benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

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

a class of differential quadratic programming problems a class of differential quadratic programming problems in finite dimensional spaces is introduced in this paper first the existence of solutions for the differential quadratic programming problem is established under some suitable assumptions second an algorithm for solving the differential quadratic programming problem is given and the convergence analysis for the algorithm is shown finally some numerical experiments are reported to verify the validity of the proposed algorithm

on the numerical integration of the schrodinger equation a finite difference method for solving the schrodinger eigenvalue equation is generalised in order to treat a larger number of potentials results are shown for asymmetrical two well oscillators

finite difference order doubling in two dimensions an order doubling process previously used to obtain eighth order eigenvalues from the fourth order numerov method is applied to the perturbed oscillator in two dimensions a simple method of obtaining high order finite difference operators is reported and an odd parity boundary condition is found to be effective in facilitating the smooth operation of the order doubling process

error exponents for dimension matched vector multiple access channels with additive noise we analyze a class of vector multiple access channels with additive noise where the sum of the dimensions of the transmitted signals matches that of the received signal we first focus on the case without power constraints in the poltyrev sense using point process techniques we find the poltyrev capacity region for noise processes that are independent and identically distributed over channel uses for each rate vector strictly in the poltyrev capacity region we study for each subset of the transmitters the exponent of the decay in block length of the smallest possible probability that decoding results in error for each transmitter in that subset in the case of independent and identically distributed gaussian noise with arbitrary positive definite covariance matrix we derive random coding exponents for each type of error event these are lower bounds to the true error exponents this also leads to random coding error exponents in the traditional power constrained case where the power constraint at each transmitter is defined by an arbitrary positive definite matrix

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods

convex envelopes of some quadratic functions over the n dimensional unit simplex in this paper we discuss convex envelopes over the unit simplex for some quadratic functions the discussion exploits the combinatorics related to quadratic functions over the unit simplex after recalling the definition of the underlying convex graph of a quadratic function i e the graph whose nodes are the vertices of the unit simplex and whose edges correspond to the edges of the unit simplex along which the function is strictly convex it is shown that the analytical formula of the convex envelope for quadratic functions whose underlying graph is triangle free is the maximum of a finite number of functions each of which is related to a spanning forest of the graph the special case of an underlying star graph is discussed and computational experiments based on a decomposition of a general quadratic function into a sum of quadratic functions with an underlying star graph are presented

semidefinite relaxations for quadratically constrained quadratic programming a review and comparisons at the intersection of nonlinear and combinatorial optimization quadratic programming has attracted significant interest over the past several decades a variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps the primary purpose of this paper is to present a systematic comparison of sdp relaxations for qcqp using theoretical analysis it is shown that the recently developed doubly nonnegative relaxation is equivalent to the shor relaxation when the latter is enhanced with a partial first order relaxation linearization technique these two relaxations are shown to theoretically dominate six other sdp relaxations a computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations while providing relaxation gaps averaging 3 as opposed to gaps of up to 19 for weaker relaxations on 700 randomly generated problems with up to 60 variables an sdp relaxation derived from lagrangian relaxation after the addition of redundant nonlinear constraints to the primal achieves gaps averaging 13 in a few cpu seconds

boolean functions and finite difference eigenvalues finite difference calculations involving the expectation values of boolean functions are shown to yield detailed information about the properties of the energy levels and eigenfunctions for double well potentials

convex relaxations for nonconvex quadratically constrained quadratic programming matrix cone decomposition and polyhedral approximation we present a decomposition approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming qcqp we first develop a general conic program relaxation for qcqp based on a matrix decomposition scheme and polyhedral piecewise linear underestimation by employing suitable matrix cones we then show that the convex conic relaxation can be reduced to a semidefinite programming sdp problem in particular we investigate polyhedral underestimations for several classes of matrix cones including the cones of rank 1 and rank 2 matrices the cone generated by the coefficient matrices the cone of positive semidefinite matrices and the cones induced by rank 2 semidefinite inequalities we demonstrate that in general the new sdp relaxations can generate lower bounds at least as tight as the best known sdp relaxations for qcqp moreover we give examples for which tighter lower bounds can be generated by the new sdp relaxations we also report comparison results of different convex relaxation schemes for nonconvex qcqp with convex quadratic linear constraints nonconvex quadratic constraints and 0 1 constraints

the hvt technique and the treatment of two basic inequalities the quantum mechanical hypervirial theorems hvt technique is used in treating two basic inequalities relating the ground state mean square radius of the orbit of a particle in a central potential and its kinetic energy respectively to the spacing of the two lowest energy levels e for quite a wide class of those potentials the parameters of which also lead to a sufficiently small dimensionless quantity s the difference between the two sides of each inequality is of order s3 and higher while e is of order s and higher and thus it is expected in general to be quite small

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

improved upper bounds on the reliability function of the gaussian channel a new lower bound on the distance distribution of spherical codes is derived this yields two new upper bounds on the reliability function of the gaussian channel these bounds outperform previously known bounds and imply a new range of rates for which the exact value of the reliability function is known

exact random coding exponents and universal decoders for the asymmetric broadcast channel this paper contains two main contributions concerning the asymmetric broadcast channel the first is an analysis of the exact random coding error exponents for both users and the second is the derivation of universal decoders for both users these universal decoders are certain variants of the maximum mutual information universal decoder and they achieve the corresponding random coding exponents of optimal decoding in addition we introduce some lower bounds which involve optimizations over very few parameters unlike the original exact exponents which involve minimizations over auxiliary probability distributions numerical results for the binary symmetric broadcast channel show improvements over previously derived error exponents for the same model

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

on the reliability function of the discrete memoryless relay channel bounds on the reliability function for the discrete memoryless relay channel are derived using the method of types two achievable error exponents are derived based on partial decode forward and compress forward which are well known superposition block markov coding schemes the derivations require combinations of the techniques involved in the proofs of csiszar korner marton s packing lemma for the error exponent of channel coding and marton s type covering lemma for the error exponent of source coding with a fidelity criterion the decode forward error exponent is evaluated on sato s relay channel from this example it is noted that to obtain the fastest possible decay in the error probability for a fixed effective coding rate one ought to optimize the number of blocks in the block markov coding scheme assuming the blocklength within each block is large an upper bound on the reliability function is also derived using ideas from haroutunian s lower bound on the error probability for point to point channel coding with feedback

copositivity and constrained fractional quadratic problems we provide completely positive and copositive optimization formulations for the constrained fractional quadratic problem cfqp and standard fractional quadratic problem stfqp based on these formulations semidefinite programming relaxations are derived for finding good lower bounds to these fractional programs which can be used in a global optimization branch and bound approach applications of the cfqp and stfqp related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

generalized bilinear programming part i models applications and linear programming relaxation abstract the evolution of the bilinear programming problem is reviewed and a new more general model is discussed the model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed the class of problems under consideration contains traditional bilinear programs general quadratic programs and bilinearly constrained and quadratically constrained extensions of these problems we describe how several important applications from the literature including the multiple modular design problem can be modeled as generalized bilinear programs finally we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound

a parametric linear relaxation algorithm for globally solving nonconvex quadratic programming abstract in this article we present a parametric linear relaxation algorithm for globally solving the nonconvex quadratic programming nqp in this algorithm a new parametric linearized technique is proposed for generating parametric linear relaxation programming plrp of the nqp which can be used to determine the lower bound of global minimum value of the nqp to improve the convergent speed of the proposed algorithm a pruning operation is employed to compress the investigated region by subdividing subsequently the initial domain and solving subsequently a series of parametric linear relaxation programming problems over the subdivided domain the proposed algorithm is convergent to the global minimum of the nqp finally an engineering problem for the design of heat exchanger network and some test examples are used to verify the effectiveness of the proposed algorithm

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

a spatial branch and cut method for nonconvex qcqp with bounded complex variables we develop a spatial branch and cut approach for nonconvex quadratically constrained quadratic programs with bounded complex variables cqcqp linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of cqcqp these valid inequalities are derived from the convex hull description of a nonconvex set of 2 times 2 positive semidefinite hermitian matrices subject to a rank one constraint we propose branching rules based on an alternative to the rank one constraint that allows for local measurement of constraint violation closed form bound tightening procedures are used to reduce the domain of the problem we apply the algorithm to solve the alternating current optimal power flow problem with complex variables as well as the box constrained quadratic programming problem with real variables

on convex relaxations for quadratically constrained quadratic programming we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints let mathcal f denote the feasible region for the linear constraints we first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on mathcal f is dominated by an alternative methodology based on convexifying the range of the quadratic form genfrac 0 0pt 1 x genfrac 0 0pt 1 x t for x in mathcal f we next show that the use of alpha bb underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms finally we show that the use of a large class of d c difference of convex underestimators is dominated by a relaxation that combines semidefiniteness with rlt constraints

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

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

global optimization algorithm for mixed integer quadratically constrained quadratic program abstract mixed integer quadratic programs with quadratic constraints miqqp occur frequently in various areas of engineering practice and management science but most solution methods for this kind of problems are often designed for its special cases in this paper we present a simple global optimization algorithm for solving problem miqqp we first convert problem miqqp into an equivalent generalized bilinear programming problem with integer variables eiqqp we next show that replacing the quadratic objective and constraint functions with their convex envelopes is dominated by an alternative methodology based on convexifying the range of the bilinear terms on the feasible region finally by incorporating the reduction correction techniques and sampling strategies into the branch and bound scheme the proposed algorithm is developed for solving miqqp convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and minlplib2 are carried out to validate the performance of the proposed algorithm

error bounds for monomial convexification in polynomial optimization convex hulls of monomials have been widely studied in the literature and monomial convexifications are implemented in global optimization software for relaxing polynomials however there has been no study of the error in the global optimum from such approaches we give bounds on the worst case error for convexifying a monomial over subsets of this implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately our main error bounds depend primarily on the degree of the monomial making them easy to compute since monomial convexification studies depend on the bounds on the associated variables in the second part we conduct an error analysis for a multilinear monomial over two different types of box constraints as part of this analysis we also derive the convex hull of a multilinear monomial over

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

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

on polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

error exponent for multiple access channels upper bounds the problem of bounding the reliability function of a multiple access channel mac is studied two new upper bounds on the error exponent of a two user discrete memoryless dm mac are derived the first bound sphere packing is an upper bound on the exponent of the average probability of error and is the first bound of this type that is zero outside the capacity region and thus results in a tighter sphere packing exponent when compared with the tightest known exponent derived by haroutunian the second bound minimum distance is an upper bound on the exponent of the maximal as opposed to average probability of error to obtain this bound first an upper bound on the minimum bhattacharyya distance between codeword pairs is derived for a certain class of two user dm macs an upper bound on the exponent of maximal probability of error is derived as a consequence of the upper bound on the minimum bhattacharyya distance we analytically evaluate the sphere packing bound for uniform composition codes for an additive and nonsymmetric channel and show that it is tight near the boundary of the capacity region i e equal to the random coding lower bound

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

polynomial type eigenfunctions the differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial the problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form properties of the eigenvalues of this matrix are investigated since they are functions of a parameter occurring in the positive definite quadratic form the asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

on the improvement of convergence of hill determinants we propose a simple approach to analytically determine a parameter of a convergence factor exp gx2 4 in the hill determinant method it provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

a matrix method for power series potentials several useful techniques are described for treating power series potentials these include a moving row matrix nested multiplication method for constructing the hamiltonian matrix a scaled folding algorithm to find the eigenvalues and an energy shift method which gives expectation values and eigencolumn elements two test problems are treated using the techniques in particular it is found to be possible to treat a triple well problem using only a simple single oscillator basis set

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

error exponents of optimum decoding for the interference channel exponential error bounds for the finite alphabet interference channel ifc with two transmitter receiver pairs are investigated under the random coding regime our focus is on optimum decoding as opposed to heuristic decoding rules that have been used in previous works like joint typicality decoding decoding based on interference cancellation and decoding that considers the interference as additional noise indeed the fact that the actual interfering signal is a codeword and not an independent and identically distributed i i d noise process complicates the application of conventional techniques to the performance analysis of the optimum decoder using analytical tools rooted in statistical physics we derive a single letter expression for error exponents achievable under optimum decoding and demonstrate strict improvement over error exponents obtainable using suboptimal decoding rules but which are amenable to more conventional analysis

a duality bounds algorithm for non convex quadratic programs with additional multiplicative constraints this article presents a duality bounds algorithm for globally solving a non convex quadratic programming problem p that contains several additional multiterm multiplicative constraints to our knowledge little progress has been made so far for globally solving p the algorithm uses a branch and bound scheme where lagrange duality theory is used to obtain the lower bounds as a result the lower bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently convergence of the algorithm is proved and a solved sample problem is given to show the feasibility of the proposed algorithm

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

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube