Partial Regularization Research Articles

An Extension of the Auxiliary Problem Principle to Nonsymmetric Auxiliary Operators

To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average.

Partial regularization of the sum of two maximal monotone operators

On analyse ici un algorithme qui recherche un zero de la somme de deux operateurs maximaux monotones. On resoud une sequence de problemes regularises dont la solution converge vers la solution cherchee quand le parametre de regularisation tend vers zero. On evite alors de se restreindre a la convergence ergodique. Le resultat principal est une generalisation d'un theoreme de Brezis qui a considere des operateurs de la forme 1 + A + B. On peut alors raffiner ces resultats dans le cas fortement monotone. Finalement, on propose un schema general d'algorithme de decomposition pour la programmation convexe

Classical phase-space structure of the single-particle motion in cranked potentials

The phase-space structure of the single-particle motion in three-dimensional cranked Buck-Pilt potentials of lemniscatoidic shape has been investigated in terms of Poincare surfaces of section, the mean value of positive maximal Ljapunov exponents and the chaotic fraction of phase space as a function of the rotational frequency. Due to the rotation of a partial regularization of the particle, motion was found for certain values of deformation and frequency.

Search for generalized, quantizable dynamical models

A precisation of the quantization procedure using a separable Hilbert space and coherent states (of minimal dispersion in phase space) liberates quantum formalism of the difficulties connected with an infinite volume and continuous spectra. A regularization with or without a final transition with subsidiary masses to infinity is investigated, and consequences of a possible breakdown of unitarity are discussed. Gauge theories, even supplemented only by a global supersymmetry, bring about a partial regularization, i.e. some compensations of infinities, so that the theory acquires a status of superrenormalizability. Some alternatives of strings and superstrings, based on the old ideas of Peierls and McManus or Yukawa (nonlocal and bilocal fields) are investigated after a generally covariant reformulation. The resulting quantum fields no longer satisfy field equations of traditional form but are describable by an operator of evolution in time with a generalized prescription of ordering in time, and are expressible in a new Picture intermediate between the Schrödinger and the Interaction Pictures.

Critical point approximation through exact regularization

We present several iterative methods for finding the critical points and/or the minima of a functional which is essentially the difference between two convex functions. The underlying idea relies upon partial and exact regularization of the functional, which allows us to preserve the local feature in a large number of applications, as well as to obtain some convergence results. These methods are further applied to some differential problems of the semilinear elliptic type arising in plasma physics and fluid mechanics.

On Representations of Semi-Infinite Programs which Have No Duality Gaps

Duality gaps which may occur in semi-infinite programs are shown to be interpretable as a phenomenon of an improper representation of the constraint set, uTPi ≧ ci, i ε I. Thus, any semi-infinite system of linear inequalities has a canonically closed equivalent (with interior points) which has no duality gap. With respect to the original system of inequalities, duality gaps may be closed by adjoining additional linear inequalities to the original system. Also, for consistent, but not necessarily canonically closed programs, a partial regularization of original data removes duality gaps that may occur. In contrast, a new “weakly consistent” duality theorem without duality gap may have a value determined by an inequality which is strictly redundant with respect to the constraint set defined by the total inequality system.