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.     

A Rotating Quantum Vacuum

We investigate how a uniformly rotating frame is defined as the rest frame of an observer rotating with constant angular velocity around the z axis of an inertial frame. Assuming this frame to be a Lorentz one, we second quantize a free massless scalar field in the rotating frame and obtain that creation-annihilation operators of the field are not the same as those of an inertial frame. This leads to a new vacuum state—a rotating vacuum. After this, introducing an apparatus device coupled linearly with the field, we obtain that there is a strong correlation between the number of Trocheries-Takeno particles (in a given state) obtained via canonical quantization and the response function of the rotating detector. Finally, we analyze polarization effects in circular accelerators in the proper frame of the electron, making a connection with the inertial frame point of view.     

A new smoothing-regularization approach for a maximum-likelihood estimation problem

We consider the problem min _i=1 ^m (aⁱ,x–b_iloga ⁱ, z) subject tox 0 which occurs as a maximum-likelihood estimation problem in several areas, and particularly in positron emission tomography. After noticing that this problem is equivalent to mind(b, Ax) subject tox 0, whered is the Kullback-Leibler information divergence andA, b are the matrix and vector with rows and entriesa ⁱ,b _i, respectively, we suggest a regularized problem mind(b, Ax) + d(v, Sx), where is the regularization parameter,S is a smoothing matrix, andv is a fixed vector. We present a computationally attractive algorithm for the regularized problem, establish its convergence, and show that the regularized solutions, as goes to 0, converge to the solution of the original problem which minimizes a convex function related tod(v, Sx). We give convergence-rate results both for the regularized solutions and for their functional values.The research of A. N. Iusem was partially supported by CNPq Grant No. 301280/86-MA.     

A Class of Fejér Convergent Algorithms,Approximate Resolvents and the Hybrid Proximal-Extragradient Method

A new framework for analyzing Fejér convergent algorithms is presented. Using this framework, we define a very general class of Fejér convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents, which preserves some useful features of the exact resolvent and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng’s Modified Forward-Backward splitting method, and Korpelevich’s method. We show that methods, based on families of approximate resolvents, fall within the aforementioned class of Fejér convergent methods. We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method, which is a generalization of the classical Proximal Point Algorithm.     

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–?ojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.     

A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed   relative residual error tolerance converges Q$Q$-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance.     

Analytic center of spherical shells and its application to analytic center machine

The two-case pattern recognition problem aims to find the best way of linearly separate two different classes of data points with a good generalization performance. In the context of learning machines proposed to solve the pattern recognition problem, the analytic center machine (ACM) uses the analytic center cutting plane method restricted to spherical shells. In this work we prove existence and uniqueness of the analytic center of a spherical surface, which guarantees the well definedness of ACM problem. We also propose and analyze new primal, dual and primal-dual formulations based on interior point methods for the analytic center machine. Further, we provide a complexity bound on the number of iterations for the primal approach. F.M.P. Raupp was partially supported by CNPq Grant 475647/2006-8 and FAPERJ/CNPq through PRONEX-Computational Modeling. B.F. Svaiter was partially supported by CNPq Grants 300755/2005-8, 475647/2006-8 and by FAPERJ/CNPq through PRONEX-Optimization.