primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

on implementing a primal dual interior point method for conic quadratic optimization based on the work of the nesterov and todd on self scaled cones an implementation of a primal dual interior point method for solving large scale sparse conic quadratic optimization problems is presented the main features of the implementation are it is based on a homogeneous and self dual model it handles rotated quadratic cones directly it employs a mehrotra type predictor corrector extension and sparse linear algebra to improve the computational efficiency finally the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems this is a novel feature for our implementation computational results are also presented to document that the implementation can solve very large problems robustly and efficiently

the berry phase for spin in the majorana representation a continuous cyclic sequence of quantum states has an associated geometric or berry phase for spin j such a sequence is described by a cyclic change in the 2j 1 coefficients of the basis states the berry phase is analysed here for the general case that is the coefficients are allowed to vary in an arbitrary cyclic manner the result is expressed in geometric terms specifically in the democratic representation due to majorana this uniquely characterizes the spin state up to overall phase by the positions of 2j dots on the unit sphere of directions in real space if the positions are denoted by unit vectors where each traces out a parametrized loop on the sphere and the berry phase is given by an integral of combinations of these vectors

a one step smoothing newton method for second order cone programming a new smoothing function for the second order cone programming is given by smoothing the symmetric perturbed fischer burmeister function based on this new function a one step smoothing newton method is presented for solving the second order cone programming the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration this algorithm does not have restrictions regarding its starting point and is q quadratically convergent numerical results suggest the effectiveness of our algorithm

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

solving a class of asymmetric variational inequalities by a new alternating direction method abstract the augmented lagrangian method also referred to as an alternating direction method solves a class of variational inequalities vi via solving a series of sub vi problems the method is effective whenever the subproblems can be solved efficiently however the subproblem to be solved in each iteration of the augmented lagrangian method itself is still a vi problem it is essentially as difficult as the original one the only difference is that the dimension of the subproblems is lower in this paper we propose a new alternating direction method for solving a class of monotone variational inequalities in each iteration the method solves a convex quadratic programming with simple constrains and a well conditioned system of nonlinear equations for such easier subproblems existing efficient numerical softwares are applicable the effectiveness of the proposed method is demonstrated with an illustrative example

weak coherent states related to the multiplicative group mathbb c generalized coherent states which are associated with the multiplicative group of non zero complex numbers are introduced they have the property of temporal stability and they form a total set in some subspace of the full hilbert space they are highly non classical states in the sense that in general their bargmann functions have zeros which are related to regions where their wigner functions take negative values states with bargmann functions sin az and exp bz az k are examples of the general formalism and they are discussed in detail

an inexact smoothing method for soccps based on a one parametric class of smoothing function in this paper we introduce a one parametric class of smoothing functions which contains the fischer burmeister smoothing function as special case based on this class of smoothing functions an inexact smoothing method for solving second order cone complementarity problems soccps is proposed in each iteration the corresponding linear system is solved only approximately under mild assumptions it is proved that the proposed method has global convergence and local superlinear convergence properties preliminary numerical results indicate that the method is effective for large scale problems

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

on the positivity of symmetric polynomial functions part i general results abstract we prove that a real symmetric polynomial inequality of degree d 2 holds on r n if and only if it holds for elements with at most d 2 distinct non zero components which may have multiplicities we establish this result by solving a cauchy problem for ordinary differential equations involving the symmetric power sums this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function in the final section extensions of our results to the whole space r n are outlined the main results are theorems 5 1 and 5 2 with corollaries 2 1 and 5 2 and the corresponding results for r n from the last subsection part ii will contain a discussion on the ordered vector space h d n in general as well as on the particular cases of degrees d 4 and d 5 finite test sets for positivity in the homogeneous case and other sufficient criteria

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

finite termination of a newton type algorithm based on a new class of smoothing functions for the affine variational inequality problem in this paper we propose a new class of smoothing functions some favorable properties of the functions are investigated by using the proposed functions the affine variational inequality problem avi is reformulated as a system of parameterized smooth equations a newton method with a projection type testing procedure is proposed to solve the equations under mild assumptions we show that the algorithm may find a maximally complementary solution to the monotone avi in a finite number of iterations preliminary numerical results indicate that the proposed smoothing functions are valuable

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

fractional monodromy in systems with coupled angular momenta we present a one parameter family of systems with fractional monodromy which arises from a 1 2 diagonal action of a dynamical symmetry so 2 i n a regime of adiabatic separation of slow and fast motions we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi quantum spectra

discriminants of cyclic homogeneous inequalities of three variables abstract in this article we study the structure of the cone of semidefinite forms it is a closed semialgebraic set but usually is not basic closed semialgebraic set a discriminant is a defining equation of an irreducible component of algebraic boundary of this cone we calculate discriminants using new tools characteristic variety and local cones a characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions local cone is a subcone of the psd cone which corresponds to a maximal ideal this theory works well for a family of polynomials which are invariant under an action of a finite group after we construct an abstract general theory we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d we calculate some discriminants for d 3 4 5 and 6 and we show that this theory derives many new results

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions

exploiting symmetries in sdp relaxations for polynomial optimization in this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations our special focus is on constrained problems especially when the symmetric group is acting on the variables in particular we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute the geometric quotient

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

a projection based prediction correction method for structured monotone variational inequalities in this paper we propose a modified prediction correction method for structured monotone variational inequalities this method is based on a reformulation of the normal prediction correction method each iteration of the new method contains two predictions and a correction at each iteration a new strategy is applied to obtain a proper step length all the computing processes are easily implemented and the global convergence is also presented under mild assumptions numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

a two stage prediction correction method for solving monotone variational inequalities in this paper we present a two stage prediction correction method for solving monotone variational inequalities the method generates the two predictors which should satisfy two acceptance criteria we also enhance the method with an adaptive rule to update prediction step size which makes the method more effective under mild assumptions we prove the convergence of the proposed method our proposed method based on projection only needs the function values so it is practical and the computation load is quite tiny some numerical experiments were carried out to validate its efficiency and practicality

a smoothing method for second order cone complementarity problem in this paper the second order cone complementarity problem is studied based on a perturbed symmetrically smoothing function which has coerciveness under proper conditions we present a smoothing newton method for this problem the boundedness of the level set can be obtained from the coerciveness which plays an important role in the convergence analysis furthermore the proposed algorithm for the reformulation has no restrictions on the starting point and solves only one system of equations preliminary numerical results indicate that the algorithm is effective

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

a modified augmented lagrangian method for a class of monotone variational inequalities abstract the augmented lagrangian method is an attractive approach for solving linearly constrained convex optimization problems ccop the method is iterative and the task in each iteration is to solve a convex optimization sub problem whose dimension is equal to the number of variables for large problems solving such a sub problem can be still very expensive in this paper we propose a modified augmented lagrangian method for a class of monotone variational inequalities which include ccop the computational load in each iteration of the proposed method is quite small in order to demonstrate the ability and efficiency of the proposed method we present some numerical results for convex nonlinear programming and traffic network equilibrium problems

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

a heuristic algorithm for network equilibration abstract in this paper a heuristic algorithm different from the frank wolfe and its modified methods is introduced for network equilibration by using the column generation technique and the network equilibrium conditions the new method need not enumerate initially all feasible paths for all origin destination o d pairs but can give all paths used between each o d pair and the path flows accordingly while the new algorithm obtains an optimal traffic assignment some convergence issues of the new method is discussed in this paper numerical experiments show that the new method is efficient and robust

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

phase retrieval problem in quantum chaos and its relation to the origin of irreversibility i abstract the purpose of this paper are i to study characteristics features of phase in quantum dynamics corresponding to chaos and ii to discuss implication of these features for the origin of irreversibility the first is done by focusing attention on topological defects of phase in the husimi representation we find that the process of generating topological defects corresponds to the folding of morse smale horseshoe dynamics moreover the distribution of topological defects changes from regular to random under the time evolution this change is a distinctive feature of quantum dynamics corresponding to chaos the second is done in terms of phase retrieval problem we present numerical results which strongly suggest the following the randomness in the distribution of topological defects is the cause of difficulty in retrieving phase based on these results we present the following hypothesis the loss of phase information in statistical mechanics can be understood in terms of the randomness of phase in quantum dynamics corresponding to chaos

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac