on g inverses and the nonsingularity of a bordered matrixabc0 abstract we consider the bordered matrix m a b c 0 with the rank additivity condition and g c 1 c 2 c 3 c 4 which is a g inverse of m we give the necessary and sufficient conditions for c 1 to be a g inverse outer inverse reflexive g inverse and moore penrose inverse of a as a special case we give necessary and sufficient conditions for m to be nonsingular and explicit expressions of c i in m 1 moreover we give a method for computing any outer inverse of a matrix

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria let h r be a c1 function on a real hilbert space h and let 0 be a positive damping parameter for any control function e r r which tends to zero as t we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator hbfc x t x t x t e t x t 0 we show that if e t does not tend to zero too rapidly as t then the term e t x t asymptotically acts as a tikhonov regularization which forces the trajectories to converge to a particular equilibrium indeed in the main result of this paper it is established that when is convex and s argmin under the key assumption that e is a slow control i e 0 e t dt then each trajectory of the hbfc system strongly converges as t to the element of minimal norm of the closed convex set s as an application we consider the damped wave equation with neumann boundary condition formula

schemes for finding minimum norm solutions of variational inequalities abstract consider the variational inequality vi of finding a point x such that x f i x t and i s x x x 0 x f i x t where t s are nonexpansive self mappings of a closed convex subset c of a hilbert space and f i x t is the set of fixed points of t assume that the solution set of this vi is nonempty this paper introduces two schemes one implicit and one explicit that can be used to find the minimum norm solution of vi namely the unique solution x to the quadratic minimization problem x arg min x x 2

convergence of a hybrid projection proximal point algorithm coupled with approximation methods in convex optimization in order to minimize a closed convex function that is approximated by a sequence of better behaved functions we investigate the global convergence of a general hybrid iterative algorithm which consists of an inexact relaxed proximal point step followed by a suitable orthogonal projection onto a hyperplane the latter permits to consider a fixed relative error criterion for the proximal step we provide various sets of conditions ensuring the global convergence of this algorithm the analysis is valid for nonsmooth data in infinite dimensional hilbert spaces some examples are presented focusing on penalty barrier methods in convex programming we also show that some results can be adapted to the zero finding problem for a maximal monotone operator

proximal point algorithm controlled by a slowly vanishing term applications to hierarchical minimization let phi 0 mathbb r n to mathbb r cup infty be a closed convex function and phi 1 mathbb r n to mathbb r be a finite convex function that are bounded from below our goal is to build an algorithm which first minimizes the map phi 0 and secondly the map phi 1 over the set s 0 rm argmin phi 0 for that purpose we define the following proximal type algorithm x n 1 x n lambda n in partial eta n phi 0 varepsilon n phi 1 x n 1 leqno mathcal a 1 where lambda n is a positive step sequence eta n is a summable error sequence and varepsilon n is a control sequence tending toward 0 partial eta denotes the eta approximate subdifferential when varepsilon n is a slow control i e sum n 0 infty varepsilon n infty we prove that under adequate conditions the sequence x n defined by mathcal a 1 tends toward an element of s 1 rm argmin s 0 phi 1 more generally given finite convex functions phi 2 ldots phi n mathbb r n to mathbb r let us define the sets s i i in 1 ldots n by the recursive relation s i rm argmin s i 1 phi i we introduce an extension of algorithm mathcal a 1 to minimize hierarchically each function phi i on the set s i 1 for i in 1 ldots n

on the rationality of cycle integrals of meromorphic modular forms we derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms moreover we prove the modularity of a completion of the generating function of such traces the theoretical framework for these results is an extension of the shintani theta lift to meromorphic modular forms of positive even weight

regularized algorithms for hierarchical fixed point problems abstract let c be a nonempty closed convex subset of a real hilbert space h let s c c be a non expansive mapping and t i i 1 c c be an infinite family of non expansive mappings the purpose of this paper is to find the minimum norm solution of the following general hierarchical fixed point problem find x n 1 f i x t n such that x s x x x 0 x n 1 f i x t n we introduce an explicit regularized algorithm with strong convergence for finding the minimum norm solution of the above hierarchical fixed point problem

hybrid methods for a class of monotone variational inequalities abstract we use contractions to regularize a class of monotone variational inequalities where the monotone operators are complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings both implicit and explicit schemes are shown to be strongly convergent an application in hierarchical minimization is included

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

arithmetic properties of coefficients of half integral weight maass poincare series zagier 23 proved that the generating functions for the traces of level 1 singular moduli are weight 3 2 modular forms he also obtained generalizations for twisted traces and for traces of special non holomorphic modular functions using properties of kloosterman salie sums and a well known reformulation of salie sums in terms of orbits of cm points we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half integral weight poincare series in kohnen s 0 4 plus space these results imply the aforementioned results of zagier and they provide exact formulas for such traces

asymptotic control of pairs of oscillators coupled by a repulsion with nonisolated equilibria ii the singular case let phi h rightarrow mathbb r be a mathcal c 1 function on a real hilbert space h and let gamma 0 be a positive damping parameter for any singular repulsive potential v h setminus 0 to mathbb r i e such that lim z to 0 v z infty and any control function varepsilon mathbb r to mathbb r setminus 0 which tends to zero as t to infty we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators left begin array l ddot x gamma dot x nabla phi x varepsilon t nabla v x y 0 vspace 1mm ddot y gamma dot y nabla phi y varepsilon t nabla v x y 0 end array right leqno rm hbfc 2 sing this system is the singular version of the regular rm hbfc 2 reg system studied in a cabot and m o czarnecki siam j control optim 41 2002 pp 1254 1280 where the potential v is defined on the whole space h the purpose of this paper is to obtain whenever possible the same existence and convergence results in the singular case as in the regular case considered by a cabot and m o czarnecki this study is mainly motivated by a better convergence behavior of rm hbfc 2 sing with the same sharp condition on the control varepsilon exhibited in h attouch and m o czarnecki j differential equations 179 2002 pp 278 310 and by a cabot and m o czarnecki precisely when h mathbb r and if varepsilon is a slow control i e int 0 infty varepsilon t dt infty then the trajectories x and y converge to extremal points of the set s lambda in mathbb r nabla phi lambda 0 of the equilibria of phi the awkward case in a cabot and m o czarnecki where the trajectories may have the same limit disappears of importance from a physical point of view we thus can consider actual for example electromagnetic repulsive potentials

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

vi constrained hemivariational inequalities distributed algorithms and power control in ad hoc networks we consider centralized and distributed algorithms for the numerical solution of a hemivariational inequality hvi where the feasible set is given by the intersection of a closed convex set with the solution set of a lower level monotone variational inequality vi the algorithms consist of a main loop wherein a sequence of one level strongly monotone hvis are solved that involve the penalization of the non vi constraint and a combination of proximal and tikhonov regularization to handle the lower level vi constraints minimization problems possibly with nonconvex objective functions over implicitly defined vi constraints are discussed in detail the methods developed in the paper are then used to successfully solve a new power control problem in ad hoc networks

a bundle method for a class of bilevel nonsmooth convex minimization problems we consider the bilevel problem of minimizing a nonsmooth convex function over the set of minimizers of another nonsmooth convex function standard convex constrained optimization is a particular case in this framework corresponding to taking the lower level function as a penalty of the feasible set we develop an explicit bundle type algorithm for solving the bilevel problem where each iteration consists of making one descent step for a weighted sum of the upper and lower level functions after which the weight can be updated immediately convergence is shown under very mild assumptions we note that in the case of standard constrained optimization the method does not require iterative solution of any penalization subproblems not even approximately and does not assume any regularity of constraints e g the slater condition we also present some computational experiments for minimizing a nonsmooth convex function over a set defined by linear complementarity constraints

real and complex monotone communication games noncooperative game theoretic tools have been increasingly used to study many important resource allocation problems in communications networking smart grids and portfolio optimization in this paper we consider a general class of convex nash equilibrium problems neps where each player aims at solving an arbitrary smooth convex optimization problem differently from most of current works we do not assume any specific structure for the players problems and we allow the optimization variables of the players to be matrices in the complex domain our main contribution is the design of a novel class of distributed asynchronous best response algorithms suitable for solving the proposed neps even in the presence of multiple solutions the new methods whose convergence analysis is based on variational inequality vi techniques can select among all the equilibria of a game those that optimize a given performance criterion at the cost of limited signaling among the players this is a major departure from existing best response algorithms whose convergence conditions imply the uniqueness of the ne some of our results hinge on the use of vi problems directly in the complex domain the study of these new kind of vis also represents a noteworthy innovative contribution we then apply the developed methods to solve some new generalizations of single input single output siso and multiple input multiple output mimo games in cognitive radio systems showing a considerable performance improvement over classical pure noncooperative schemes

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

primal and dual convergence of a proximal point exponential penalty method for linear programming we consider the diagonal inexact proximal point iteration frac u k u k 1 lambda k in partial varepsilon k f u k r k nu k where f x r ctx r exp a i x b i r is the exponential penalty approximation of the linear program min ctx ax b we prove that under an appropriate choice of the sequences k e k and with some control on the residual k for every r k 0 the sequence u k converges towards an optimal point u of the linear program we also study the convergence of the associated dual sequence i k exp a i u k b i r k towards a dual optimal solution