lagrange multipliers theorem and saddle point optimality criteria in mathematical programming we prove a version of lagrange multipliers theorem for nonsmooth functionals defined on normed spaces applying these results we extend some results about saddle point optimality criteria in mathematical programming

kkt conditions for weak compact convex sets theorems of the alternative and optimality conditions abstract we present several equivalent conditions for the karush kuhn tucker conditions for weak compact convex sets using them we extend several existing theorems of the alternative in terms of weak compact convex sets such extensions allow us to express the kkt conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints furthermore several new equivalent optimality conditions for optimization problems with inequality constraints are obtained

on a class of compactly epi lipschitzian sets the paper is devoted to the study of the so called compactly epi lipschitzian sets these sets are needed for many aspects of generalized differentiation particulary for necessary optimality conditions stability of mathematical programming problems and calculus rules for subdifferentials and normal cones we present general conditions under which sets defined by general constraints are compactly epi lipschitzian this allows us to show how the compact epi lipschitzness properties behave under set intersections

necessary conditions for constrained optimization problems with semicontinuous and continuous data we consider nonsmooth constrained optimization problems with semicontinuous and continuous data in banach space and derive necessary conditions sl without constraint qualification in terms of smooth subderivatives and normal cones these results in different versions are set in reflexive and smooth banach spaces

classical optimality conditions under weaker assumptions in this article an optimization problem that incorporates equality constraints inequality constraints and an abstract constraint is considered we prove the classical optimality conditions first order and second order necessary and sufficient under weaker assumptions functions involved are continuous around and differentiable at the optimal point instead of continuously differentiable or strictly differentiable and or twice differentiable the proof is based on the exact expressions for the contingent cone and the second order contingent hull to the feasible set established in this article it is interesting to note that our second order lagrange multiplier rule for minimization problems incorporating a more generalized abstract constraint has an improved appearance direct applications to max type minimization problems and pareto optimality are mentioned

equivalence among various derivatives and subdifferentials of the distance function abstract for a nonempty closed set c in a normed linear space x with uniformly g teaux differentiable norm it is shown that the distance function d c is strictly differentiable at x x c iff it is regular at x iff its modified upper or lower dini subdifferential at x is a singleton iff its upper or lower dini subdifferential at x is nonempty iff its upper or lower dini derivative at x is subadditive moreover if x is a hilbert space then d c is frechet differentiable at x x c iff its frechet subdifferential at x is nonempty many characteristics of proximally smooth sets and convex closed sets in a hilbert space are also given

reeb orbits and the minimal discrepancy of an isolated singularity let a be an affine variety inside a complex n dimensional vector space which has an isolated singularity at the origin the intersection of a with a very small sphere turns out to be a contact manifold called the link of a any contact manifold contactomorphic to the link of a is said to be milnor fillable by a if the first chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy which is an important invariant in birational geometry we define an invariant of the link up to contactomorphism using conley zehnder indices of reeb orbits and then we relate this invariant with the minimal discrepancy as a result we show that the standard contact five dimensional sphere has a unique milnor filling up to normalization proving a conjecture by seidel

constraint qualifications in semi infinite systems and their applications in nonsmooth semi infinite problems with mixed constraints in this paper we present several constraint qualifications and we show that conditions guarantee the nonvacuity of the karush kuhn tucker multipliers set for nonsmooth semi infinite programming problems with mixed constraints the relationships with various constraint qualifications are investigated all results are given in terms of the michel penot subdifferential

the lagrange multiplier rule for multifunctions in banach spaces we study general constrained multiobjective optimization problems with objectives being closed multifunctions in banach spaces in terms of the coderivatives and normal cones we provide generalized lagrange multiplier rules as necessary optimality conditions of the above problems in an asplund space setting sharper results are presented

nondifferentiable multiplier rules for optimization and bilevel optimization problems in this paper we study optimization problems with equality and inequality constraints on a banach space where the objective function and the binding constraints are either differentiable at the optimal solution or lipschitz near the optimal solution necessary and sufficient optimality conditions and constraint qualifications in terms of the michel penot subdifferential are given and the results are applied to bilevel optimization problems

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

boundedness of certain sets of lagrange multipliers in vector optimization in this paper we establish lagrange multiplier rules in terms of michel penot subdifferential for nonsmooth vector optimization problem a constraint qualification or regularity condition in terms of michel penot subdifferential is given and under this regularity condition the boundedness of certain sets of lagrange multipliers are discussed

a multiplier rule on a metric space a multiplier rule is proved for constrained minimization problems defined on a metric spaces the proof requires a generalization of the values of a derivative in the classical case that the metric space is a normed space

log canonical thresholds in positive characteristic in this paper we study the singularities of pairs in arbitrary characteristic via jet schemes for a smooth variety x in characteristic 0 ein lazarsfeld and musta showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on x via this correspondence one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation we now extend this result to positive characteristic in particular we prove musta s log canonical threshold formula avoiding the use of log resolutions making the formula available also in positive characteristic as a consequence we get a comparison theorem via reduction modulo p and a version of inversion of adjunction in positive characteristic

necessary conditions for the bolza problem in this paper we provide necessary conditions for the bolza problem which are obtained by using a generalization of frechet derivative and by reducing the original problem to a convex problem the investigation of this convex problem is carried out by approximating the given functions by smooth ones

a nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions abstract fuzzy optimization conditions in terms of the frechet subdifferential for reflexive spaces were investigated by borwein treiman and zhu 1998 in 1 to achieve the nondegenerate form it is well known that some qualification conditions should be assumed in this paper we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in asplund spaces in particular reflexive spaces for optimization problems with semi continuous and continuous data the results are even new in finite dimensional frameworks

on higher dimensional complex plateau problem let x be a compact connected strongly pseudoconvex cr manifold of real dimension 2n 1 in mathbb c n it has been an interesting question to find an intrinsic smoothness criteria for the complex plateau problem for n ge 3 and n n 1 yau found a necessary and sufficient condition for the interior regularity of the harvey lawson solution to the complex plateau problem by means of kohn rossi cohomology groups on x in 1981 for n 2 and n ge n 1 the first and third authors introduced a new cr invariant g 1 1 x of x the vanishing of this invariant will give the interior regularity of the harvey lawson solution up to normalization for n ge 3 and n n 1 the problem still remains open in this paper we generalize the invariant g 1 1 x to higher dimension as g lambda n 1 x and show that if g lambda n 1 x 0 then the interior has at most finite number of rational singularities in particular if x is calabi yau of real dimension 5 then the vanishing of this invariant is equivalent to give the interior regularity up to normalization

the fuzzy intersection rule in variational analysis with applications the fuzzy intersection rule for frechet normal cones in asplund spaces was established by mordukhovich and the author using the extremal principle which appears more convenient to apply in some applications in this paper we present a complete discussion of this rule in various aspects we show that the fuzzy intersection rule is another characterization of the asplund property of the space various applications are considered as well in particular a complete set of fuzzy calculus rules for general lower semicontinuous functions are established

on the fuzzy intersection rule abstract the paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory new forms of the rule and further applications are presented

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

the ekeland variational principle for set valued maps involving coderivatives in this paper we use the frechet clarke and mordukhovich coderivatives to obtain variants of the ekeland variational principle for a set valued map f and establish optimality conditions for set valued optimization problems our technique is based on scalarization with the help of a marginal function associated with f and estimates of subdifferentials of this function in terms of coderivatives of f

the ekeland variational principle for henig proper minimizers and super minimizers abstract in this paper we consider for the first time approximate henig proper minimizers and approximate super minimizers of a set valued map f with values in a partially ordered vector space and formulate two versions of the ekeland variational principle for these points involving coderivatives in the sense of ioffe clarke and mordukhovich as applications we obtain sufficient conditions for f to have a henig proper minimizer or a super minimizer under the palais smale type conditions the techniques are essentially based on the characterizations of henig proper efficient points and super efficient points by mean of the henig dilating cones and the hiriart urruty signed distance function

metric regularity tangential distances and generalized differentiation in banach spaces abstract in this paper we establish new generalized differentiation rules in general banach spaces regarding normal cones to set images under functions coderivatives of compositions of set valued mappings as well as calculus results for normal compactness of sets and their images in addition to the metric regularity of mappings our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations basic properties as well as relations to similar notions some related results are also established

ideal adic semi continuity problem for minimal log discrepancies we discuss the ideal adic semi continuity problem for minimal log discrepancies by musta we study the purely log terminal case and prove the semi continuity of minimal log discrepancies when a kawamata log terminal triple deforms in the ideal adic topology

arcs on determinantal varieties we study arc spaces and jet schemes of generic determinantal varieties using the natural group action we decompose the arc spaces into orbits and analyze their structure this allows us to compute the number of irreducible components of jet schemes log canonical thresholds and topological zeta functions

on almost holomorphic lagrangian fibrations we prove that the pull back of an ample line bundle by an almost holomorphic lagrangian fibration is nef as an application we show birational semi rigidity of lagrangian fibrations

constraint qualifications and kkt conditions for bilevel programming problems in this paper we consider the bilevel programming problem blpp which is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem we extend well known constraint qualifications for nonlinear programming problems such as the abadie constraint qualification the kuhn tucker constraint qualification the zangwill constraint qualification the arrow hurwicz uzawa constraint qualification and the weak reverse convex constraint qualification to blpps and derive a karash kuhn tucker kkt type necessary optimality condition under these constraint qualifications without assuming the lower level problem satisfying the mangasarian fromovitz constraint qualification relationships among various constraint qualifications are also given

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

element by element a posteriori error estimation and improvement of stress solutions for two dimensional elastic problems a quantitative method of a posteriori error estimation based on finite element solutions is developed the proposed method can estimate the error of solutions obtained by the finite element method for two dimensional elastic problems with 4 node elements since the error is estimated element by element the proposed method does not require a large memory or long computing time not only rectangular elements but also arbitrarily shaped 4 node elements can be used in this method for estimating the error in the finite element computation with high accuracy the finite element solutions are improved by adding the estimated errors onto the original solutions the proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used

a unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization in this paper we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now we establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations with the help of quasi efficient solutions a generalized subdifferential of a vector mapping is introduced which generates a number of approximate subdifferentials frequently used in optimization in a unifying way the generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization an application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

necessary conditions for constrained optimization problems in smooth banach spaces and applications we derive necessary optimality conditions for constrained minimization problems with lower semicontinuous inequality constraints continuous equality constraints and a set constraint in smooth banach spaces then we apply these necessary optimality conditions to derive subdifferential characterizations of singular normal vectors to the epigraph and the graph of a function subdifferential calculus and necessary optimality conditions for mathematical programs with equilibrium constraints

generalized e quasi solutions in multiobjective optimization problems existence results and optimality conditions abstract this paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant we introduce the concept of generalized e quasi solution that extends other well known approximate efficiency notions of the literature and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets moreover we prove several existence results and a bound for the generalized e quasi solution set under convexity assumptions and by using asymptotic analysis tools finally we develop optimality conditions for a particular case of these kinds of e quasi solutions in nonsmooth convex and nonconvex problems

a fractional programming algorithm based on conic quasi newton trust region method for unconstrained minimization in this paper we present a new algorithm for quasi newton type trust region subproblem with a conic model solving unconstrained optimization problems in the algorithm we propose to use the method of dealing with fractional programming into the conic model as the conic model is of fractional form and the method can find the real solution of subproblem not an approximate solution like the available method this new approach can be easily generalized to any optimization method which its approximate subproblem with fractional expression whose approximate effectiveness to the class of highly vibrating objective functions is superior to that of the normal quadratic model this idea is what the author want to develop through the conic model as a special example in the paper the preliminary numerical test shows that the new method is more effective

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

multiplier rules under mixed assumptions of differentiability and lipschitz continuity in this paper we study nonlinear programming problems with equality inequality and abstract constraints where some of the functions are frechet differentiable at the optimal solution some of the functions are lipschitz near the optimal solution and the abstract constraint set may be nonconvex we derive fritz john type and karush kuhn tucker kkt type first order necessary optimality conditions for the above problem where frechet derivatives are used for the differentiable functions and subdifferentials are used for the lipschitz continuous functions constraint qualifications for the kkt type first order necessary optimality conditions to hold include the generalized mangasarian fromovitz constraint qualification the no nonzero abnormal multiplier constraint qualification the metric regularity of the constraint region and the calmness constraint qualification

dimensions of jet schemes of log singularities we characterize kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes our main result is theorem 2 4

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

epsilon maximum principle of pontryagin type and perturbation analysis of convex optimal control problems in this paper we study the first order behaviour of the optimal value function associated with a convex parametric problem of optimal control a formula for the subdifferential of the optimal value function is derived without assuming the existence of optimal solutions to the unperturbed problem the so called epsilon maximum principle is a key ingredient in the writing of our main sensitivity result

the linear nonconvex generalized gradient and lagrange multipliers a lagrange multiplier rule that uses small generalized gradients is introduced it includes both inequality and set constraints the generalized gradient is the linear generalized gradient it is smaller than the generalized gradients of clarke and mordukhovich but retains much of their nice calculus its convex hull is the generalized gradient of michel and penot if a function is lipschitz the tools used in the proof of this lagrange multiplier result are a coderivative a chain rule and a scalarization formula for this coderivative many smooth and nonsmooth lagrange multiplier results are corollaries of this result it is shown that the technique in this paper can be used for cases of equality inequality and set constraints if one considers the generalized gradient of mordukhovich an open question is does a lagrange multiplier result hold when one has equality constraints and uses the linear generalized gradient

a sharp lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints it is shown that a lagrange multiplier rule that uses approximate jacobians holds for mathematical programming problems involving lipschitzian functions finitely many equality constraints and convex set constraints it is sharper than the corresponding lagrange multiplier rules for the convex valued subdifferentials such as those of clarke optimization and nonsmooth analysis 2nd ed siam 1990 and michel and penot differential integral equations 5 1992 pp 433 454 the lagrange multiplier result is obtained by means of a controllability criterion and the theory of fans developed by a d ioffe math oper res 9 1984 pp 159 189 math programming 58 1993 pp 137 145 as an application necessary optimality conditions are derived for a class of constrained minimax problems an example is discussed to illustrate the nature of the multiplier rule

the first order necessary conditions for nonsmooth variational and control problems the first order necessary conditions for nonsmooth problems of lagrange bolza multiple integral and optimal control are given the necessary conditions are expressed in terms of differential inclusions we deal with some new differentials that generalize the differential of frechet in particular the pontryagin maximum principle is proved without the assumption of continuous differentiability

on incomplete lagrange function and saddle point optimality criteria in mathematical programming an incomplete lagrange function is used to study saddle point optimality criteria for a class of nonlinear programming problems under generalized convexity assumptions this study is further extended to certain class of fractional and generalized fractional programming problems as well

the arc space of a toric variety abstract the nash problem on arc families is affirmatively answered for a toric variety by ishii and kollar s paper which also shows the negative answer for general case the nash problem is one of questions about the relation between arc families and valuations in this paper the relation is described clearly for a toric variety the arc space of a toric variety admits an action of the group scheme determined by the torus each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation the dominant relation among the orbits is described in terms of the lattice points as a corollary we obtain the answer to the embedded version of the nash problem for an invariant ideal on a toric variety

a lagrange multiplier rule with small convex valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints it is shown that a lagrange multiplier rule involving the michel penot subdifferentials is valid for the problem minimizef0 x subject tofi x 0 i 1 m fi x 0 i m 1 n x q where all functionsf are lipschitz continuous andq is a closed convex set the proof is based on the theory of fans

birationally rigid hypersurfaces we prove that for n 4 all smooth hypersurfaces of degree n in n are birationally superrigid first discovered in the case n 4 by iskovskikh and manin in a work that started this whole direction of research this property was later conjectured to hold in general by pukhlikov the proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

necessary conditions in nonsmooth optimization the paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in banach spaces a lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints a general maximum principle for nonsmooth optimal control problems with state constraints and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the lipschitz continuity assumptions are removed a summary of relevant background results from analysis is provided

towards boundedness of minimal log discrepancies by the riemann roch theorem we introduce an approach via the riemann roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension after reducing it to the case of a gorenstein terminal singularity firstly we prove that the minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded secondly we recover the characterization of a gorenstein terminal three fold singularity by reid and the sharp bound for its minimal log discrepancy by markushevich without explicit classification finally we provide the sharp bound for a special four fold singularity whose general hyperplane section has a terminal piece

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

new approximation method in the proof of the maximum principle for nonsmooth optimal control problems with state constraints abstract traditional proofs of the pontryagin maximum principle pmp require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory when arbitrary values of control variable are inserted into the dynamic equations sussmann has drawn attention to the fact that the pmp remains valid when the dynamics are differentiable with respect to the state variable merely when the minimizing control is inserted into the dynamic equations this weakening of earlier hypotheses has been referred to as the lojasiewicz refinement arutyunov and vinter showed that these extensions of early versions of the pmp can be simply proved by finite dimensional approximations application of a lagrange multiplier rule in finite dimensions and passage to the limit this paper generalizes the finite dimensional approximation technique to a problem with state constraints where the use of needle variations of the optimal control had not been successful moreover the cost function and endpoint constraints are not assumed to be differentiable but merely locally lipschitz continuous the maximum principle is expressed in terms of michel penot subdifferential

saddle point optimality criteria involving p b d invexity and p b d pseudoinvexity in interval valued optimisation problems in this paper under new generalised convexity assumptions we investigate some connections between an lu optimal solution of a variational control problem governed by interval valued multiple inte

inversion of adjunction for local complete intersection varieties we extend inversion of adjunction and semicontinity of minimal log discrepancies from the case of a smooth ambient variety to the case of local complete intersection varieties

a variational method for finite element stress recovery and error estimation abstract a variational method for obtaining smoothed stresses from a finite element derived non smooth stress field is presented the method is based on minimizing a functional involving discrete least squares error plus a penalty constraint that ensures smoothness of the stress field an equivalent accuracy criterion is developed for the smoothing analysis which results in a c1 continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis application of the smoothing analysis to residual error estimation is also demonstrated

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

motivic integration over deligne mumford stacks abstract the aim of this article is to develop the theory of motivic integration over deligne mumford stacks and to apply it to the birational geometry of deligne mumford stacks

on the equivalence of some basic principles in variational analysis the primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization considering a broad class of banach spaces admitting smooth renorms with respect to some bornology we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions an extremal principle for nonconvex sets and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks further refinements of the equivalence result are obtained in the case of a frechet differentiable norm based on the new enhanced sum rule we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces

a unified separation theorem for closed sets in a banach space and optimality conditions for vector optimization using the technique of variational analysis and in terms of normal cones we establish unified separation results for finitely many closed not necessarily convex sets in banach spaces which not only cover the existing nonconvex separation results and a classical convex separation theorem but also recapture the approximate projection theorem with help of the separation result for closed sets we provide necessary and sufficient conditions for approximate pareto solutions of constrained vector optimization problems in particular we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting

error estimates and adaptive refinement for plate bending problems the zienkiewicz zhu error estimator is shown to be effective in problems of plate flexure when used in conjunction with triangular elements and an adaptive mesh generator allowing a prescribed size of elements to be developed very fast adaptive convergence for results of specified accuracy is achieved

on solving simple bilevel programs with a nonconvex lower level program in this paper we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint by using the value function of the lower level program we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint to deal with such a nonsmooth and nonconvex optimization problem we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint we show that if the sequence of penalty parameters is bounded then any accumulation point is a stationary point of the nonsmooth optimization problem and if the generated sequence is convergent and the extended mangasarian fromovitz constraint qualification holds at the limit then the limit point is a stationary point of the nonsmooth optimization problem we apply the smoothing projected gradient algorithm to the bilevel program if a calmness condition holds and to an approximate bilevel program otherwise preliminary numerical experiments show that the algorithm is efficient for solving the simple bilevel program

the self equilibration of residuals and complementary a posteriori error estimates in the finite element method it is demonstrated that the residual in a compatible displacement finite element solution can be partitioned into local self equilibrating systems on each element an a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved

application of the zienkiewicz zhu error estimator for plate and shell analysis the global error in energy norm is computed by summing all the elemental errors the method is well suited for adaptive mesh refinement as the error contribution for each element is known demonstration examples using uniform and adaptive refinement schemes for both the plate and shell analysis show the effectiveness of the method proposed

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

lagrange multipliers for set valued optimization problems associated with coderivatives in this paper we investigate a vector optimization problem p where objective and constraints are given by set valued maps we show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of p as solutions of a scalar optimization problem sp with single valued objective and constraint functions then applying some classical or recent results in optimization theory to sp and using estimates of subdifferentials of marginal functions we obtain optimality conditions for p expressed in terms of lagrange or sequential lagrange multipliers associated with various coderivatives of the set valued data

a fuzzy necessary optimality condition for non lipschitz optimization in asplund spaces in this paper we consider a general optimization problem in the broad class of asplund spaces we derive a new necessary optimality condition in the so called lagrangian fuzzy form without standard lipschitz conditions we also give a chain rule for frechet subdifferentials and subdifferential representations of frechet normals to constrained sets