Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.     

Interior-point methods for nonconvex nonlinear programming: cubic regularization

In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization step with an appropriately chosen regularization parameter. This equivalence allows us to use the favorable theoretical results of Griewank (The modification of Newton’s method for unconstrained optimization by bounding cubic terms, 1981), Nesterov and Polyak (Math. Program., Ser. A 108:177–205, 2006), Cartis et al. (Math. Program., Ser. A 127:245–295, 2011; Math. Program., Ser. A 130:295–319, 2011), but its application at every iteration of the algorithm, as proposed by these papers, is computationally expensive. We propose a hybrid method: use a Newton direction with a line search on iterations with positive definite Hessians and a cubic step, found using a sufficiently large LM perturbation to guarantee a steplength of 1, otherwise. Numerical results are provided on a large library of problems to illustrate the robustness and efficiency of the proposed approach on both unconstrained and constrained problems.     

An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal–dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

Solving Sum of Ratios Fractional Programs via Concave Minimization

This article presents an algorithm for globally solving a sum of ratios fractional programming problem. To solve this problem, the algorithm globally solves an equivalent concave minimization problem via a branch-and-bound search. The main work of the algorithm involves solving a sequence of convex programming problems that differ only in their objective function coefficients. Therefore, to solve efficiently these convex programming problems, an optimal solution to one problem can potentially be used to good advantage as a starting solution to the next problem.     

Natural time analysis of critical phenomena: The case of acoustic emissions in triaxially deformed Etna basalt

Acoustic emissions exhibit complex correlations between space, time, and magnitude. As such, they present a unique example for a complex time series. We apply the recently introduced method of natural time analysis, which enables the detection of long-range temporal correlations even in the presence of heavy tails and find that the acoustic emissions exhibits features similar to that of other equilibrium or non-equilibrium critical systems such as worldwide seismicity as presented in the Centennial earthquake catalogue which includes global seismicity events with magnitude _Mw>7.0$M_w>7.0$. These results support the universal behaviour of the fracturing processes from the laboratory to the global scale.     

Admissible points of a convex polyhedron

In this paper, we present several new properties of the admissible points of a convex polyhedron. These properties can be classified into two categories. One category concerns the characterization and generation of these points. The other category concerns the circumstances under which these points are efficient solutions for linear multiple-objective programs with nonnegative criteria matrices. To illustrate some of the potential utility of these properties, we describe their application to two practical problems. These problems are the linear multiple-objective problem with interval coefficients and the problem of optimizing over the efficient set.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.     

Fast and accurate reflectivity calculations for shaped core optical fibres

A novel scalar (i.e. paraxial) free space radiation mode method is presented for calculating the normal incidence facet reflectivity of the main guided mode of a step-index optical fibre with generally shaped smooth core. Numerical results are presented and discussed for several cases of the Nth order super-elliptical core shape, including the circular, elliptical, rectangular-type and slab limiting cases.