a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

on the schrodinger equation for the gaussian potential a exp λr2 the energy eigenvalues of the attractive radial gaussian potential for various eigenstates are very accurately determined within the framework of the hypervirial pade scheme

dirac equation with hulthen potential an algebraic approach the energy levels of the dirac equation with scalar and vector hulthen type potentials are obtained by means of algebraic perturbation calculations which are based upon the dynamical group structure so 2 1 numerical results are given for the particular case when the strengths of the vector and scalar potentials are equal

on the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm in this paper we focus on the problem of identifying the index sets p x i xi 0 n x i fi x 0 and c x i xi fi x 0 for a solution x of the monotone nonlinear complementarity problem ncp f the correct identification of these sets is important from both theoretical and practical points of view such an identification enables us to remove complementarity conditions from the ncp and locally reduce the ncp to a system which can be dealt with more easily we present a new technique that utilizes a sequence generated by the proximal point algorithm ppa using the superlinear convergence property of ppa we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution

simple variational approaches to eigenvalues in quantum theory the authors develop a very simple variational procedure for obtaining quite accurate analytical expressions for the eigenvalues of quantum mechanical models it consists of finding the minimum value of a properly built functional form for the energy in the phase space the method enables one to make use of available information about the analytic structure of the eigenenergies results are shown for the linear confirming potential model

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

group theoretic approach for a dirac particle in coulomb like potentials the energy levels of the dirac equation with a kohn sham ks potential are obtained using algebraic perturbation theory based on the dynamical group structure so 2 1 without making any non relativistic approximation it has been shown that this formalism reproduces the exact analytical result for the eigenvalues of the dirac equation with both vector and scalar coulomb potentials the lowest order results obtained from the analytical formulae are found to be in excellent agreement with exact numerical results

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

local behavior of an iterative framework for generalized equations with nonisolated solutions an iterative framework for solving generalized equations with nonisolated solutions is presented for generalized equations with the structure where is a multifunction and f is single valued the framework covers methods that at each step solve subproblems of the type the multifunction approximates f around s besides a condition on the quality of this approximation two other basic assumptions are employed to show q superlinear or q quadratic convergence of the iterates to a solution a key assumption is the upper lipschitz continuity of the solution set map of the perturbed generalized equation moreover the solvability of the subproblems is required conditions that ensure these assumptions are discussed in general and by means of several applications they include monotone mixed complementarity problems karush kuhn tucker systems arising from nonlinear programs and nonlinear equations particular results deal with error bounds and upper lipschitz continuity properties for these problems

proper complex random processes with applications to information theory the covariance of complex random variables and processes when defined consistently with the corresponding notion for real random variables is shown to be determined by the usual complex covariance together with a quantity called the pseudo covariance a characterization of uncorrelatedness and wide sense stationarity in terms of covariance and pseudo covariance is given complex random variables and processes with a vanishing pseudo covariance are called proper it is shown that properness is preserved under affine transformations and that the complex multivariate gaussian density assumes a natural form only for proper random variables the maximum entropy theorem is generalized to the complex multivariate case the differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum if and only if the random vector is proper gaussian and zero mean the notion of circular stationarity is introduced for the class of proper complex random processes a discrete fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain an application of the theory is presented

a class of gap functions for variational inequalities recently auchmuty 1989 has introduced a new class of merit functions or optimization formulations for variational inequalities in finite dimensional space we develop and generalize auchmuty s results and relate his class of merit functions to other works done in this field especially we investigate differentiability and convexity properties and present characterizations of the set of solutions to variational inequalities we then present new descent algorithms for variational inequalities within this framework including approximate solutions of the direction finding and line search problems the new class of merit functions include the primal and dual gap functions introduced by zuhovickii et al 1969a 1969b and the differentiable merit function recently presented by fukushima 1992 also the descent algorithm proposed by fukushima is a special case from the class of descent methods developed in this paper through a generalization of auchmuty s class of merit functions we extend those inherent in the works of dafermos 1983 cohen 1988 and wu et al 1991 new algorithmic equivalence results relating these algorithm classes to each other and to auchmuty s framework are also given

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

on finite convergence of proximal point algorithms for variational inequalities in this paper we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem vip where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations by using this result we obtain that the well known proximal point algorithm possesses finite convergence if the solution set of vip is weakly sharp as an extension we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g sharpness

shannon theoretic approach to a gaussian cellular multiple access channel with fading shannon theoretic limits on the achievable throughput for a simple infinite cellular multiple access channel mac model wyner 1994 in the presence of fading are presented in this model which is modified to account for flat fading the received signal at a given cell site s antenna is the sum of the faded signals transmitted from all users within that cell plus an attenuation factor spl alpha spl isin 0 1 times the sum of the faded signals received from the adjacent cells accompanied by gaussian additive noise this model serves as a tractable model providing considerable insight into complex and analytically intractable real world cellular communications both linear and planar cellular arrays are considered with exactly k active users in each cell we assume a hyper receiver jointly decoding all of the users incorporating the received signals from all of the active cell sites the hyper receiver is assumed to be aware of the codebooks and realizations of the fading processes of all the users in the system in this work we consider the intracell time division multiple access tdma and the wideband wb protocols we focus on the maximum reliably transmitted equal rate bounds to this rate are found for the intracell tdma protocol by incorporating information theoretic inequalities and the chebyshev markov moment theory as applied to the limiting distribution of the eigenvalues of a quadratic form of tridiagonal random matrices we demonstrate our results for the special case where the amplitudes of the fading coefficients are drawn from a rayleigh distribution i e rayleigh fading for this special case we observe the rather surprising result that fading may increase the maximum equal rate for a certain range of spl alpha as compared to the nonfaded case in this setting the wb strategy which achieves the maximum reliable equal rate of the model is proved to be superior to the tdma scheme an upper bound to the maximum equal rate of the wb scheme is also obtained this bound is asymptotically tight when the number of users is large k spl gt 1 the asymptotic bound shows that the maximum equal rate of the wb scheme in the presence of fading is higher than the rate which corresponds to the nonfaded case for any intercell interference factor spl alpha spl isin 0 1 signal to noise ratio snr values this result is found to be independent of the statistics of the fading coefficients

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

a simple approximation for eigenvalues in quantum theory an approximation method for the determination of energy levels in quantum theory is discussed which starts from a scaled set of eigenstates of a solvable model it is shown that the lowest approximation fulfils certain relations which also hold exactly the approximation is tested by comparison with numerically computed eigenvalues in several cases the errors turn out to be moderate in most of these and depend very little on typical coupling constants

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

a unified approach to error bounds for structured convex optimization problems error bounds which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems in this paper we present a new framework for establishing error bounds for a class of structured convex optimization problems in which the objective function is the sum of a smooth convex function and a general closed proper convex function such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning signal processing and statistics using our framework we show that a number of existing error bound results can be recovered in a unified and transparent manner to further demonstrate the power of our framework we apply it to a class of nuclear norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity type regularity condition we then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition we believe that our approach will find further applications in the study of error bounds for structured convex optimization problems

variational perturbation theory for summing divergent tunnelling amplitudes we present a method for evaluating divergent series with factorially growing coefficients of equal sign the method is based on an analytic continuation of variational perturbation theory from the regime of alternating signs we demonstrate its power first by applying it to the exactly known partition function of the anharmonic oscillator in zero space time dimensions the simple integral then we consider the quantum mechanical case of one space time dimension and derive the imaginary part of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant g including the non analytic tunnelling regime at small g as a highlight of the theory we extract from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble some figures in this article are in colour only in the electronic version

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

an algebraic approach to the exponential cosine screened coulomb potential an analytical formula is obtained for the energy eigenvalues of the exponential cosine screened coulomb potential by a method that is a combination of algebraic perturbation theory based on the lie algebra of the so 2 1 group and the scaling variational method the results are found to be in excellent agreement with pade approximation calculations

a derivative free algorithm for least squares minimization we develop a framework for a class of derivative free algorithms for the least squares minimization problem these algorithms are designed to take advantage of the problem structure by building polynomial interpolation models for each function in the least squares minimization under suitable conditions global convergence of the algorithm is established within a trust region framework promising numerical results indicate the algorithm is both efficient and robust numerical comparisons are made with standard derivative free software packages that do not exploit the special structure of the least squares problem or that use finite differences to approximate the gradients

spectral efficiency of joint multiple cell site processors for randomly spread ds cdma systems a chip interleaved randomly spread direct sequence code division multiple access ds cdma scheme is considered employed in two variants of wyner s infinite linear cell array model with flat fading focusing on the asymptotic setup in which both the number of users per cell and the processing gain go to infinity while their ratio the ldquocell loadrdquo goes to some finite constant the spectral efficiencies of the optimum and linear minimum mean squared error mmse joint multicell receivers are investigated a dramatic performance enhancement as compared to single cell site processing is demonstrated the asymptotic behavior of the two receivers in extreme signal to noise ratio snr regimes and in a high cell load regime are analyzed as well the impact of chip interleaving versus symbol interleaving is also investigated chip level interleaving is found beneficial in several cases of interests and is conjectured to be beneficial in general

ergodic sum capacity of macrodiversity mimo systems in flat rayleigh fading the prospect of base station cooperation leading to joint combining at widely separated antennas has led to increased interest in macrodiversity systems where both sources and receive antennas are geographically distributed in this scenario analytical investigation of channel capacity is extremely challenging for finite size systems since the channel matrices have a very general form where each path may have a different power hence in this paper we consider the ergodic sum capacity of a macrodiversity multiple input multiple output system with arbitrary numbers of sources and receive antennas operating over rayleigh fading channels for this system we compute the exact ergodic capacity for a system with at most two transmit antennas and a compact approximation for the general system which is shown to be very accurate over a wide range of cases finally we compare our results with previous asymptotic results and bounds results are verified by monte carlo simulations and the impact on capacity of various channel power profiles is investigated

the proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem in this paper we consider a proximal point algorithm ppa for solving monotone nonlinear complementarity problems ncp ppa generates a sequence by solving subproblems that are regularizations of the original problem it is known that ppa has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems however it is not always easy to solve subproblems or to check those criteria in this paper we adopt the generalized newton method proposed by de luca facchinei and kanzow to solve subproblems and adopt some ncp functions to check the criteria then we show that the ppa converges globally provided that the solution set of the problem is nonempty moreover without assuming the local uniqueness of the solution we show that the rate of convergence is superlinear in a genuine sense provided that the limit point satisfies the strict complementarity condition

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

shannon theoretic approach to a gaussian cellular multiple access channel we obtain shannon theoretic limits for a very simple cellular multiple access system in our model the received signal at a given cell site is the sum of the signals transmitted from within that cell plus a factor spl alpha 0 spl les spl alpha spl les 1 times the sum of the signals transmitted from the adjacent cells plus ambient gaussian noise although this simple model is scarcely realistic it nevertheless has enough meat so that the results yield considerable insight into the workings of real systems we consider both a one dimensional linear cellular array and the familiar two dimensional hexagonal cellular pattern the discrete time channel is memoryless we assume that n contiguous cells have active transmitters in the one dimensional case and that n sup 2 contiguous cells have active transmitters in the two dimensional case there are k transmitters per cell most of our results are obtained for the limiting case as n spl rarr spl infin the results include the following 1 we define c sub n c spl circ sub n as the largest achievable rate per transmitter in the usual shannon theoretic sense in the one and two dimensional cases respectively assuming that all signals are jointly decoded we find expressions for limn spl rarr spl infin c sub n and limn spl rarr spl infin c spl circ sub n 2 as the interference parameter spl alpha increases from 0 c sub n and c spl circ sub n increase or decrease according to whether the signal to noise ratio is less than or greater than unity 3 optimal performance is attainable using tdma within the cell but using tdma for adjacent cells is distinctly suboptimal 4 we suggest a scheme which does not require joint decoding of all the users and is in many cases close to optimal

impact of antenna correlation on the capacity of multiantenna channels this paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels the analysis is not restricted to the popular separable correlation model but rather it embraces a more general representation that subsumes most of the channel models that have been treated in the literature for arbitrary signal to noise ratios snr the characterization is conducted in the regime of large numbers of antennas for the low and high snr regions in turn we uncover compact capacity expansions that are valid for arbitrary numbers of antennas and that shed insight on how antenna correlation impacts the tradeoffs among power bandwidth and rate