Global optimization of general nonconvex problems with intermediate polynomial substructures

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.     

A LOW-FREQUENCY ELECTROMAGNETIC NEAR-FIELD INVERSE PROBLEM FOR A SPHERICAL SCATTERER

The interior low-frequency electromagnetic dipole excitation of a dielectric sphere is uti- lized as a simplified but realistic model in various biomedical applications. Motivated by these considerations, in this paper, we investigate analytically a near-field inverse scatter- ing problem for the electromagnetic interior dipole excitation of a dielectric sphere. First, we obtain, under the low-frequency assumption, a closed-form approximation of the series of the secondary electric field at the dipole＇s location. Then, we utilize this derived approx- imation in the development of a simple inverse medium scattering algorithm determining the sphere＇s dielectric permittivity. Finally, we present numerical results for a human head model, which demonstrate the accurate determination of the complex permittivity by the developed algorithm.     

Rumor spreading on random regular graphs and expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well‐studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d ≥ 3, i.e., the underlying graph is drawn uniformly at random from the set of all d ‐regular graphs with n vertices. We show that with probability 1 ‐ o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d lnn rounds, where Particularly, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo‐random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1 + o(1))Clnn with probability 1 ‐ o(1), where \begin{align*}C = \lim_{d\to\infty}C_d = \frac{1}{\ln2} + 1\end{align*}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Nontrivial Solutions for Resonant Hemivariational Inequalities

We study a resonant semilinear elliptic hemivariational inequality. Under some assumptions of strong resonance on the Clarke subdifferential of the superpotential, and using nonsmooth critical point theory, the existence of a nontrivial solution of the problem is shown. This paper has been partially supported by the State Committee for Scientific Research of Poland (KBN) under research grants no. 2 P03A 003 25 and no. 4 T07A 027 26.     

Scattering relations for point‐source excitation in chiral media

A spherical electromagnetic wave propagating in a chiral medium is scattered by a bounded chiral obstacle which can have any of the usual properties. Reciprocity and general scattering theorems, relating the scattered fields due to scattering of waves from a point source put in any two different locations are established. Applying the general scattering theorem for appropriate locations and polarizations of the point source we prove an associated forward scattering theorem. Mixed scattering relations, relating the scattered fields due to a plane wave and the far‐field patterns due to a spherical wave, are also established. Copyright © 2005 John Wiley & Sons, Ltd.     

Minimal varieties of residuated lattices

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated bounded-lattice.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 1, 2003; accepted in final form April 27, 2004.     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

On logarithmic Sobolev inequalities for higher order fractional derivatives

On ${\mathbb{R}}^n$, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f ∈ $H^s({\mathbb{R}}^n)$ satisfies

$\underset{{\mathbb{R}}^n}{\int }|f(x){|}^2\ln (\frac{{|f(x)|}^2}{6f{6}_2^2})\mathrm{d}x+(n+\frac{n}{s}\ln \alpha +\ln \frac{s\mathrm{\Gamma }(\frac{n}{2})}{\mathrm{\Gamma }(\frac{n}{2s})})6f{6}_2^2?\frac{{\alpha }^2}{{\pi }^s}6{(?\mathrm{\Delta })}^{s/2}f{6}_2^2$

with $\alpha >0$ be any number and where the operators ${(?\mathrm{\Delta })}^{s/2}$ in Fourier spaces are defined by $\stackrel{?}{{(?\mathrm{\Delta })}^{s/2}f}(k)\text{:}={(2\pi |k|)}^s\stackrel{?}{f}(k)$. To cite this article: A. Cotsiolis, N.K. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A genetic approach for strategic resource allocation planning

Enterprises often implement a measurement system to monitor their march towards their strategic goals. Although this way it is possible to assess the progress of each goal, there is no structured way to reconsider resource allocation to those goals and to plan an optimal (or near optimal) allocation scheme. In this study we propose a genetic approach to match each goal with an autonomous entity (agent) with a specific resource sharing behavior. The overall performance is evaluated through a set of functions and genetic algorithms are used to eventuate in approximate optimal behavior’s schemes. To outline the strategic goals of the enterprise we used the balanced scorecard method. Letting agents deploy their sharing behavior over simulation time, we measure the scorecard’s performance and detect distinguished behaviors, namely recommendations for resource allocation.     

Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential (hemivariational inequality). Using the degree theory for multivalued perturbations of (S)₊-operators and the spectrum of a class of weighted eigenvalue problems for the scalar p-Laplacian, we prove the existence of at least three distinct nontrivial solutions, two of which have constant sign.