hopf transcritical bifurcation in toxic phytoplankton zooplankton model with delay abstract in this paper a phytoplankton zooplankton model with toxic liberation delay is considered firstly the critical values of hopf bifurcation transcritical bifurcation and hopf transcritical bifurcation are given and to give more detailed information about the periodic oscillations the direction and stability of hopf bifurcation is studied by using the normal form theory and center manifold theorem then we give the detailed bifurcation set by calculating the universal unfoldings near the hopf transcritical bifurcation point finally we show that the plankton system may exhibit quasi periodic oscillations which are verified both theoretically and numerically and explain the experimental observed fluctuation phenomenon of plankton population

maximization and minimization of the rank and inertia of the hermitian matrix expression a bx bx with applications abstract we give in this paper a group of closed form formulas for the maximal and minimal ranks and inertias of the linear hermitian matrix function a bx bx with respect to a variable matrix x as applications we derive the extremal ranks and inertias of the matrices x x where x is a solution to the matrix equation axb c and then give necessary and sufficient conditions for the matrix equation axb c to have hermitian definite and re definite solutions in addition we give closed form formulas for the extremal ranks and inertias of the difference x 1 x 2 where x 1 and x 2 are hermitian solutions of two matrix equations a 1 x 1 a 1 c 1 and a 2 x 2 a 2 c 2 and then use the formulas to characterize relations between hermitian solutions of the two equations

fractional high order methods for the nonlinear fractional ordinary differential equation abstract in this paper we consider the nonlinear fractional order ordinary differential equation 0 d t y t f y t t 0 n 1 n y i 0 y 0 i i 0 1 2 n 1 where f y t satisfies the l condition i e f y 1 t f y 2 t l y 1 y 2 in t 0 t fractional order linear multiple step methods are introduced the high order 2 6 approximations of the fractional order ordinary differential equation with an initial value are proposed the consistence convergence and stability of the fractional high order methods are proved finally some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods

elastic mode algorithms for mathematical programs with equilibrium constraints global convergence and stationarity properties the elastic mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that for a finite value of the penalty parameter local solutions satisfying first and second order necessary optimality conditions for the original problem are also first and second order points of the elastic mode formulation here we study global convergence properties of methods based on this formulation which involve generating an exact or inexact first or second order point of the formulation for nondecreasing values of the penalty parameter under certain regularity conditions on the active constraints we establish finite or asymptotic convergence to points having a certain stationarity property such as strong stationarity m stationarity or c stationarity numerical experience with these approaches is discussed in particular our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic mode formulation is solved inexactly

on inertia and schur complement in optimization partitioned symmetric matrices in particular the hessian of the lagrangian play a fundamental role in nonlinear optimization for this type of matrices s p han and o fujiwara recently presented an inertia theorem under a certain regularity assumption we prove that this theorem is true without any regularity assumption then we consider matrix extensions preserving the sign of the determinant such extensions are shown to be related with the positive definiteness of some schur complement under a regularity assumption this shows from the viewpoint of linear algebra the equivalence of strong stability in the sense of m kojima and strong regularity in the sense of s m robinson finally we discuss the inertia of a typical one parameter family of symmetric matrices occurring in various places in optimization augmented lagrangians focal point theory etc

on iterated minimization in nonconvex optimization in dynamic programming and decomposition methods one often applies an iterated minimization procedure the problem variables are partitioned into several blocks say x and y treating y as a parameter the first phase consists of minimization with respect to the variable x in a second phase the minimization of the resulting optimal value function depending on y is considered in this paper we treat this basic idea on a local level it turns out that strong stability in the sense of kojima in the first phase is a natural assumption in order to show that the iterated local minima of the parametric problem lead to a local minimum for the whole problem we use a generalized version of a positive definiteness criterion of fujiwara han mangasarian

on optimality conditions for structured families of nonlinear programming problems optimality conditions for families of nonlinear programming problems inrn are studied from a generic point of view the objective function and some of the constraints are assumed to depend on a parameter while others are held fixed techniques of differential topology are used to show that under suitable conditions certain strong second order conditions are necessary for optimality except possibly for parameter values lying in a negligible set

on structure and computation of generalized nash equilibria we consider generalized nash equilibrium problems gnep from a structural and computational point of view in gnep the players feasible sets may depend on the other players strategies moreover the players may share common constraints in particular the latter leads to the stable appearance of nash equilibria which are fritz john fj points but not karush kuhn tucker kkt points basic in our approach is the representation of fj points as zeros of an appropriate underdetermined system of nonsmooth equations here additional nonsmooth variables are taken into account we prove that the set of fj points together with corresponding active lagrange multipliers generically constitutes a lipschitz manifold its dimension is n 1 j 0 where n is the number of players and j 0 is the number of active common constraints in a structural analysis of nash equilibria the number n 1 j 0 plays a crucial role in fact the latter number encodes both the possible degeneracies for the players parametric subproblems and the dimension of the set of nash equilibria in particular in the nondegenerate case the dimension of the set of nash equilibria equals locally n 1 j 0 for the computation of fj points we propose a nonsmooth projection method npm which aims at nding solutions of an underdetermined system of nonsmooth equations npm is shown to be well dened for gnep local convergence of npm is conjectured for gnep under generic assumptions and its proof is challenging however we indicate special cases known from the literature in which convergence holds

primal dual properties of sequential gradient restoration algorithms for optimal control problems 2 general problem abstract the problem of minimizing a functional subject to differential constraints nondifferential constraints initial constraints and final constraints is considered in connection with sequential gradient restoration algorithms sgra for optimal control problems both the primal formulation and the dual formulation are presented depending on whether the primal formulation is used or the dual formulation is used one obtains a primal sequential gradient restoration algorithm psgra or a dual sequential gradient restoration algorithm dsgra for the problem under consideration it is found convenient to split the control vector into an independent control vector and a dependent control vector the latter having the same dimension as the nondifferential constraint vector this modification enhances the computational efficiency of both the primal formulation and the dual formulation the system of lagrange multipliers associated with i the gradient phase of sgra and ii the restoration phase of sgra is examined for each phase it is shown that the lagrange multipliers are endowed with a duality property they minimize a special functional quadratic in the multipliers subject to the multiplier differential equations and boundary conditions for given state control and parameter these duality properties have considerable computational implications they allow one to reduce the auxiliary optimal control problems associated with i and ii to mathematical programming problems involving a finite number of parameters as unknowns

global stabilization of nonlinear systems with inputs subject to magnitude and rate bounds a parametric optimization approach a bounded feedback control design approach is proposed for the global asymptotic stabilization of a class of nonlinear systems with stable free dynamics the control inputs and their derivatives are constrained to take values on sets defined by a cartesian product of eta dimensional closed balls mathcal b mathbf r eta p which are defined by means of a p norm and a radius vector parameter mathbf r in order to derive the bounded control stabilizer the resulting procedure implies that gains as state functions are obtained from the solution to a set of c parameterized nonlinear programming problems in general the resulting closed loop system could be implicitly defined i e consisting of a system of differential equations plus a set of nonlinear algebraic equations required to compute the control special interest is focused on an important class of homogeneous systems that includes a class of globally asymptotically stabilizable systems by linear feedback and bilinear systems for those systems the problem of inputs subject to globally bounded rates is also addressed

on eigenvalue variations of rayleigh quotient matrix pencils of a definite pencil abstract let a b be a definite matrix pencil of order n i e both a and b are n n hermitian and c a b def min xe c n x 2 1 x h a ib x 0 suppose y is an n l matrix with full column rank whose column vectors span an approximate invariant subspace for a b this note investigates the relation between eigenvalues of a b and those of y h ay y h by our result for the spectral norm improves sun s linear algebra appl 139 253 267 1990 we also present bounds in frobenius norms and in general unitarily invariant norms

on eigenvalues of a rayleigh quotient matrix this note deals with the following problem let a be an n nhermitian matrix and q and q be two n m n m 1 matricesboth with orthonormal column vectors how do the eigenvalues of the m m hermitian matrix qhaq differ from those of the m m hermitian matrix qhaq we give a positive answer to one of the unsolved problems raised recently by sun

a conjugate decomposition of the euclidean space abstract given a closed convex cone k in the n dimensional real euclidean space rn and an n n real matrix a that is positive definite on k we show that each vector in rn can be decomposed into a component that lies in k and another that lies in the conjugate cone induced by a and such that the two vectors are conjugate to each other with respect to a at as a consequence of this decomposition we establish the following characterization of positive definite matrices an n n real matrix a is positive definite if and only if it is positive definite on some closed convex cone k in rn and a at 1 exists and is positive semidefinite on the polar cone k0 if k is a subspace of rn then k0 is its orthogonal complement k unk other applications include local duality results for nonlinear programs and other characterizations of positive definite and semidefinite matrices

a quantitative formulation of sylvester s law of inertia iii using elementary matrix algebra we establish the following theorems 1 3 let h be any n n hermitian matrix and let m be any n m complex matrix suppose that i m m has r eigenvalues in the interval a1 b1 ii h has s eigenvalues in a2 b2 a2 0 then m hm has at least r s n eigenvalues in a1a2 b1b2 3 1 let h be any n n hermitian matrix with in h let m be any real n m matrix and let m dimker m let 1 1 1 denote the inertia of m hm then m n m 1 and m n m 1 when m is a square matrix these inequalities are simply m 1 and m 1

a restricted signature normal form for hermitian matrices quasi spectral decompositions and applications abstract in recent years a number of results on the relationships between the inertias of hermitian matrices and the inertias of their principal submatrices have appeared in the literature in this paper we study restricted congruence transformations of hermitian matrices m that at the same time induce a congruence transformation of a given principal submatrix a of m such transformations lead to the concept of the restricted signature normal form of m in particular by means of this normal form we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type for some applications a special class of almost unitary restricted congruence transformations turns out to be useful we show that with such transformations m can be reduced to a quasi diagonal form which in particular displays the eigenvalues of a moreover this quasi spectral decomposition is used to derive a generalized signature formula and to study hermitian matrix pencils

dynamics of linear discrete systems connected to local essentially non linear attachments the dynamics of a linear periodic substructure weakly coupled to an essentially non linear attachment are studied the essential non linearizable non linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena e g passive one way irreversible transfer of energy from the substructure to the attachment as a specific application the dynamics of a finite linear chain of coupled oscillators with a non linear end attachment is examined in the absence of damping it is found that the dynamical effect of the non linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain when damping exists energy pumping phenomena are realized in the system it is shown that energy pumping strongly depends on the topological structure of the non linear normal modes nnms of the underlying undamped system this is due to the fact that energy pumping is caused by the excitation of certain damped invariant nnm manifolds that are analytic continuations for weak damping of nnms of the underlying undamped system the bifurcations of the nnms of the undamped system help explain resonance capture cascades in the damped system this is a series of energy pumping phenomena occurring at different frequencies with sudden lower frequency transitions between sequential events the observed multi frequency energy pumping cascades are particularly interesting from a practical point of view since they indicate that non linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure in compatibility with the design objectives

structure of the first order solution set for a class of nonlinear programs wit h parameters this paper studies the set of first order solutions of a family of nonlinear programs in which the inequality constraints are fixed but the right hand side of the equality constraints varies it is shown that under certain conditions this set is a topological manifold the results are applied to the problem of describing the set of first order pareto optima in a pure exchange economy with monotone utility functions

a new predictor corrector method for fractional differential equations we present a new predictor corrector method to solve non linear fractional differential equations involving caputo derivative the proposed method is compared with the fractional adams method numerous illustrative examples discussed here demonstrate that the new method is more accurate and time efficient a detailed error analysis given points to the higher accuracy of the new method furthermore the proposed method when applied to fractional analog of chaotic system introduced by bhalekar and daftardar gejji unravels the underlying rich dynamics of the system

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

on the interaction of the responses at the resonance frequencies of a nonlinear two degrees of freedom system this paper describes the dynamic behaviour of a coupled system which includes a nonlinear hardening system driven harmonically by a shaker the shaker is modelled as a linear single degree of freedom system and the nonlinear system under test is modelled as a hardening duffing oscillator the mass of the nonlinear system is much less than the moving mass of the shaker and thus the nonlinear system has little effect on the shaker dynamics the nonlinearity is due to the geometric configuration consisting of a mass suspended on four springs which incline as they are extended following experimental validation the model is used to explore the dynamic behaviour of the system under a range of different conditions of particular interest is the situation when the linear natural frequency of the nonlinear system is less than the natural frequency of the shaker such that the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the shaker it is found that for some values of the system parameters a complicated frequency response curve for the nonlinear system can occur closed detached curves can appear as a part of the overall amplitude frequency response these detached curves can lie outside or inside the main resonance curve and a physical explanation for their occurrence is given

on the inertias of symmetric matrices and bounded self adjoint operators abstract let s be a hermitian matrix and s 1 a principal submatrix with r less rows and columns let and 1 be the numbers of positive eigenvalues of s and s 1 one of poincare s inequalities says that 1 r 1 we tighten these inequalities as follows let and 1 be dimkers and dimkers 1 and let d dimkers kers 1 then 1 r d 1 1 d

a new fractional operator of variable order application in the description of anomalous diffusion abstract in this paper a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of caputo type the properties in term of the laplace and fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed the new formulation is efficient in modeling a class of concentrations in the complex transport process

on ostrowski s generalization of sylvester s law of inertia abstract ostrowski s quantitative form of sylvester s law of inertia is derived directly from the courant fischer theorem

stability results for constrained calculus of variations problems an analysis of the twisted elastic loop problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history constrained variational problems have been much less studied particularly the theory of stability which determines which solutions are physically realizable in this paper we develop stability exchange results appropriate for parameter dependent calculus of variations problems with two particular features either the parameter appears in the boundary conditions or there are isoperimetric constraints in particular we identify an associated distinguished bifurcation diagram which encodes the direction of stability exchange at folds we apply the theory to a twisted elastic loop which can naturally be formulated as a calculus of variations problem with both isoperimetric constraints and parameter dependent boundary conditions in combination with a perturbation expansion that classifies certain pitchfork bifurcations as sub or super critical the distinguished diagram for the twisted loop provides a classification of the stability properties of all equilibria in particular an unanticipated sensitive dependence of stability properties on the ratio of twisting to bending stiffness is revealed

a note on weakly active constraints in connection with nonconvex quadratic programming abstract in general it is very difficult to determine a true local minimizer in nonconvex quadratic programming the main problems arise if we have so called weakly active constraints we discuss an efficient method for checking the local optimality or determining a direction of descent

convergence of petviashvili s iteration method for numerical approximation of stationary solutions of nonlinear wave equations we analyze a heuristic numerical method suggested by v i petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations the method is used to construct numerically the solitary wave solutions such as solitons lumps and vortices in a space of one and higher dimensions assuming that the stationary solution exists we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest the theory is illustrated with examples of physical interest such as generalized korteweg de vries benjamin ono zakharov kuznetsov kadomtsev petviashvili and klein gordon equations

review v i arnol d s m gusein zade and a n varchenko singularities of differentiable maps vol 1 the classification of critical sets caustics and wave fronts

on generic one parametric semi infinite optimization we consider differentiable semi infinite optimization problems depending on a real parameter for generic one parametric families we classify the corresponding set of generalized critical points into eight types five of these types also occur in problems with a finite number of inequality constraints whereas the other three types are typical for the semi infinite case we discuss types 7 and 8 in detail while at points of type 6 the singularity is due to the fact that in the associated lower level problem a lagrange multiplier corresponding to an active inequality constraint vanishes at points of type 7 and 8 the gradients of the active constraints in the lower level problem are linearly dependent if the total number of active constraints in the lower level problem does not exceed the lower level dimension the point is of type 7 otherwise it is of type 8 moreover we distinguish between points of type 8a and 8b where a point is of type 8a if the mangasarian fromovitz constraint qualification holds in the lower level problem and of type 8b otherwise at points of type 8a the set of generalized critical points is not smooth but it does not exhibit a turning point the linear and quadratic indices remain constant when passing along a point of type 8a points of type 7 and type 8b are relative boundary points of the set of generalized critical points

definiteness and semidefiniteness of quadratic forms revisited abstract it is shown how the schur complement theory can be used for the derivation of criteria for the definiteness or semidefiniteness of the restriction of a quadratic form to the null space of a matrix

bilevel optimization on the structure of the feasible set we consider bilevel optimization from the optimistic point of view let the pair x y denote the variables the main difficulty in studying such problems lies in the fact that the lower level contains a global constraint in fact a point x y is feasible if y solves a parametric optimization problem l x in this paper we restrict ourselves to the special case that the variable x is one dimensional we describe the generic structure of the feasible set m moreover we discuss local reductions of the bilevel problem as well as corresponding optimality criteria finally we point out typical problems that appear when trying to extend the ideas to higher dimensional x dimensions this will clarify the high intrinsic complexity of the general generic structure of the feasible set m and corresponding optimality conditions for the bilevel problem u

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

determining subspaces on which a matrix is nonnegative definite abstract we give methods for finding a subspace l of r n of high dimension on which a given real symmetric n n matrix a is nonnegative positive definite the subspace l is constructed as the intersection of some hyperplanes the first method assumes that a has few negative nonpositive eigenvalues and selects the hyperplanes from a given collection which includes the coordinate planes the second method finds a subspace l of maximal dimension it proceeds by selecting some hyperplanes from the given collection and then successively constructing additional hyperplanes until a maximal l is determined both methods rely on inertia theory some new inertia results are derived numerical examples are given

a note on differentiability of global optimal values the differentiability of global optimal values for smooth constrained programs with respect to general perturbations is considered the strong whitney c2 topology is used to the space of objective functions and constraints it will be shown generically the global optimal value is twice continuously differentiable with respect to the function space of objective functions and constraints an application of this genericity argument to a nonsmooth objective function a function of maximum type is also discussed

an improved pc scheme for nonlinear fractional differential equations a novel computationally effective fractional predictorcorrector pc scheme is proposed to solve fractional differential equations involving caputo derivative the properties of the caputo derivative are used to reduce the fractional differential equation into a volterra integral equation to design high order numerical solution of fdes the simpsons 3 8 rule is applied to the volterra type integral equation the scheme is capable of handling both linear and nonlinear fractional differential equations a detailed error analysis and stability analysis of the numerical scheme are rigorously established the proposed scheme is compared with the pc schemes of literature for illustrating the effectiveness of the algorithm

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

on the effects of system parameters on the response of a harmonically excited system consisting of weakly coupled nonlinear and linear oscillators this paper discusses the dynamic behaviour of a nonlinear two degree of freedom system consisting of a harmonically excited linear oscillator weakly connected to a nonlinear attachment having linear and cubic restoring forces the effects of the system parameters on the shape of the frequency response curve are investigated in particular those yielding the appearance and disappearance of outer and inner detached resonance curves in contrast to the case when the linear stiffness of the attachment is zero it is found that multivaluedness occurs at low frequencies as the resonant peak bends to the right it is also found that as the coefficient of the linear term increases the range of parameters yielding detached curves reduces compared to the case when the attached system has no linear stiffness term this range of parameters corresponds to smaller values of the damping and nonlinear coefficients approximate analytical expressions for the jump up and jump down frequencies of the system under investigation are also derived

critical sets in parametric optimization we deal with one parameter families of optimization problems in finite dimensions the constraints are both of equality and inequality type the concept of a generalized critical point g c point is introduced in particular every local minimum kuhn tucker point and point of fritz john type is a g c point under fairly weak even generic conditions we study the set consisting of all g c points due to the parameter the set is pieced together from one dimensional manifolds the points of can be divided into five characteristic types the subset of nondegenerate critical points first type is open and dense in nondegenerate means strict complementarity nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints a nondegenerate critical point is completely characterized by means of four indices the change of these indices along is presented finally the kuhn tucker subset of is studied in more detail in particular in connection with the failure of the mangasarian fromowitz constraint qualification

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

local duality of nonlinear programs it is shown that the second order sufficient necessary optimality condition for the dual of a nonlinear program is equivalent to the inverse of the hessian of the lagrangian being positive definite semidefinite on the normal cone to the local primal constraint surface this compares with the hessian itself being positive definite semidefinite on the tangent cone on the local primal constraint surface for the corresponding second order condition for the primal problem we also show that primal second order sufficiency necessity and dual second order necessity sufficiency is essentially equivalent to the hessian of the lagrangian being positive definite this follows from the following interesting linear algebra result a necessary and sufficient condition for a nonsingular symmetric n times n matrix to be positive definite is that for some subspace of r n the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace

restricted quadratic forms inertia theorems and the schur complement the starting point of this investigation is the properties of restricted quadratic forms x ax x s rm where a is an m m real symmetric matrix and s is a subspace the index theory of hestenes 1951 and maddocks 1985 that treats the more general hilbert space version of this problem is first specialized to the finite dimensional context and appropriate extensions valid only in finite dimensions are made the theory is then applied to obtain various inertia theorems for matrices and positivity tests for quadratic forms expressions for the inertias of divers symmetrically partitioned matrices are described in particular an inertia theorem for the generalized schur complement is given the investigation recovers links and extends several formerly disparate results in the general area of inertia theorems

response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning dynamical system under investigation in the current work is comprised of harmonically forced linear oscillator with attached nonlinear energy sink external forcing frequency detuning near the main resonance 1 1 is included in the system investigation the detailed study of the periodic and quasiperiodic regimes is done in the work via adaptive averaging method local bifurcations of the periodic regimes are revealed and fully described in the space of system parameters amplitude of excitation damping and frequency detuning novel analytical approach for predictions of strongly modulated response smr is presented this approach provides a sufficient condition for the smr existence contrary to the previous studies various possibilities of coexistence of the response regimes are predicted analytically and demonstrated numerically among those is a coexistence of two distinct periodic regimes together with the smr all findings of the simplified analytic model are verified numerically and considerable agreement is observed

a parametric view on the mangasarian fromovitz constraint qualification in this paper we consider the behaviour of the multipliers for one parametric semi infinite programs near critical parameters where the mangasarian fromovitz constraint qualification can be violated the growth of the multipliers when the parameter approaches such a critical parameter is characterized by a parametric constraint qualification which is introduced here it is equivalent to a bound on the growth of the multipliers

characteristic polynomials of distance matrices of one dimensional sets abstract properties of the eigenvalues of the distance matrix of a one dimensional point set are derived from identities involving the characteristic polynomial and some related polynomials called the ant sym din and sof polynomials of the point set let a b denote the concatenation of the lists a and b and m s i j m m 1 i n 1 j the spin of the m by n matrix m the ant and sym polynomials come from a factorization of the characteristic polynomial of the distance matrix of the set a s a obtained by reflecting a about the origin the roots of ant are the eigenvalues with antisymmetric eigenvectors and the roots of sym are the eigenvalues with symmetric eigenvectors given a square matrix m and a vector a we say that v 0 is an eigenvector of m relative to a iff mv v ka and a v 0 some basic properties of relative eigenvectors are developed the roots of din are the eigenvalues of the distance matrix of a relative to the vector of 1 s and the roots of sof are the eigenvalues relative to a itself some simple recursions for these polynomials obtained using expansion by minors are elaborated into an extensive series of identities relating polynomials of lists to polynomials of concatenations of the lists these identities are then used to derive a linear time algorithm for computing the polynomials and proving some results about location distinctness and interlacing of eigenvalues

finite difference methods with non uniform meshes for nonlinear fractional differential equations in this article finite difference methods with non uniform meshes for solving nonlinear fractional differential equations are presented where the non equidistant stepsize is non decreasing the rectangle formula and trapezoid formula are proposed based on the non uniform meshes combining the above two methods we then establish the predictor corrector scheme the error and stability analysis are carefully investigated at last numerical examples are carried out to verify the theoretical analysis besides the comparisons between non uniform and uniform meshes are given where the non uniform meshes show the better performance when dealing with the less smooth problems

linear rate convergence of the alternating direction method of multipliers for convex composite programming in this paper we aim to prove the linear rate convergence of the alternating direction method of multipliers admm for solving linearly constrained convex composite optimization problems under a mild calmness condition which holds automatically for convex composite piecewise linear quadratic programming we establish the global q linear rate of convergence for a general semi proximal admm with the dual step length being taken in 0 1 51 2 2 this semi proximal admm which covers the classic one has the advantage to resolve the potentially nonsolvability issue of the subproblems in the classic admm and possesses the abilities of handling the multi block cases efficiently we demonstrate the usefulness of the obtained results when applied to two and multi block convex quadratic semidefinite programming

the c index a new stability concept for quadratic programs with complementarity constraints we introduce nondegeneracy and the c index for c stationary points of a qpcc that is for a mathematical program with a quadratic objective function and linear complementarity constraints the c index characterizes the qualitative local behavior of a qpcc around a nondegenerate c stationary point the article focuses on the structure of the c stationary set of qpccs depending on a real parameter we show that for generic qpcc data the c index changes exactly at turning points of the c stationary set and that it changes by exactly one to illustrate this concept we introduce and analyze two homotopy methods for finding c stationary points numerical results illustrate that for randomly generated test problems the two homotopy methods very often identify b stationary points

characterization of strong stability for c stationary points in mpcc we study mathematical programs with complementarity constraints mpcc special focus will be on c stationary points under the linear independence constraint qualification we characterize strong stability of c stationary points in the sense of kojima by means of first and second order information of the defining functions it turns out that strong stability of c stationary points allows a possible degeneracy of bi active lagrange multipliers some relations to other stationarity concepts such as a m s and b stationarity are shortly discussed

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

moreau s decomposition in banach spaces moreau s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics in this paper it is extended to reflexive banach spaces and in the context of generalized proximity measures this extension unifies and significantly improves upon existing results

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

the negative signature of some hermitian matrices abstract it is proved that the range of the negative signature of the hermitian completions of the partial matrix a b b c d d e is an interval and explicit formulae for the bounds of this interval are obtained as an application the solvability of the problem of lifting with prescribed negative signature of defect is settled

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method abstract we establish in this paper a group of closed form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic hermitian matrix function x q x p x with respect to the variable matrix x by using a linearization method and some known formulas for extremum ranks and inertias of linear hermitian matrix functions where both p and q are complex hermitian matrices and x is the conjugate transpose of x we then derive the global maximum and minimum ranks and inertias of the two quadratic hermitian matrix functions 1 x q 1 x p 1 x and 2 x q 2 x p 2 x subject to a consistent matrix equation a x b respectively by using some pure algebraic operations of matrices and their generalized inverses as consequences we establish necessary and sufficient conditions for the solutions of the matrix equation a x b to satisfy the quadratic hermitian matrix equalities x p 1 x q 1 and x p 2 x q 2 respectively and for the quadratic matrix inequalities x p 1 x q 1 and x p 2 x q 2 in the lowner partial ordering to hold respectively in addition we give complete solutions to four lowner partial ordering optimization problems on the matrix functions 1 x and 2 x subject to a x b examples are also presented to illustrative applications of the equality constrained quadratic optimizations in some matrix completion problems

conjugate cone characterization of positive definite and semidefinite matrices abstract positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in rn these results are obtained by generalizing moreau s polar decomposition to a conjugate decomposition some typical results are the matrix a is positive definite if and only if for some closed convex cone k a is positive definite on k and a at 1 exists and is semidefinite on the polar cone k the matrix a is positive semidefinite if and only if for some closed convex cone k such that either k is polyhedral or a at k is closed a is positive semidefinite on both k and the conjugate cone ka s xt a at s 0 x k and a at x 0 for all x in k such that xtax 0

a note on the augmented hessian when the reduced hessian is semidefinite certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming let h be an n n symmetric matrix a an m n matrix and z a basis for the null space of a in a typical optimization context h is the hessian of a smooth function and a is the jacobian of a set of constraints when the reduced hessian zthz is positive definite augmented lagrangian methods rely on the known existence of a finite bar rho ge 0 such that for all rho bar rho the augmented hessian h rho ata is positive definite in this note we analyze the case when zthz is positive semidefinite i e singularity is allowed and show that the situation is more complicated in particular we give a simple necessary and sufficient condition for the existence of a finite bar rho so that h rho ata is positive semidefinite for rho ge bar rho a corollary of our result is that if h is nonsingular and indefinite while zthz is positive semidefinite and singular no such bar rho exists

on the hessian of lagrangian and second order optimality conditions for a constrained minimization problem the restriction of a hessian of lagrangian to a tangent space of the feasible set can be used to detect whether a karush kuhn tucker point is a local minimum maximum or saddle point of the problem it is shown in this paper that the restriction of the hessian to a normal space with respect to the indefinite inner product induced by the hessian can be used to characterize a karush kuhn tucker point for the wolfe dual from this result and by an inertia theorem in s p han and o fujiwara an inertia theorem and its application to nonlinear programs manuscript march 1984 we deduce that under a regularity condition the hessian is positive semidefinite if and only if the considered karush kuhn tucker point satisfies the second order necessary condition for a local minimum point of the primal problem and at the same time satisfies the second order necessary condition for a local maximum point of thee dual similar results on the positive definiteness of the hes

shape stress trade off design of membrane structures for specified sequence of boundary shapes a new method is presented for parametrically generating optimal shapes and stresses of a membrane structure supported by a frame the problem considered here is a so called trade off design problem where the weighted sum of the deviations of the initial stress and the shape from the desired distribution is to be minimized the membrane is discretized by using the finite element method and optimal shapes are expanded with respect to a parameter defining the shape of the boundary frame an approximate method is presented without exact calculation of the hessian of the stresses with respect to the nodal coordinates the explicit developability conditions derived by the authors are used to guarantee the curved surface to be reduced to a set of plane cutting patterns after removing the pretensioning force in the examples trade off designs are found for an hp type membrane structure and the efficiency of the proposed method is discussed

an inertia theorem for symmetric matrices and its application to nonlinear programming abstract the relationship between the inertia of a symmetric real matrix and the inertia of its projection to a subspace is studied the result is then used to characterize a karush kuhn tucker point of a nonlinear programming problem by the inertia of a hessian of lagrangian with respect to both primal and dual variables a theorem of kojima on the determinant of a matrix in nonlinear programming is also derived as a direct consequence of the result

a bifurcation analysis of the nonlinear parametric programming problem the structure of solutions to the nonlinear parametric programming problem with a one dimensional parameter is analyzed in terms of the bifurcation behavior of the curves of critical points and the persistence of minima along these curves changes in the structure of the solution occur at singularities of a nonlinear system of equations motivated by the fritz john first order necessary conditions it has been shown that these singularities may be completely partitioned into seven distinct classes based upon the violation of one or more of the following a complementarity condition a constraint qualification and the nonsingularity of the hessian of the lagrangian on a tangent space to apply classical bifurcation techniques to these singularities a further subdivision of each case is necessary the structure of curves of critical points near singularities of lowest zero codimension within each case is analyzed as well as the persistence of minima along curves emanating from these singularities bifurcation behavior is also investigated or discussed for many of the subcases giving rise to a codimension one singularity

lifting mathematical programs with complementarity constraints we present a new smoothing approach for mathematical programs with complementarity constraints based on the orthogonal projection of a smooth manifold we study regularity of the lifted feasible set and since the corresponding optimality conditions are inherently degenerate introduce a regularization approach involving a novel concept of tilting stability a correspondence between the c index in the original problem and the quadratic index in the lifted problem is shown in particular a local minimizer of the mathematical program with complementarity constraints may numerically be found by minimization of the lifted smooth problem we report preliminary computational experience with the lifting approach

novel algorithms to estimate nonlinear fdes applied to fractional order nutrient phytoplankton zooplankton system abstract in this paper dynamical behavior of a mathematical model for the interaction of nutrient phytoplankton and its predator zooplankton is investigated numerically stability analysis of the phytoplankton zooplankton model is studied by using the fractional routh hurwitz stability conditions we have studied the local stability of the equilibrium points then a new numerical algorithm as well as its modification for solving fractional differential equations fdes based on the polynomial interpolation is proposed the algorithms are designed to estimate linear and nonlinear fdes and they have the capability to apply for solving fractional order systems the convergence order and stability of the fractional higher order algorithms are proved and stability regions of the algorithms are achieved extensive numerical simulation results are provided and compared with the literature for illustrating the effectiveness and applicability of the presented algorithms to solve fractional differential equations the obtained analytical results are validated by numerical simulations and the global dynamics of the model system is studied

new approaches to the fractional dynamics of schistosomiasis disease model abstract in this paper schistosomiasis fractional order dynamic model is examined via exponential law kernel sense and mittag leffler kernel in liouville caputo sense some special solutions for two operators are obtained using the iterative scheme through laplace transform and sumudu picard integration technique the uniqueness and existence of solution for both operators are established the numerical solutions for both operators approve that the desirable results can be obtained when alpha value is less than 1

change of generalized indices of stationary solutions to multiparametric optimization we study the local change of the generalized index which is a modification of the morse index and the stationary index for the multiparametric optimization under the regular value condition the change of the generalized index around a triplet x v t is locally bounded by the dimension of the parameter vector t where x is a variable vector and v a vector of the lagrange multiplier space we also discuss the local change of the generalized index around a pair x t

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

a modified standard embedding with jumps in nonlinear optimization the paper deals with a combination of pathfollowing methods embedding approach and feasible descent direction methods so called jumps for solving a nonlinear optimization problem with equality and inequality constraints since the method that we propose here uses jumps from one connected component to another one more than one connected component of the solution set of the corresponding one parametric problem can be followed numerically it is assumed that the problem under consideration belongs to a generic subset which was introduced by jongen jonker and twilt there already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers in this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set the topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed furthermore this new method is applied to a so called modified standard embedding which is a particular construction without equality constraints finally an algorithmic version of this new method as well as computational results are presented

equalities and inequalities for inertias of hermitian matrices with applications abstract the inertia of a hermitian matrix is defined to be a triplet composed of the numbers of the positive negative and zero eigenvalues of the matrix counted with multiplicities respectively in this paper we show some basic formulas for inertias of 2 2 block hermitian matrices from these formulas we derive various equalities and inequalities for inertias of sums parallel sums products of hermitian matrices submatrices in block hermitian matrices differences of outer inverses of hermitian matrices as applications we derive the extremal inertias of the linear matrix expression a bxb with respect to a variable hermitian matrix x in addition we give some results on the extremal inertias of hermitian solutions to the matrix equation ax b as well as the extremal inertias of a partial block hermitian matrix

local duality of nonlinear semidefinite programming recently chan and sun chan z x d sun constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming siam j optim 19 370 376 reported for semidefinite programming sdp that the primal dual constraint nondegeneracy is equivalent to the dual primal strong second order sufficient condition ssosc this result is responsible for a number of important results in stability analysis of sdp in this paper we study duality of this type in nonlinear semidefinite programming nsdp we introduce the dual ssosc at a karush kuhn tucker kkt triple of nsdp and study its various characterizations and relationships to the primal nondegeneracy although the dual ssosc is nothing but the ssosc for the wolfe dual of the nsdp it suggests new information for the primal nsdp for example it ensures that the inverse of the hessian of the lagrangian function exists at the kkt triple and the inverse is positive definite on some normal space it also ensures the primal nondegeneracy some of our results generalize the corresponding classical duality results in nonlinear programming studied by fujiwara et al fujiwara o s p han o l mangasarian 1984 local duality of nonlinear programs siam j control optim 22 162 169 for the convex quadratic sdp qsdp we have complete characterizations for the primal and dual ssosc our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual ssosc but also the primal and dual nondegeneracy at its solution suggesting that it is a well conditioned qsdp

bifurcation problems in nonlinear parametric programming the nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters this system which is motivated by the fritz john first order necessary conditions contains all fritz john and all karush kuhn tucker points as well as local minima and maxima saddle points feasible and nonfeasible critical points necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition the linear dependence of the gradients of the active constraints and the singularity of the hessian of the lagrangian on a tangent space any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution for problems with one parameter we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity

inertia law for spectral stability of solitary waves in coupled nonlinear schrodinger equations spectral stability analysis for solitary waves is developed in the context of the hamiltonian system of coupled nonlinear schrodinger equations the linear eigen value problem for a non self adjoint operator is studied with two self adjoint matrix schrodinger operators sharp bounds on the number and type of unstable eigen values in the spectral problem are found from the inertia law for quadratic forms associated with the two self adjoint operators symmetry breaking stability analysis is also developed with the same method

relative eigenvalues of hermitian matrices abstract motivated by the interest in a more thorough understanding of the relationship between the eigenvalues of a hermitian matrix m and those of a matrix of the form a ma we study the relative eigenvalues of a hermitian matrix with respect to a subspace a geometrically meaningful result and some of its useful corollaries are established as a consequence some inertia theorems given in earlier work are also strengthened

adaptive pinning synchronization in fractional order complex dynamical networks abstract synchronization of general complex dynamical networks with fractional order dynamical nodes is addressed in this paper based on the stability theory of fractional order differential systems and adaptive pinning control some sufficient local asymptotical synchronization criteria and global asymptotical ones are derived respectively which succeed in solving the problem about how many nodes are need to be controlled and how much coupling strength should be applied to ensure the synchronization of the entire fractional order networks the obtained results are more general and effective than those reported moreover the coupling configuration matrices and the inner coupling matrices are not assumed to be symmetric and irreducible finally a numerical simulation is presented to demonstrate the validity and feasibility of the proposed synchronization criteria

a novel high order algorithm for the numerical estimation of fractional differential equations abstract this paper uses polynomial interpolation to design a novel high order algorithm for the numerical estimation of fractional differential equations the riemann liouville fractional derivative is expressed by using the hadamard finite part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral the detailed error analysis is presented and it is established that the convergence order of the algorithm is o h 4 asymptotic expansion of the error for the presented algorithm is also investigated some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

morse programs a topological approach to smooth constrained optimization the purpose of this paper is to give a geometrical answer to the question to the strong second order sufficiency conditions hold at any local minimum point for almost all nonlinear programs our idea is to reduce the nonlinear programming problem to a finite family of well behaved nonlinear programs by perturbing the objective function in a linear fashion and perturbing the right hand side of the constraints by adding a constant each of the well behaved nonlinear programs will consist of minimizing a morse function on a manifold with boundary where the morse function has no critical points on the boundary

respiration effect on plankton oxygen dynamics in view of non singular time fractional derivatives abstract in this work we investigate a model of oxygen phytoplankton zooplankton dynamics with fractional order derivatives in different types known as the caputo fabrizio and atangana baleanu derivatives having nonsingular kernels exponential and mittag leffler in order to detail the underlying structure of the oxygen plankton interaction the model is detailed in analytical and numerical techniques hypothetical set of parameters are used in numerical simulations to guarantee that in natural sense the system in biologically meaningful region all results obtained are compared with each other under different parameters and different fractional orders with simulations

on the augmented lagrangian it is well known that for a nonlinear program any karush kuhn tucker point that satisfies the second order sufficient condition is also a strict local minimum of the augmented lagrangian with an underlying penalty parameter sufficiently large a main theme of the paper is to strengthen the theorem and bring to light an interesting connection among general critical points of a nonlinear program and its augmented lagrangian when a penalty parameter is sufficiently large it will be shown that a karush kuhn tucker point of a nonlinear program is also a critical point of its augmented lagrangian with the same index and the same nullity the aforementioned theorem is just the special case when the index and the nullity are zeros this result is established through an extension of a lemma due to finsler and debreu which is interesting in its own right consequently under a constraint qualification any karush kuhn tucker point can also be shown to satisfy a second order necessary condition for a local maximum of the generalized wolfe dual induced from the augmented lagrangian interestingly this relation holds without requiring the karush kuhn tucker point itself to satisfy any second order optimality condition