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

ideal qn spaces we continue to investigate an ideal version of qn space a jqn space introduced by p das and d chandra 8 following r filipow and m staniszewski 15 we show that an ideal j on contains an isomorphic copy of the ideal fin fin on if and only if every topological space is a jqn space if j does not contain an isomorphic copy of the ideal fin fin then the baire space is not a jqn space however if p c then there is an ideal j not containing an isomorphic copy of the ideal fin fin and there is a jqn space which is not a qn space we prove few results related to an ideal version of an s1 space indeed we show that there is no ideal j such that the notion of an s1 j space is trivial consequently the ideal version of scheepers conjecture does not hold for ideals containing an isomorphic copy of fin fin

newton type methods for optimization problems without constraint qualifications we consider equality constrained optimization problems where a given solution may not satisfy any constraint qualification but satisfies the standard second order sufficient condition for optimality based on local identification of the rank of the constraints degeneracy via the singular value decomposition we derive a modified primal dual optimality system whose solution is locally unique nondegenerate and thus can be found by standard newton type techniques using identification of active constraints we further extend our approach to mixed equality and inequality constrained problems and to mathematical programs with complementarity constraints mpcc in particular for mpcc we obtain a local algorithm with quadratic convergence under the second order sufficient condition only without any constraint qualifications not even the special mpcc constraint qualifications

a note on a new ideal abstract in this paper we study a new ideal wr the main result is the following an ideal is not weakly ramsey if and only if it is above wr in the kat tov order weak ramseyness was introduced by laflamme in order to characterize winning strategies in a certain game we apply result of natkaniec and szuca to conclude that wr is critical for ideal convergence of sequences of quasi continuous functions we study further combinatorial properties of wr and weak ramseyness answering a question of filipow et al we show that wr is not 2 ramsey but every ideal on isomorphic to wr is mon every sequence of reals contains a monotone subsequence indexed by an i positive set

new bifurcation theorems via the second order optimality conditions we derive new sufficient conditions for bifurcation relying on subtle second order necessary optimality conditions for abnormal equality constrained optimization problems we relate these conditions to the known ones and demonstrate the cases when the new conditions are easier to verify

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

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

multiparameter local bifurcation based on the linear part abstract in this paper we give a result in multiparameter local bifurcation theory this result is a generalization of the hopf bifurcation theorem and of a previous result by j k hale our result can be applied to fredholm operators with arbitrary index

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

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

on ideal equal convergence ii abstract there are generalizations and complementations of results from our joint papers with filipow 15 and 16 we study relationships between ideal equal convergence and various kinds of ideal convergences of sequences of real functions furthermore we consider ideal version of the bounding number on sets from coideals

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

second order necessary conditions for nonlinear optimization problems with abstract constraints the degenerate case in this paper we derive second order necessary conditions for optimality for an optimization problem with abstract constraints in banach spaces results for the nondegenerate case derived earlier h gfrerer siam j control optim 45 2006 pp 972 997 are extended to the degenerate case for the mathematical programming problem where the constraints are given by equality and finitely many inequality constraints our approach applies to the degenerate case when the equality constraints are not regular our results appear to be new even in this special case our second order necessary conditions are contained in the gap between the standard necessary and sufficient conditions where the only difference is the change from a nonstrict to a strict inequality our results are formulated in such a way to be applicable also to vector optimization problems

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

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

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

the pth order optimality conditions for inequality constrained optimization problems in this paper we present pth order necessary conditions for optimality for an optimization problem with inequality constraints in the finite dimensional spaces models of this type arise as the discretization of optimal control problems calculus of variations problems and other problems the paper addresses the degenerate nonregular case when the active constraint gradients are linearly dependent in this case the classical necessary conditions are either inapplicable or trivially satisfied with the zero multiplier corresponding to the objective function the proposed high order optimality conditions give new and nontrivial conditions for nonregular cases and reduce to the classical conditions for regular cases

on attraction of newton type iterates to multipliers violating second order sufficiency conditions assuming that the primal part of the sequence generated by a newton type e g sqp method applied to an equality constrained problem converges to a solution where the constraints are degenerate we investigate whether the dual part of the sequence is attracted by those lagrange multipliers which satisfy second order sufficient condition sosc for optimality or by those multipliers which violate it this question is relevant at least for two reasons one is speed of convergence of standard methods the other is applicability of some recently proposed approaches for handling degenerate constraints we show that for the class of damped newton methods convergence of the dual sequence to multipliers satisfying sosc is unlikely to occur we support our findings by numerical experiments we also suggest a simple auxiliary procedure for computing multiplier estimates which does not have this undesirable property finally some consequences for the case of mixed equality and inequality constraints are discussed

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

sensitivity analysis for cone constrained optimization problems under the relaxed constraint qualifications we present the local sensitivity analysis for cone constrained optimization problems under the cq type conditions significantly weaker than those traditionally used in this context our basic sensitivity results are established under the first or second order sufficient optimality conditions combined with the estimate of the distance to the feasible set of the perturbed problem we demonstrate how such an estimate can be obtained under the assumptions weaker than robinson s cq and establish the corresponding sensitivity results finally we apply our results to sensitivity analysis and relaxation schemes for mathematical programs with complementarity constraints

second order necessary optimality conditions for problems without a priori normality assumptions in this article we derive second order necessary conditions of optimality for an abstract optimization problem with equality and inequality constraints and constraints in the form of an inclusion into a given closed set an important feature is that our optimality conditions dispense with any a priori normality assumptions such as robinsons constraint qualification and remain informative even for abnormal points moreover our optimality conditions take into account the second order effect of the curvature of the set in the inclusion constraints

ideal equal baire classes abstract for any borel ideal we characterize ideal equal baire system generated by the families of continuous and quasi continuous functions i e the families of ideal equal limits of sequences of continuous and quasi continuous functions

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

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

ideal invariant injections abstract for an ideal i on we introduce the notions of i invariant and bi i invariant injections from to we study injections that are invariant with respect to selected classes of ideals we show some applications to ideal convergence

collective motions of spatio temporal systems abstract a new mathematical approach to the theory of collective motions and quasiparticles is presented this approach is based on some recent results in the theory of commuting non selfadjoint operators

second order conditions for optimization problems with constraints using a projective approach new necessary conditions and new sufficient conditions for optimization problems with explicit or implicit constraints are examined they are compared to previous ones a particular emphasis is given to mathematical programming problems with non polyhedral constraints this case occurs in particular when the constraints are defined in functional spaces

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

three kinds of convergence and the associated i baire classes abstract we consider ideal versions of pointwise discrete and equal convergence of sequences of functions defining in a natural way ideal pointwise discrete equal baire classes of functions we show that these classes are equal to their classical counterparts for ideals for which there is a winning strategy in a game introduced by laflamme 1996 10 in the proofs we make extensive use of a characterization in terms of filters f which are diagonalizable by f universal sets of a winning strategy this article extends results of laczkovich and reclaw 2009 9 and debs and saint raymond 2009 5

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

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

pointwise versus equal quasi normal convergence via ideals abstract we prove a characterization showing when the ideal pointwise convergence does not imply the ideal equal aka quasi normal convergence the characterization is expressed in terms of a cardinal coefficient related to the bounding number b we also prove a characterization showing when the ideal equal limit is unique

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

the structure of the zero sets of nonlinear mappings near generic multiparameter eigenvalues abstract let x and y be real banach spaces let f r p x y be a c l mapping l 2 and let l df 0 for r p suppose also that f 0 0 for all and at some point 0 dim n l 0 n 0 codim r l 0 m 0 we examine the structure of the zero set of f in a neighbourhood of 0 0 and give conditions under which this zero set is diffeomorphic to the zero set of the lower order terms in the taylor expansion of f near 0 0 these results tell us about the structure of the set of non trivial solutions bifurcating from zero when there are several parameters and the dimension of the null space of the linearization is greater than 1 finally we consider the genericity of the conditions used