The Hencky strain energy ‖log U‖2 measures the geodesic distance of the deformation gradient to SO(n) in the canonical left-invariant Riemannian metric on GL(n)

The well-known isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set of rigid rotations in a canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n) invariant, isotropy requires the Riemannian metric to be right-O(n) invariant. The latter two conditions are satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters. In deriving the result, geodesics on GL(n) have to be parametrized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

We call a metric m-quasi-Einstein if \({Ric_X^m}\) (a modification of the m-Bakry–Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contain Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field X such that the left-invariant Riemannian metric on a quadratic Lie group is m-quasi-Einstein is a Killing vector field. Then we construct infinitely many non-trivial m-quasi-Einstein metrics on solvable quadratic Lie groups G(n) for m finite.     

On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g ₀) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g ₀) is not conformally equivalent to the unit sphere S ⁿ and that the set of all such metrics is compact. When (M, g ₀) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2. This work was supported by the Australian Research Council.     

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.     

Nonlinear coordinate representations of smooth optimization problems

Nonlinear coordinate representations of smooth optimization problems are investigated from the point of view of variable metric algorithms. In other words, nonlinear coordinate systems, in the sense of differential geometry, are studied by taking into consideration the structure of smooth optimization problems and variable metric methods.Both the unconstrained and constrained cases are discussed. The present approach is based on the fact that the nonlinear coordinate transformation of an optimization problem can be replaced by a suitable Riemannian metric belonging to the Euclidean metric class. In the case of equality and inequality constraints, these questions are related closely to the right inverses of full-rank matrices; therefore, their characterization is a starting point of the present analysis. The main results concern a new subclass of nonlinear transformations in connection with the common supply of coordinates to two Riemannian manifolds, one immersed in the other one. This situation corresponds to the differentiable manifold structure of nonlinear optimization problems and improves the insight into the theoretical background of variable metric algorithms. For a wide class of variable metric methods, a convergence theorem in invariant form (not depending on coordinate representations) is proved. Finally, a problem of convexification by nonlinear coordinate transformations and image representations is studied.This research was supported by the Hungarian National Research Foundation, Grant Nos. OTKA-2568 and OTKA-2116, and by the Project Trasporti of the Italian National Research Council (CNR).     

S-curvature of isotropic Berwald metrics

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(√nL) iterations. Among them, several potential reduction algorithms combine both theoretical (O(√nL) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(√nL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least Θ(√nL) iterations to find an optimal solution. The research of the author was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory.     

Partial regularity of mass-minimizing rectifiable sections

Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M ⁿ. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section σ: M → B is the mass of the image σ(M) as a rectifiable n-current in B. Theorem 1. For any homology class of sections of B, there is a mass-minimizing rectifiable current T representing that homology class which is the graph of a C¹ section on an open dense subset of M. Mathematics Subject Classifications (2000): 49F20, 49F22, 49F10, 58A25, 53C42, 53C65.     

Polynomiality of primal-dual affine scaling algorithms for nonlinear complementarity problems

This paper provides an analysis of the polynomiality of primal-dual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu’s scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primal-dual affine scaling algorithms generates an approximate solution (given a precision ε) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial ofn, ln(1/ε) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in Jansen et al., SIAM Journal on Optimization 7 (1997) 126–140. Research supported in part by Grant-in-Aids for Encouragement of Young Scientists (06750066) from the Ministry of Education, Science and Culture, Japan. Research supported by Dutch Organization for Scientific Research (NWO), grant 611-304-028     

On dominatedl 1 metrics

We introduce and study a classl ₁ ^dom (ρ) ofl ₁-embeddable metrics corresponding to a given metric ρ. This class is defined as the set of all convex combinations of ρ-dominated line metrics. Such metrics were implicitly used before in several constuctions of low-distortion embeddings intol _p -spaces, such as Bourgain’s embedding of an arbitrary metric ρ onn points withO(logh) distortion. Our main result is that the gap between the distortions of embedding of a finite metric ρ of sizen intol ₂ versus intol ₁ ^dom (ρ) is at most , and that this bound is essentially tight. A significant part of the paper is devoted to proving lower bounds on distortion of such embeddings. We also discuss some general properties and concrete examples. Research by J. M. supported by Charles University grants No. 158/99 and 159/99. Part of the work by Y. R. was done during his visit at the Charles University in Prague partially supported by these grants, by the grant GAČR 201/99/0242, and by Haifa University grant for Promotion of Research.     

New Self-Concordant Barrier for the Hypercube

We introduce a new barrier function to build new interior-point algorithms to solve optimization problems with bounded variables. First, we show that this function is a (3/2)n-self-concordant barrier for the unitary hypercube [0,1] ⁿ , assuring thus the polynomial property of related algorithms. Second, using the Hessian metric of that barrier, we present new explicit algorithms from the point of view of Riemannian geometry applications. Third, we prove that the central path defined by the new barrier to solve a certain class of linearly constrained convex problems maintains most of the properties of the central path defined by the usual logarithmic barrier. We present also a primal long-step path-following algorithm with similar complexity to the classical barrier. Finally, we introduce a new proximal-point Bregman type algorithm to solve linear problems in [0,1] ⁿ and prove its convergence. P.R. Oliveira was partially supported by CNPq/Brazil.     

Obstruction to Positive Curvature on Homogeneous Bundles

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F) were discovered recently [J. Differential Geom. 65:273–287, 2003; Invent. Math. 148:117–141, 2002]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h) × F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [Proc. Amer. Math. Soc. 66(2):321–326, 1977].Supported in part by NSF grant DMS–0303326.     

Empirical Bayes estimation in a multiple linear regression model

Summary Estimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+ε is considered when β has a completely unknown and unspecified distribution and the error-vector ε has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in practice. UsingX, Y and the information contained in the observation-vectors obtained fromn independent past experiences of the problem, (empirical Bayes) estimators for β are exhibited. These estimators are compared with the optimal estimator and are shown to be asymptotically optimal. Estimators asymptotically optimal with rates nearO(n ⁻¹) are constructed. Supported in part by a Natural Sciences and Engineering Research Council of Canada grant.     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

Graded metrics adapted to splittings

Homogeneous graded metrics over split ℤ₂-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesR ^Δ′ (X,Y)² = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed. Partially supported by DGICYT grants #PB94-0972, and SAB94-0311; IVEI grant 95-031; CONACyT grant #3189-E9307.     

Distance-Regular (0,α)-Reguli

We introduce distance-regular (0,α)-reguli and show that they give rise to (0,α)-geometries with a distance-regular point graph. This generalises the SPG-reguli of Thas [14] and the strongly regular (α,β)-reguli of Hamilton and Mathon [9], which yield semipartial geometries and strongly regular (α,β)-geometries, respectively. We describe two infinite classes of examples, one of which is a generalisation of the well-known semipartial geometry T_n^*(B) arising from a Baer subspace PG(n, q) in PG(n, q²). Research Fellow supported by the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/13367/Tonesi Research assistant of the Fund for Scientific Research Flanders (FWO-Vlaanderen).     

Pinching, Pontrjagin classes, and negatively curved vector bundles

We prove several finiteness results for the class ℳ_a,b,π,n of n-manifolds that have fundamental groups isomorphic to π and that can be given complete Riemannian metrics of sectional curvatures within {a,b} where a≤b<0. In particular, if M is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class ℳ_a,b,π1(M),n are total spaces of vector bundles over M. Furthermore, given a word-hyperbolic group π and an integer n there exists a positive ε=ε(n,π) such that the tangent bundle of any manifold in the class ℳ_{-1-ε, -1, π, n} has zero rational Pontrjagin classes. Oblatum 27-IX-1999 & 17-XI-2000?Published online: 5 March 2001     

Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems

We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair (x,y)∈ℝ ²ⁿ satisfying y=f(x) and x _i y _i =0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ ⁿ to ℝ ⁿ . The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features; (a) it traces a trajectory in ℝ ³ⁿ which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary (not necessarily positive) point in ℝ ²ⁿ in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including (not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it. Received April 9, 1997 / Revised version received September 2, 1998? Published online May 28, 1999