and cover times of random walks on some special graphs

We give general bounds (and in some cases exact values) for the expected hitting and cover times of the simple random walk on some special undirected connected graphs using symmetry and properties of electrical networks. In particular we give easy proofs for an N–1H_N-1 lower bound and an N² upper bound for the cover time of symmetric graphs and for the fact that the cover time of the unit cube is Φ(NlogN). We giver a counterexample to a conjecture of Freidland about a general bound for hitting times. Using the electric approach, we provide some genral upper and lower bounds for the expected cover times in terms of the diameter of the graph. These bounds are tight in many instances, particularly when the graph is a tree. © 1994 John Wiley & Sons, Inc.