error bounds for 2 regular mappings with lipschitzian derivatives and their applications we obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature specifically we assume that the constraints mapping has a lipschitzian derivative and satisfies a certain 2 regularity condition at the point under consideration this setting directly subsumes the classical regular case and the twice differentiable 2 regular case for which error bounds are known but it is significantly richer than either of these two cases when applied to a certain equation based reformulation of the nonlinear complementarity problem our results yield an error bound under an assumption more general than b regularity the latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available we also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems

complementarity constraint qualification via the theory of 2 regularity we exhibit certain second order regularity properties of parametric complementarity constraints which are notorious for being irregular in the classical sense our approach leads to a constraint qualification in terms of 2 regularity of the mapping corresponding to the subset of constraints which must be satisfied as equalities around the given feasible point while no qualification is required for the rest of the constraints under this 2 regularity assumption we derive constructive sufficient conditions for tangent directions to feasible sets defined by complementarity constraints a special form of primal dual optimality conditions is also obtained we further show that our 2 regularity condition always holds under the piecewise mangasarian fromovitz constraint qualification but not vice versa relations with other constraint qualifications and optimality conditions are also discussed it is shown that our approach can be useful when alternative ones are not applicable

laurent interpolation for rational matrix functions and a local factorization principle abstract a regular rational matrix function is constructed when a finite part of the laurent series of the function and of its inverse are given at a finite number of points provided an obvious necessary condition is satisfied the construction is such that the function will have at most one extra pole or zero possibly of high multiplicity this construction is further applied to develop local principles for factorizations of rational matrix functions

asymptotic expansions for linear homogeneous divide and conquer recurrences algebraic and analytic approaches collated among all sequences that satisfy a divide and conquer recurrence those which are rational with respect to a numeration system are certainly the most basic and the most essential nevertheless until recently this specific class of sequences has not been systematically studied from the asymptotic standpoint we recall how a mechanical process designed by the author permits to compute their asymptotic expansions the process is based on linear algebra and involves computing jordan normal forms joint spectral radii and solving dilation equations the main contribution of the present article is the comparison between our algebraic method and the classical analytic number theory approach moreover we develop new ways to compute the fourier series of the periodic functions involved in the expansion the article comes with an extended bibliography

existence of minimal nonsquare j symmetric factorizations for self adjoint rational matrix functions in this paper we show that any rational matrix function having hermitian values on the imaginary axis and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors our proof is based on a construction which shows that any such function can be extended preserving its mcmillan degree to a function that admits j symmetric factorization with square factors also we consider other properties of the factors in j symmetric factorizations particular attention is given to the study of the common invariant zero structure of these factors

generalized convexity and efficiency for non regular multiobjective programming problems with inequality type constraints abstract for multiobjective problems with inequality type constraints the necessary conditions for efficient solutions are presented these conditions are applied when the constraints do not necessarily satisfy any regularity assumptions and they are based on the concept of 2 regularity introduced by izmailov in general the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the karush kuhn tucker points set so it is necessary to introduce generalized convexity notions in the multiobjective non regular case we give the notion of 2 kkt pseudoinvex ii problems this new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions

generalized convexity in non regular programming problems with inequality type constraints abstract convexity plays a very important role in optimization for establishing optimality conditions different works have shown that the convexity property can be replaced by a weaker notion the invexity in particular for problems with inequality type constraints martin defined a weaker notion of invexity the karush kuhn tucker invexity hereafter kkt invexity that is both necessary and sufficient to obtain karush kuhn tucker type optimality conditions it is well known that for this result to hold the problem has to verify a constraint qualification i e it must be regular or non degenerate in non regular problems the classical optimality conditions are totally inapplicable meaningful results were obtained for problems with inequality type constraints by izmailov they are based on the 2 regularity condition of the constraints at a feasible point in this work we generalize martin s result to non regular problems by defining an analogous concept the 2 kkt invexity and using the characterization of the tangent cone in the 2 regular case and the necessary optimality condition given by izmailov

stability of invariant maximal semidefinite subspaces ii applications self adjoint rational matrix functions algebraic riccati equations abstract the stability of various factorizations of self adjoint rational matrix functions and matrix polynomials as well as of hermitian solutions of symmetric matrix algebraic riccati equations is studied in the first part of this paper results on stability of certain classes of invariant subspaces of a matrix which is self adjoint in an indefinite inner product were obtained these results serve as the main tools in the investigation

nonsquare spectral factors via factorizations of a unitary function abstract we consider the problem of parametrizing the set of nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis the tool here is the use of factorizations of a certain type of the function w 1w where w resp w is the minimal square spectral factor that has all its poles and zeros in the right resp left halfplane

abnormal equality constrained optimization problems sensitivity theory for the equality constrained optimization problem we consider the case when the customary regularity of constraints can be violated under the assumptions substantially weaker than those previously used in the literature we develop a reasonably complete local sensitivity theory for this class of problems including upper and lower bounds for the rate of change of the optimal value function subject to parametric perturbations as well as the estimates and the description of asymptotic behavior of solutions of the perturbed problems

tangent vectors to a zero set at abnormal points we are concerned with sharp characterization of the contingent cone to the set defined by a finite number of equality constraints in the absence of classical regularity constrained qualification

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

stability of invariant maximal semidefinite subspaces i abstract we study stability in classes of subspaces which are invariant under a self adjoint matrix in an indefinite inner product and have various maximality and semidefiniteness properties with respect to this indefinite inner product descriptions of all subspaces in such a class for which these properties are stable in one or another way are obtained

necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications we derive first and second order necessary optimality conditions for set constrained optimization problems under the constraint qualification type conditions significantly weaker than robinson s constraint qualification our development relies on the so called 2 regularity concept and unifies and extends the previous studies based on this concept specifically in our setting constraints are given by an inclusion with an arbitrary closed convex set on the right hand side thus for the second order analysis some curvature characterizations of this set near the reference point must be taken into account

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

optimality conditions for irregular inequality constrained problems we consider feasible sets given by conic constraints where the cone defining the constraints is convex with nonempty interior we study the case where the feasible set is not assumed to be regular in the classical sense of robinson and obtain a constructive description of the tangent cone under a certain new second order regularity condition this condition contains classical regularity as a special case while being weaker when constraints are twice differentiable assuming that the cone defining the constraints is finitely generated we also derive a special form of primal dual optimality conditions for the corresponding constrained optimization problem our results subsume optimality conditions for both the classical regular and second order regular cases while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

uniform random generation of strings in a context free language let s be the set of all strings of length n generated by a given context free grammar a uniform random generator is one which produces strings from s with equal probability in generating these strings care must be taken in choosing the disjuncts that form the right hand side of a grammar rule so that the produced string will have the specified length uniform random generators have applications in studying the complexity of parsers in estimating the average efficiency of theorem provers for the propositional calculus in establishing a measure of ambiguity of a grammar etc two methods are presented for generating uniform random strings in an unambiguous context free language the first method will generate a random string of length n in linear time but must use a precomputed table of size o n r 1 where r is the number of nonterminals in the grammar used to specify the language the second method precomputes part of the table and calculates the other entries as they are called for it requi

a matricial boundary value problem which appears in the transport theory abstract we study the boundary value problem t a i p 0 p 0 where t t a a p 2 p p t tp t invertible the motivation to consider such problem comes from the transport theory the behavior of values of for which there is a nontrivial solution exceptional values is investigated using the indicator function this is an analytic hermitian valued function of the real parameter t which reflects in particular the characteristic properties of exceptional values

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

on hamiltonian rational transfer functions abstract a sequel to a previous paper fuhrmann 1983 this paper contains a detailed study of rational transfer functions having hamiltonian symmetry a detailed realization procedure is described that reflects this symmetry as a corollary the williamson canonical form for hamiltonian maps in symplectic spaces is derived

unitary solutions of a class of algebraic riccati equations and factorization abstract we consider the algebraic riccati equation xd 1 x xd 2 d 3 x d 1 0 where d 2 and d 3 are self adjoint we are interested in contractive and unitary solutions x of this equation stability properties of such solutions under perturbations of the coefficients are studied further the connection between unitary solutions of such equations and symmetric factorizations of nonnegative rational matrix functions is investigated

high order approximations and generalized necessary conditions for optimality in this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a banach space the equality constraints are given in operator form as q x in x f x 0 where f x rightarrow y is an operator between banach spaces the inequality constraints are given by smooth functionals or by closed convex sets models of this type are common in the optimal control problem the paper addresses the case when the frechet derivative f x is not onto and hence the classical lyusternik theorem does not apply to describe the tangent space to q in this case the classical euler lagrange type necessary conditions are trivially satisfied generating abnormal cases a high order generalization of the lyusternik theorem derived earlier u ledzewicz and h schattler nonlinear anal 34 1998 pp 793 815 is used to calculate high order tangent cones to the equality constraint at points x in q where f x is not onto combining these with high order approximating cones related to the other constraints of the problem feasible cones respectively cones of decrease a high order generalization of the dubovitskii milyutin theorem is given and then applied to derive generalized necessary conditions for optimality these conditions reduce to classical conditions for normal cases but they give new and nontrivial conditions for abnormal cases

uniform random generation of decomposable structures using floating point arithmetic the em recursive method formalized by nijenhuis and wilf cite niwi78 and systematized by flajolet van cutsem and zimmermann cite flziva94 is extended here to floating point arithmetic the resulting adz method enables one to generate decomposable data structures both labelled or unlabelled uniformly at random in expected o n 1 epsilon time and space after a preprocessing phase of o n 2 epsilon time which reduces to o n 1 epsilon for context free grammars

a new dichotomic algorithm for the uniform random generation of words in regular languages we present a new algorithm for generating uniformly at random words of any regular language l when using floating point arithmetics its bit complexity is o qlg 2n in space and o qnlg 2n in time where n stands for the length of the word and q stands for the number of states of a finite deterministic automaton of l we implemented the algorithm and compared its behavior to the state of the art algorithms on a set of large automata from the vlts benchmark suite both theoretical and experimental results show that our algorithm offers an excellent compromise in terms of space and time requirements compared to the known best alternatives in particular it is the only method that can generate long paths in large automata

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

bifurcation theorems via second order optimality conditions abstract we present a new approach to bifurcation study that relies on the theory of second order optimality conditions for abnormal constrained optimization problems developed earlier by the first author this theory does not subsume the primal description of the feasible set in terms of tangent vectors or in any other way as a result we obtain new sufficient conditions for bifurcation which are to some extent complementary with respect to the known bifurcation theory

the theory of 2 regularity for mappings with lipschitzian derivatives and its applications to optimality conditions we study local structure of a nonlinear mapping near points where standard regularity and or smoothness assumptions need not be satisfied we introduce a new concept of 2 regularity a certain kind of second order regularity for a once differentiable mapping whose derivative is lipschitz continuous under this 2 regularity condition we obtain the representation theorem and the covering theorem i e stability with respect to right hand side perturbations under assumptions that are weaker than those previously employed in the literature for results of this type these results are further used to derive a constructive description of the tangent cone to a set defined by 2 regular equality constraints and optimality conditions for related optimization problems the class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation based reformulation optimality conditions for mathematical programs with equivalently reformulated complementarity constraints are also discussed

brief necessary and sufficient conditions for existence of j spectral factorization for para hermitian rational matrix functions for a para hermitian rational matrix function g l j c li p a 1b where j j is invertible and which has no poles and zeros on the imaginary line we give necessary and sufficient conditions in terms of a b c and j for the existence of a j spectral factorization as well as an algorithm to obtain the j spectral factor in case it exists

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

critical solutions of nonlinear equations stability issues it is known that when the set of lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton this set may contain so called critical multipliers this special subset of lagrange multipliers defines to a great extent stability pattern of the solution in question subject to parametric perturbations criticality of a lagrange multiplier can be equivalently characterized by the absence of the local lipschitzian error bound in terms of the natural residual of the optimality system in this work taking the view of criticality as that associated to the error bound we extend the concept to general nonlinear equations not necessarily with primal dual optimality structure among other things we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor asymptotically thin classes of perturbations critical solutions can be stable under rich classes of perturbations this fact is quite remarkable considering that in the case of nonisolated solutions critical solutions usually form a thin subset within all the solutions we also note that the results for general equations lead to some new insights into the properties of critical lagrange multipliers i e solutions of equations with primal dual structure

optimality conditions in mathematical programming and composite optimization new second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented they are derived from a general second order fermat s rule for the minimization of a function over an arbitrary subset of a banach space the necessary conditions are more accurate than the recent results of kawasaki 1988 and cominetti 1989 but more importantly in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality we point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function our conditions are especially important when one deals with functional constraints when the cone defining the constraints is polyhedral we recover the classical conditions of ben tal zowe 1982 and cominetti 1990

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered