a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

new extremal principles with applications to stochastic and semi infinite programming this paper develops new extremal principles of variational analysis that are motivated by applications to constrained problems of stochastic programming and semi infinite programming without smoothness and or convexity assumptions these extremal principles concern measurable set valued mappings multifunctions with values in finite dimensional spaces and are established in both approximate and exact forms the obtained principles are instrumental to derive via variational approaches integral representations and upper estimates of regular and limiting normals cones to essential intersections of sets defined by measurable multifunctions which are in turn crucial for novel applications to stochastic and semi infinite programming

the secq linear regularity and the strong chip for an infinite system of closed convex sets in normed linear spaces we consider a finite or infinite family of closed convex sets with nonempty intersection in a normed space a property relating their epigraphs with their intersection s epigraph is studied and its relations to other constraint qualifications such as the linear regularity the strong chip and jameson s g property are established with suitable continuity assumption we show how this property can be ensured from the corresponding property of some of its finite subfamilies

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

tangential extremal principles for finite and infinite systems of sets ii applications to semi infinite and multiobjective optimization this paper contains selected applications of the new tangential extremal principles and related results developed in mordukhovich and phan math program 2011 to calculus rules for infinite intersections of sets and optimality conditions for problems of semi infinite programming and multiobjective optimization with countable constraints

generalized semi infinite optimization a first order optimality condition and examples we consider a generalized semi infinite optimization problem gsip of the form gsip min f x xem where m x n hi x 0i l m g x y 0 y y x and all appearing functions are continuously differentiable furthermore we assume that the sety x is compact for allx under consideration and the set valued mappingy is upper semi continuous the difference with a standard semi infinite problem lies in thex dependence of the index sety we prove a first order necessary optimality condition of fritz john type without assuming a constraint qualification or any kind of reduction approach moreover we discuss some geometrical properties of the feasible setm

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

wave equations with concentrated nonlinearities in this paper we address the problem of wave dynamics in the presence of concentrated nonlinearities given a vector field v on an open subset of and a discrete set with n elements we define a nonlinear operator v y on which coincides with the free laplacian when restricted to regular functions vanishing at y and which reduces to the usual laplacian with point interactions placed at y when v is linear and represented by a hermitian matrix we then consider the nonlinear wave equation and study the corresponding cauchy problem giving an existence and uniqueness result when v is lipschitz the solution of such a problem is explicitly expressed in terms of the solutions of two cauchy problems one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part the main properties of the solution are given and when y is a singleton the mechanism and details of blow up are studied

a class of nonlinear schrodinger equations with concentrated nonlinearity abstract we consider the nonlinear schrodinger equation in dimension one with a nonlinearity concentrated in a finite number of points detailed results on the local existence of the solution in fractional sobolev spaces h are given we also prove the conservation of the l 2 norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space h 1 finally we prove the existence of blow up solutions for strongly attractive interaction

a posteriori error estimates for nonconforming approximations of steklov eigenvalue problems this paper deals with a posteriori error estimators for the non conforming crouzeix raviart finite element approximations of the steklov eigenvalue problem first we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue next we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges finally based on these results we introduce two simpler equivalent error estimators the analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms

local optimization of dynamic programs with guaranteed satisfaction of path constraints an algorithm is proposed for locating a feasible point satisfying the kkt conditions to a specified tolerance of feasible inequality path constrained dynamic programs pcdp within a finite number of iterations the algorithm is based on iteratively approximating the pcdp by restricting the right hand side of the path constraints and enforcing the path constraints at finitely many time points the main contribution of this article is an adaptation of the semi infinite program sip algorithm proposed in mitsos 2011 to pcdp it is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first order kkt conditions of the pcdp to a specified tolerance the main assumptions are i availability of a nonlinear program nlp local solver that generates a kkt point of the constructed approximation to pcdp at each iteration if this problem is indeed feasible ii existence of a slater point of the pcdp that also satisfies the first order kkt conditions of the pcdp to a specified tolerance iii all kkt multipliers are nonnegative and uniformly bounded with respect to all iterations the performance of the algorithm is analyzed through two numerical case studies

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

spatial structure of the focusing singularity of the nonlinear schro dinger equation a geometrical analysis the cubic nonlinear schrodinger equation has solutions that become singular in finite time for spatial dimension d geq 2 numerical simulations demonstrate that in three dimensions the blowup is self similar and symmetric with the structure described by a nonlinear nonautonomous profile equation in two dimensions the blowup is again believed to be symmetric but the self similarity is weakly broken with structure described by the profile equation in the limit as d tends to 2 from above this paper gives a proof of the existence of a locally unique solution to the profile equation for d 2 and sufficiently close to 2 satisfying the boundary and global conditions associated with the blowup solution dynamical systems methods are used to transform previously derived asymptotic analysis into constructions of manifolds of solutions satisfying the relevant boundary conditions and to follow these manifolds to show that they have a transverse intersection

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

variational analysis in semi infinite and infinite programming ii necessary optimality conditions this paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development canovas et al siam j optim 20 2009 pp 1504 1526 from the viewpoint of robust lipschitzian stability the main results establish necessary optimality conditions for broad classes of semi infinite and infinite programs where objectives are generally described by nonsmooth and nonconvex functions on banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets the results obtained are new in both smooth and nonsmooth settings of semi infinite and infinite programming we illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

lagrange multipliers in nonsmooth semi infinite optimization problems using the variational analysis technique in terms of the epi coderivative we provide lagrange multiplier rules for a class of semi infinite optimization problems where all functions are lower semicontinuous or locally lipschitz

subdifferentials of nonconvex supremum functions and their applications to semi infinite and infinite programs with lipschitzian data the paper is devoted to the subdifferential study and applications of the supremum of uniformly lipschitzian functions over arbitrary index sets with no topology based on advanced techniques of variational analysis we evaluate major subdifferentials of the supremum functions in the general framework of asplund in particular reflexive spaces with no convexity or relaxation assumptions the results obtained are applied to deriving new necessary optimality conditions for nonsmooth and nonconvex problems of semi infinite and infinite programming

tangential extremal principles for finite and infinite systems of sets i basic theory in this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets the results obtained unified under the name of tangential extremal principles combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets the first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi infinite programming and multiobjective optimization

subdifferentials of value functions and optimality conditions for dc and bilevel infinite and semi infinite programs the paper concerns the study of new classes of parametric optimization problems of the so called infinite programming that are generally defined on infinite dimensional spaces of decision variables and contain among other constraints infinitely many inequality constraints these problems reduce to semi infinite programs in the case of finite dimensional spaces of decision variables we focus on dc infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints the main results establish efficient upper estimates of certain subdifferentials of intrinsically nonsmooth value functions in dc infinite programs based on advanced tools of variational analysis and generalized differentiation the value marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well posedness and sensitivity in this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric dc infinite programs and their remarkable specifications finally we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization the results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi infinite counterparts

on strong and total lagrange duality for convex optimization problems abstract we give some necessary and sufficient conditions which completely characterize the strong and total lagrange duality respectively for convex optimization problems in separated locally convex spaces we also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals in the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so called farkas minkowski and locally farkas minkowski conditions for systems of convex inequalities recently used in the literature moreover we show that our new results extend some existing ones in the literature

convergence analysis of the gauss newton method for convex inclusion and convex composite optimization problems abstract using the convex process theory we study the convergence issues of the iterative sequences generated by the gauss newton method for the convex inclusion problem defined by a cone c and a smooth function f the derivative is denoted by f the restriction in our consideration is minimal and even in the classical case the initial point x 0 is assumed to satisfy the following two conditions f is lipschitz around x 0 and the convex process t x 0 defined by t x 0 f x 0 c is surjective our results are new in giving sufficient conditions which are weaker than the known ones and have a remarkable property being affine invariant ensuring the convergence of the iterative sequence with initial point x 0 the same study is also made for the so called convex composite optimization problem with objective function given as the composite of a convex function with a smooth map

a combined finite element and bayesian optimization framework for shape optimization in spectral geometry abstract we present a novel computational framework for shape optimization problems arising in spectral geometry the goal in such problems is to identify domains in r d which are the global optima of certain functions of the spectrum of elliptic operators on the domains we propose the use of a combined finite element and bayesian optimization fem bo framework in this context and demonstrate the key ideas on two concrete examples we study the polya szego conjecture on polygons and demonstrate that our proposed framework yields the theoretically proven result for triangles and quadrilaterals and also provides compelling numerical evidence for the case of pentagons we next study a variant of this conjecture for the steklov eigenvalue problem

boundary half strips and the strong chip when the subdifferential sum rule formula holds for the indicator functions iota c and iota d of two closed convex sets c and d of a locally convex space x the pair c d is said to have the strong conical hull intersection property the strong chip the specification of a well known theorem due to moreau to the case of the support functionals sigma c and sigma d subsumes the fact that the pair c d has the strong chip whenever the inf convolution of sigma c and sigma d is exact in this article we prove in the setting of euclidean spaces that if the pair c d has the strong chip while the boundary of c does not contain any half strip then the inf convolution of sigma c and sigma d is exact moreover when the boundary of a closed and convex set c does contain a half strip it is possible to find a closed and convex set d such that the pair c d has the strong chip while the inf convolution of sigma c and sigma d is not exact the validity of the converse of moreau s theorem in euclidean spaces is thus associated with the absence of half strips within the boundary of concerned convex sets

subdifferential characterization of probability functions under gaussian distribution probability functions figure prominently in optimization problems of engineering they may be nonsmooth even if all input data are smooth this fact motivates the consideration of subdifferentials for such typically just continuous functions the aim of this paper is to provide subdifferential formulae of such functions in the case of gaussian distributions for possibly infinite dimensional decision variables and nonsmooth locally lipschitzian input data these formulae are based on the spheric radial decomposition of gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other by successively adding additional hypotheses conditions are satisfied under which the probability function is locally lipschitzian or even differentiable

stability of the log log bound for blow up solutions to the critical non linear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for initial data with negative energy finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so called log log law has been proved in 10 11 we focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur we establish the stability in energy space h1 of the log log upper bound exhibited in the negative energy case and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions

constraint qualifications and optimality conditions for nonconvex semi infinite and infinite programs the paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so called infinite programming that are generally defined on infinite dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary may not be compact index sets these problems reduce to semi infinite programs in the case of finite dimensional spaces of decision variables we extend the classical mangasarian fromovitz and farkas minkowski constraint qualifications to such infinite and semi infinite programs the new qualification conditions are used for exact calculations of the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation in the further development we derive first order necessary optimality conditions for infinite and semi infinite programs which are new in both finite dimensional and infinite dimensional settings

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

constraint qualifications for convex inequality systems with applications in constrained optimization for an inequality system defined by an infinite family of proper convex functions we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well known constraint qualifications including the basic constraint qualification studied by hiriart urrutty and lemarechal and by li nahak and singer extensions of known results to more general settings are presented and applications to particular important problems such as conic programming and approximation theory are also studied

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

blow up for the 1d nonlinear schrodinger equation with point nonlinearity i basic theory abstract we consider the 1d nonlinear schrodinger equation nls with focusing point nonlinearity 0 1 i t x 2 p 1 0 where x is the delta function supported at the origin in this work we show that 0 1 shares many properties in common with those previously established for the focusing autonomous translationally invariant nls 0 2 i t p 1 0 the critical sobolev space h c for 0 1 is c 1 2 1 p 1 whereas for 0 2 it is c d 2 2 p 1 in particular the l 2 critical case for 0 1 is p 3 we prove several results pertaining to blow up for 0 1 that correspond to key classical results for 0 2 specifically we 1 obtain a sharp gagliardo nirenberg inequality analogous to weinstein 44 2 apply the sharp gagliardo nirenberg inequality and a local virial identity to obtain a sharp global existence blow up threshold analogous to weinstein 44 glassey 17 in the case c 0 and duyckaerts holmer roudenko 12 guevara 18 and fang xie cazenave 13 for 0 c 1 3 prove a sharp mass concentration result in the l 2 critical case analogous to tsutsumi 43 merle tsutsumi 36 and 4 show that minimal mass blow up solutions in the l 2 critical case are pseudoconformal transformations of the ground state analogous to merle 28

metric regularity in convex semi infinite optimization under canonical perturbations this paper is concerned with the lipschitzian behavior of the optimal set of convex semi infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function in this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping this condition consists of the slater constraint qualification together with a certain additional requirement in the karush kuhn tucker conditions for linear problems this sufficient condition turns out to be also necessary for the metric regularity and it is equivalent to some well known stability concepts

complete characterizations of local weak sharp minima with applications to semi infinite optimization and complementarity abstract in this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials or tangent cones and subderivatives or their mixture in finite dimensional spaces the results obtained not only extend previous ones in the literature but also allow us to provide new types of criteria for local weak sharpness applications of the developed theory are given to semi infinite programming and to a new class of semi infinite complementarity problems

on the stability of the feasible set in optimization problems this paper provides stability theorems for the feasible set of optimization problems posed in locally convex topological vector spaces the problems considered in this paper have an arbitrary number of inequality constraints and one constraint set different models are discussed depending on the properties of the constraint functions linear or not convex or not but at least lower semicontinuous and one closed constraint set but not necessarily convex the parameter space is formed by systems of the same type as the nominal one with the same space of variables and the same number of constraints where the constraint set can be perturbed or not equipped with the metric of the uniform convergence on the positive multiples of a fixed barrelled neighborhood of zero in finite dimensions this topology describes the uniform convergence on compact sets and in the particular case that the constraints are linear the uniform convergence of the vector coefficients the paper examines in a unified way the lower and upper semicontinuity and the closedness of the feasible set mapping the stable consistency of the constraint system with respect to arbitrary and right hand side perturbations tuy and robinson regularities and other desirable stability properties of the feasible set