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.     

Killing graphs with prescribed mean curvature

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds. M. Dajczer was partially supported by Procad, CNPq and Faperj. P. A. Hinojosa was partially supported by PADCT/CT-INFRA/CNPq/MCT Grant #620120/2004-5. J. H. de Lira was partially supported by CNPq and Funcap.     

A generalized proximal-point method for convex optimization problems in Hilbert spaces

We deal with a generalization of the proximal-point method and the closely related Tikhonov regularization method for convex optimization problems. The prime motivation behind this is the well-known connection between the classical proximal-point and augmented Lagrangian methods, and the emergence of modified augmented Lagrangian methods in recent years. Our discussion includes a formal proof of a corresponding connection between the generalized proximal-point method and the modified augmented Lagrange approach in infinite dimensions. Several examples and counterexamples illustrate the convergence properties of the generalized proximal-point method and indicate that the corresponding assumptions are sharp.     

THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS

Abstract

In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we prove that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point. The monotonic behavior of the primal and dual logarithmic barriers and of the primal and dual objective functions along the trajectory is also discussed. The existence and optimality of cluster points is established and finally, under the additional assumption of analyticity of the data functions, the convergence of the primal-dual trajectory is proved.     

Islands at infinity on manifolds of asymptotically nonnegative curvature

We introduce an invariant which measures the R-eccentricity of a point in a complete Riemannian manifold M and show that it goes to zero when the point goes to infinity, if M has asymptotically nonnegative curvature. As a consequence we show that the isometry group is compact if M has asymptotically nonnegative curvature and a point with positive sectional curvature. Both authors were partially supported by CNPq of Brazil and the second author was also partially supported by FAPERJ of Brazil.     

Statistical Structures on Metric Path Spaces

The authors extend the notion of statistical structure from Riemannian geometry to the general framework of path spaces endowed with a nonlinear connection and a generalized metric. Two particular cases of statistical data are defined. The existence and uniqueness of a nonlinear connection corresponding to these classes is proved. Two Koszul tensors are introduced in accordance with the Riemannian approach. As applications, the authors treat the Finslerian (α, β)-metrics and the Beil metrics used in relativity and field theories while the support Riemannian metric is the Fisher-Rao metric of a statistical model.     

On the convergence of the entropy-exponential penalty trajectories and generalized proximal point methods in semidefinite optimization

The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. It is proved that the primal path converges to the analytic center of the primal optimal set with respect to the entropy function, the dual path converges to a point in the dual optimal set and the primal-dual path associated to this paths converges to a point in the primal-dual optimal set. As an application, the generalized proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered. The convergence of the primal proximal sequence to the analytic center of the primal optimal set with respect to the entropy function is established and the convergence of a particular weighted dual proximal sequence to a point in the dual optimal set is obtained.     

The existence of light‐like homogeneous geodesics in homogeneous Lorentzian manifolds

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo‐Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo‐Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light‐like homogeneous geodesic. The method is illustrated in detail with an example of the Lie group of dimension 3 with an invariant metric, which does not admit any light‐like homogeneous geodesic.     

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.     

Remarks on the two-dimensional Benjamin equation

We generalize some recent results proved for the KP equation to the generalized Benjamin equation. First, we establish that the Cauchy problem cannot be solved by an iteration method. As a consequence, the flow map fails to be smooth. The second goal is to prove that the zero-mass constraint is satisfied at any non-zero time even it is not satisfied at the initial time.     

Cauchy problem of the generalized double dispersion equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions.     

On geodesic mappings of equidistant generalized Riemannian spaces

In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.     

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.     

Clifford分析中对偶的k-Hypergenic函数

研究了取值于实Clifford代数空间Cl_(n+1,0)(R)中对偶的k-hypergenic函数.首先,给出了对偶的k-hypergenic函数的一些等价条件,其中包括广义的Cauchy-Riemann方程.其次,给出了对偶的hypergenic函数的Cauchy积分公式,并且应用其证明了(1-n)-hypergenic函数的Cauchy积分公式.最后,证明了对偶的hypergenic函数的Cauchy积分公式右端的积分是U\Ω_2中对偶的hypergenic函数.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

Estimates for a solution to one differential inequality

In a domain D = Ω × (?T,T) we consider a differential inequality whose left-hand side contains a linear second-order hyperbolic operator with coefficients depending only on x ∈ ? ⁿ, n ≥ 2, and the right-hand side contains the modulus of the gradient of the sought function. We supplement the inequality with the Cauchy data on the lateral part of the boundary of D and consider the problem of estimating a solution to the differential inequality satisfying the Cauchy data. We establish the estimate under some assumptions that involves the upper bound of the sectional curvatures of the Riemannian space associated with the differential operator, the Riemannian diameter of Ω, and the length of the interval (?T,T). The result is generalized to the case of compact domains bounded from above and below by characteristic surfaces.     

Asymptotic behavior of the central path for a special class of degenerate SDP problems

This paper studies the asymptotic behavior of the central path (X(ν),S(ν),y(ν)) as ν↓0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose “degenerate diagonal blocks” of the central path are assumed to satisfy We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization t>0→(X(t⁴),S(t⁴),y(t⁴)) of the central path is analytic at t=0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class.This author was supported in part by CAPES and PRONEX-Otimização (FAPERJ/CNPq).This author was supported in part by FUNAPE/UFG, CAPES, PADCT-CNPq and PRONEX-Otimização (FAPERJ/CNPq).This author was supported in part by NSF Grants CCR-9902010, CCR-0203113 and INT-9910084 and ONR grant N00014-03-1-0401.Mathematics Subject Classification (1991): 90C20, 90C22, 90C25, 90C30, 90C33, 90C45, 90C51     

局部凸空间中的不动点定理及其应用

讨论了局部凸空间中推广的Leray-Schauder度的基本性质,建立了一些新的不动点定理,并给出了对局部凸空间Cauchy初值问题的应用.这些定理是Banach空间中相应结果的推广.     

Global existence to generalized Boussinesq equation with combined power-type nonlinearities

The Cauchy problem to the generalized Boussinesq equation with combined power-type nonlinearities is studied. Global solvability or finite time blow-up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the Nehari functional. For generalized Lienard (or generalized Bernoulli) nonlinear terms the critical energy constant is explicitly evaluated. A new method, that can be considered as a modification of the potential well method, is developed. The performed numerical experiments support the theoretical results.     

Robustness of the Hybrid Extragradient Proximal-Point Algorithm

The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error tolerance. We extend the error tolerance of this method, proving that it converges even if a summable error is added to the relative error. Furthermore, the extragradient step may be performed inexactly with a summable error. We present a convergence analysis, which encompasses other well-known variations of the proximal-point method, previously unrelated. We establish weak global convergence under mild assumptions.