eigenvalue convergence on perturbed lipschitz domains for elliptic systems with mixed general decompositions of the boundary abstract we consider eigenvalues of an elliptic operator l u x j a i j u x i where u u 1 u m t is a vector valued function and the coefficients a i j are m m matrices whose elements a i j are bounded and symmetric we perturb our domain 0 by adding a set of small measure t e to form the domain e we prescribe mixed boundary conditions on quite general decompositions of the boundary and look at the behavior of the eigenvalues of e as t e shrinks to zero we look at systems which satisfy either a strong ellipticity condition a legendre hadamard condition or in particular the system of linear elasticity

multiple probability assessments by dependent experts abstract when two or more information sources experts provide a decision maker with information on two or more random variables the decision maker using bayes s rule has an opportunity to a update a prior about the random variables and b calibrate the experts calibration is the process of adjusting the decision maker s likelihood about the experts assessments this article presents a model for this two way process and specializes to the case in which the experts assessment errors have a multivariate normal density in general we find that variables which the decision maker and the experts regard as independent a priori will be dependent a posteriori because of dependence in the assessment errors formulas for posterior densities are given for the normal model in this model the posterior density of the random variables depends on only a weighted average of the expert s means with weights that depend on the experts assessments of previously known quantities i also present a special case o

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

primal dual nonlinear rescaling method with dynamic scaling parameter update in this paper we developed a general primal dual nonlinear rescaling method with dynamic scaling parameter update pdnrd for convex optimization we proved the global convergence established 1 5 q superlinear rate of convergence under the standard second order optimality conditions the pdnrd was numerically implemented and tested on a number of nonlinear problems from cops and cute sets we present numerical results which strongly corroborate the theory

eigenvalues and resonances for domains with tubes neumann boundary conditions abstract we consider unbounded regions which consist of a bounded domain c joined to an unbounded region e by a tube t whose cross section is of small diameter on such a region we consider the laplacian with neumann boundary conditions we show that as 0 the spectral resonances converge to eigenvalues of c resonances of e or eigenvalues for a two point boundary value problem on an interval of the same length as the tube the main goal of our work is to give estimates for the rates of convergence

separation of nonconvex sets with general augmenting functions in this paper we consider two geometric optimization problems that are dual to each other and characterize conditions under which the optimal values of the two problems are equal this characterization relies on establishing separation results for nonconvex sets using general concave surfaces defined in terms of convex augmenting functions we prove separation results for augmenting functions that are bounded from below unbounded augmenting functions and asymptotic augmenting functions

the essential spectrum of neumann laplacians on some bounded singular domains in the present paper we consider neumann laplacians on singular domains of the type rooms and passages or combs and we show that in typical situations the essential spectrum can be determined from the geometric data moreover given an arbitrary closed subset s of the non negative reals we construct domains s such that the essential spectrum of the neumann laplacian on is just this set s

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

neumann spectral problem in a domain with very corrugated boundary let rn be a bounded domain we perturb it to a domain e attaching a family of small protuberances with room and passage like geometry e 0 is a small parameter peculiar spectral properties of neumann problems in so perturbed domains were observed for the first time by r courant and d hilbert we study the case when the number of protuberances tends to infinity as e 0 and they are e periodically distributed along a part of our goal is to describe the behavior of the spectrum of the operator ae e 1 e where e is the neumann laplacian in e and the positive function e is equal to 1 in we prove that the spectrum of ae converges as e 0 to the spectrum of a certain boundary value problem for the neumann laplacian in with boundary conditions containing the spectral parameter in a nonlinear manner its eigenvalues may accumulate to a finite point

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

a nonlinear programming algorithm based on non coercive penalty functions we consider first the differentiable nonlinear programming problem and study the asymptotic behavior of methods based on a family of penalty functions that approximate asymptotically the usual exact penalty function we associate two parameters to these functions one is used to control the slope and the other controls the deviation from the exact penalty we propose a method that does not change the slope for feasible iterates and show that for problems satisfying the mangasarian fromovitz constraint qualification all iterates will remain feasible after a finite number of iterations the same results are obtained for non smooth convex problems under a slater qualification condition

convergence of a hybrid projection proximal point algorithm coupled with approximation methods in convex optimization in order to minimize a closed convex function that is approximated by a sequence of better behaved functions we investigate the global convergence of a general hybrid iterative algorithm which consists of an inexact relaxed proximal point step followed by a suitable orthogonal projection onto a hyperplane the latter permits to consider a fixed relative error criterion for the proximal step we provide various sets of conditions ensuring the global convergence of this algorithm the analysis is valid for nonsmooth data in infinite dimensional hilbert spaces some examples are presented focusing on penalty barrier methods in convex programming we also show that some results can be adapted to the zero finding problem for a maximal monotone operator

a new approach to characterize the solution set of a pseudoconvex programming problem a new approach to characterize the solution set of a nonconvex optimization problem via its dual problem is proposed some properties of the lagrange function associated to the problem are investigated then characterizations of the solution set of the problem are established

efficiency and approachability of nonconvex bicriteria programs abstract in this paper we investigate equivalences between an efficient solution of a bicriteria program and a lower envelope point of a certain image set of the bicriteria program we also employ various kinds of approachabilities to characterize efficiency and proper efficiency for nonconvex bicriteria programs in particular nonlinear lagranian functions are applied to construct dual problems and to study stability for the corresponding constrained scalar optimization problems under certain conditions we show that the finite approachability proper efficiency stability and exact penalization of the relevant constrained scalar optimization problems are equivalent

characterizations of nonemptiness and compactness of the set of weakly efficient solutions for convex vector optimization and applications in this paper we give characterizations for the nonemptiness and compactness of the set of weakly efficient solutions of an unconstrained constrained convex vector optimization problem with extended vector valued functions in terms of the 0 coercivity of some scalar functions finally we apply these results to discuss solution characterizations of a constrained convex vector optimization problem in terms of solutions of a sequence of unconstrained vector optimization problems which are constructed with a general nonlinear lagrangian

a nonlinear lagrangian approach to constrained optimization problems in this paper we study nonlinear lagrangian functions for constrained optimization problems which are in general nonlinear with respect to the objective function we establish an equivalence between two types of zero duality gap properties which are described using augmented lagrangian dual functions and nonlinear lagrangian dual functions respectively furthermore we show the existence of a path of optimal solutions generated by nonlinear lagrangian problems and show its convergence toward the optimal set of the original problem we analyze the convergence of several classes of nonlinear lagrangian problems in terms of their first and second order necessary optimality conditions

generalized levitin polyak well posedness in constrained optimization in this paper we consider levitin polyak type well posedness for a general constrained optimization problem we introduce generalized levitin polyak well posedness and strongly generalized levitin polyak well posedness necessary and sufficient conditions for these types of well posedness are given relations among these types of well posedness are investigated finally we consider convergence of a class of penalty methods and a class of augmented lagrangian methods under the assumption of strongly generalized levitin polyak well posedness

domain perturbation for elliptic equations subject to robin boundary conditions abstract we prove domain perturbation theorems for linear and nonlinear elliptic equations under robin boundary conditions the theory allows very singular perturbation of domains in particular it includes cutting holes parts degenerating to a set of measure zero such as the dumbbell problem or wildly oscillating boundaries in the last case we show that the limiting problem is the dirichlet problem

a derivative free approach to constrained multiobjective nonsmooth optimization in this work we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints where objective and constraint function values can only be obtained by querying a black box we define a linesearch based solution method and we show that it converges to a set of pareto stationary points to this aim we carry out a theoretical analysis of the problem by only assuming lipschitz continuity of the functions more specifically we give new optimality conditions that take explicitly into account the bound constraints and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function finally we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems showing that our approach is promising when compared to a state of the art method from the literature

examples of ill behaved central paths in convex optimization this paper presents some examples of ill behaved central paths in convex optimization some contain infinitely many fixed length central segments others manifest oscillations with infinite variation these central paths can be encountered even for infinitely differentiable data

modified barrier functions theory and methods the nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and or the objective function into an equivalent problem the classical lagrangian which has important properties on the primal and the dual spaces the application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions mbf s and mbf methods mbfm s being classical lagrangians cl s for an equivalent problem the mbf s combine the best properties of the cl s and classical barrier functions cbf s but at the same time are free of their most essential deficiencies due to the excellent mbf properties new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed the mbfm have up to a superlinear rate of convergence and are to the classical barrier functions cbf s method as the multipliers method for augmented lagrangians is to the classical penalty function method based on the dual theory associated with mbf the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed the application of the mbf to linear lp and quadratic qp programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the cbf methods the numerical realization of the mbfm leads to the newton modified barrier method nmbm the excellent mbf properties allow us to discover that for any nondegenerate constrained optimization problem there exists a hot start from which the nmbm has a better rate of convergence a better complexity bound and is more stable than the interior point methods which are based on the classical barrier functions

a nonlinear parabolic equation with varying domain abstract this document considers a boundary value problem where the basic problem is to understand how new equilibrium solutions may appear for a given nonlinear function f as the domain omega is changed these new solutions being bifurcations from known equilibrium solutions for a given cubic function f an intuitive discussion of this phenomenon was given by hale indicating how stable non constant equilibria appear as secondary rather than primary bifurcations additional keywords continuity perturbations reprints

asymptotic analysis for penalty and barrier methods in variational inequalities we study the convergence of a class of penalty and barrier methods for solving monotone variational inequalities with constraints this class of methods is an extension of penalty and barrier methods for convex optimization to the setting of variational inequalities primal convergence is established under weaker conditions than usual the solution set is supposed either to be a compact set or for the case of interior barrier methods the sum of a compact set and a linear space dual convergence is also analyzed the analysis strongly exploits a new formula related to recession calculus

an augmented lagrangian approach with a variable transformation in nonlinear programming abstract tangent cone and regular normal cone of a closed set under an invertible variable transformation around a given point are investigated which lead to the concepts of 1 tangent cone of a set and 1 subderivative of a function when the notion of 1 subderivative is applied to perturbation functions a class of augmented lagrangians involving an invertible mapping of perturbation variables are obtained in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables a necessary and sufficient condition for the exact penalty representation under the proposed augmented lagrangian scheme is obtained for an augmenting function with an euclidean norm a sufficient condition resp a sufficient and necessary condition for an arbitrary vector resp 0 to support an exact penalty representation is given in terms of 1 subderivatives an example of the variable transformation applied to constrained optimization problems is given which yields several exact penalization results in the literature

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

conjugate duality for vector maximization problems in this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n dimensional orthant using this duality we obtain a zero gap duality for a vector maximization problem

reformulating decision theory using fuzzy set theory and shafer s theory of evidence abstract utilities and probabilities in decision theory are usually assessed by asking individuals to indicate their preferences between various uncertain choices in this paper we argue that 1 the utility of a consequence can be assessed as the membership function of the consequence in the fuzzy set satisfactory 2 the probability of an event instead of being directly assessed should be inferred from the evidence associated with that event the degree of evidence is quantified using shaferian basic probability assignments in addition we use the heisenberg uncertainty principle to argue for a change in one of the technical assumptions underlying decision theory as a result of this change some kinds of evidence will be observable in certain experiments but unobservable in others since probabilities are defined over the potential outcomes of an experiment they will only be defined over some but not all the evidence as a result the probabilities associated with different experiments could be inconsistent this formulation emphasizes the importance of new distinctions and not just new information in updating probabilities we argue that this formulation addresses many of the observed empirical deviations between decision theory and experiment it also addresses the anomalies of quantum physics we close with a brief discussion of directions for further research

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

combining probability distributions from dependent information sources inferences or decisions in the face of uncertainty should be based on all available information thus when probability distributions for an uncertain quantity are obtained from experts models or

new sequential lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs in this paper a new sequential lagrange multiplier condition characterizing optimality without a constraint qualification for an abstract nonsmooth convex program is presented in terms of the subdifferentials and the epsilon subdifferentials a sequential condition involving only the subdifferentials but at nearby points to the minimizer for constraints is also derived for a smooth convex program the sequential condition yields a limiting kuhn tucker condition at nearby points without a constraint qualification it is shown how the sequential conditions are related to the standard lagrange multiplier condition applications to semidefinite programs semi infinite programs and semiconvex programs are given several numerical examples are discussed to illustrate the significance of the sequential conditions

lagrangian regularization approach to constrained optimization problems abstract this paper proposes a new regularization approach referred to as the lagrangian regularization approach which aims to circumvent the nondifferentiability of a positively homogeneous function r m in a conceptual unconstrained reformulation for constrained optimization problems with the appropriate choices of regularizing function we obtain a family of smooth functions that include as special cases the existing penalty and barrier functions as such our approach can be used as an instrumental tool to resolve the nondifferentiability of r m as well as a unified way to construct penalty functions for convex programming cases we present its global convergence analysis

a unified augmented lagrangian approach to duality and exact penalization in this paper the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real valued function are established via a generalized augmented lagrangian and corresponding generalized augmented lagrangian problems in which no convexity is imposed on the augmenting function these results further imply a zero duality gap property between the primal problem and the generalized augmented lagrangian dual problem a necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented lagrangian is obtained in the context of constrained programs we show that generalized augmented lagrangians present a unified approach to several classes of exact penalization results some equivalences among exact penalization results are obtained

fuzzy set theory observer bias and probability theory abstract it is known that given certain seemingly implausible assumption probability functions act like fuzzy set membership functions this paper shows that these seemingly implausible assumptions are satisfied in probabilistic situations where the sole uncertainty arises from unknown but systematic observer bias now such situations could be considered as on the borderline between the domain of applicability of probability functions and the domain of fuzzy set membership functions hence the convergence between the rules for combining probabilistic functions and those for combining fuzzy set membership functions in these borderline instances is intuitively quite satisfying this convergence also suggests a number of other ways in which possibility theory and probability theory are interrelated

nonlinear lagrangian for multiobjective optimization and applications to duality and exact penalization duality and penalty methods are popular in optimization the study on duality and penalty methods for nonconvex multiobjective optimization problems is very limited in this paper we introduce vector valued nonlinear lagrangian and penalty functions and formulate nonlinear lagrangian dual problems and nonlinear penalty problems for multiobjective constrained optimization problems we establish strong duality and exact penalization results the strong duality is an inclusion between the set of infimum points of the original multiobjective constrained optimization problem and that of the nonlinear lagrangian dual problem exact penalization is established via a generalized calmness type condition

smoothing approximation to l1 exact penalty function for inequality constrained optimization abstract in this paper we propose a method to smooth l 1 exact penalty function for inequality constrained optimization it is shown that an approximate global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem under some mild conditions the method based on our smoothing function is shown to be globally convergent some numerical examples are given to illustrate the applicability of the present smoothing method

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

bifurcations caused by perturbing the domain in an elliptic equation we consider the neumann problem u u up on a continuous family of bounded domains which approach as 0 a set 0 with two connected components and analyze it as a bifurcation problem with two parameters and the bifurcation diagrams and the qualitative properties of the bifurcation sets for p odd and p even are obtained and the relations between them are studied by considering the problem u u au2 u3 for different values of a

a nonlinear lagrangian based on fischer burmeister ncp function this paper proposes a nonlinear lagrangian based on fischer burmeister ncp function for solving nonlinear programming problems with inequality constraints the convergence theorem shows that the sequence of points generated by this nonlinear lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound of solution depending on the penalty parameter is also established moreover the paper develops the dual approach associated with the proposed nonlinear lagrangian in which the related duality theorem is demonstrated furthermore it is shown that the condition number of the nonlinear lagrangian hessian at the optimal solution is proportional to the controlling penalty parameter numerical results for solving several nonlinear programming problems are reported showing that the new nonlinear lagrangian is superior over other known nonlinear lagrangians for solving some nonlinear programming problems

nonlinear rescaling vs smoothing technique in convex optimization we introduce an alternative to the smoothing technique approach for constrained optimization as it turns out for any given smoothing function there exists a modification with particular properties we use the modification for nonlinear rescaling nr the constraints of a given constrained optimization problem into an equivalent set of constraints the constraints transformation is scaled by a vector of positive parameters the lagrangian for the equivalent problems is to the correspondent smoothing penalty functions as augmented lagrangian to the classical penalty function or mbfs to the barrier functions moreover the lagrangians for the equivalent problems combine the best properties of quadratic and nonquadratic augmented lagrangians and at the same time are free from their main drawbacks sequential unconstrained minimization of the lagrangian for the equivalent problem in primal space followed by both lagrange multipliers and scaling parameters update leads to a new class of nr multipliers methods which are equivalent to the interior quadratic prox methods for the dual problem we proved convergence and estimate the rate of convergence of the nr multipliers method under very mild assumptions on the input data we also estimate the rate of convergence under various assumptions on the input data in particular under the standard second order optimality conditions the nr method converges with q linear rate without unbounded increase of the scaling parameters which correspond to the active constraints we also established global quadratic convergence of the nr methods for linear programming with unique dual solution we provide numerical results which strongly support the theory

the singularly perturbed domain and the characterization for the eigenfunctions with neumann boundary condition on considere un domaine mobile 0 parametre qui degenere partiellement quand 0 et on donne une caracterisation elaboree des comportements des fonctions propres du laplacien avec la condition limite de neumann

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

ideal weakly efficient solutions for vector optimization problems we establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded the solutions to be referred are ideal strong weakly efficient weakly pareto also several necessary and or sufficient conditions for the solution set to be non empty and compact are established moreover some characterizations of the non emptiness boundedness of the convex solution set in case the solutions are searched in a subset of the real line are also given however the solution set fails to be convex in general in addition special attention is addressed when the underlying cone is the non negative orthant and when the semi strict quasiconvexity of each component of the vector valued function is assumed our approach is based on the asymptotic description of the functions and sets some examples illustrating such an approach are also exhibited

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

a bfgs ip algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach this paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints it is assumed that the lagrangian of the problem is strongly convex the algorithm combines interior point methods for dealing with the inequality constraints and quasi newton techniques for accelerating the convergence feasibility of the iterates is progressively enforced thanks to shift variables and an exact penalty approach global and q superlinear convergence is obtained for a fixed penalty parameter global convergence to the analytic center of the optimal set is ensured when the barrier parameter tends to zero provided strict complementarity holds

exact penalty functions and stability in locally lipschitz programming in this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally lipschitz arbitrary simple constraints are also allowed assuming a weak stability condition we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function a tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied we apply these results to unify and extend some results for convex programming since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions our results provide a framework for extending these algorithms to problems with locally lipschitz functions

inequality systems and global optimization abstract solvability results for infinite inequality systems involving convex and difference of convex dc functions are given generalizations of farkas lemma are obtained these results are presented in terms of epigraphs of conjugate functions applications are given for characterizing global optimality and optimality of difference of convex dc optimization problems with convex inequality constraints solvability results for cone convex inequality systems are also given

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

decreasing functions with applications to penalization the theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions a multiplicative version of the inf convolution operation is studied for decreasing functions modified penalty functions for some constrained optimization problems are introduced that are in general nonlinear with respect to the objective function of the original problem as the perturbation function of a constrained optimization problem is decreasing the theory of decreasing functions is subsequently applied to the study of modified penalty functions the zero duality gap property and the exact penalization

sharp convergence rate of eigenvalues in a domain with a shrinking tube abstract in this paper we consider a class of singularly perturbed domains obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero we use an almgren type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues via courant fischer min max characterization and blow up analysis for scaled eigenfunctions

paper evidence measures based on fuzzy information an organized body of results pertaining to the analysis of evidence when the available knowledge is pervaded with imprecision is provided the evidence theory of dempster and shafer is extended to the case of fuzzy observations and fuzzy events upper and lower possibilities of such events are derived by iterating the generation process of upper and lower probabilities as done by dempster both probabilistic and possibilistic models are developed in parallel these evidence measures are used for decision evaluation when the available knowledge is poor the classical model of decision making under uncertainty is thus extended to the case when the consequences of a decision are only roughly described and their probabilities of occurrence modeled by intervals or fuzzy numbers

characterizing set containments involving infinite convex constraints and reverse convex constraints dual characterizations of the containment of a closed convex set defined by infinite convex constraints in an arbitrary polyhedral set in a reverse convex set defined by convex constraints and in another convex set defined by finite convex constraints are given a special case of these dual characterizations has played a key role in generating knowledge based support vector machine classifiers which are powerful tools in data classification and mining the conditions in these dual characterizations reduce to simple nonasymptotic conditions under slater s constraint qualification

dual conditions characterizing optimality for convex multi objective programs asymptotic necessary and sufficient conditions for a point to be a pareto minimum and weak minimum proper minimum for a convex multi objective program are given without a regularity condition it is further shown that in the cases of weak minimum and single objective function the asymptotic dual conditions reduce to nonasymptotic optimality conditions under slater s constraint qualification the results are applied to multi objective quadratic and linar programming problems numerical examples are given to illustrate the nature of the conditions

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

characterizations of solution sets of convex vector minimization problems complete dual characterizations of the weak and proper optimal solution sets of an infinite dimensional convex vector minimization problem are given the results are expressed in terms of subgradients lagrange multipliers and epigraphs of conjugate functions a dual condition characterizing the containment of a closed convex set defined by a cone convex inequality in a reverse convex set plays a key role in deriving the results simple lagrange multiplier characterizations of the solution sets are also derived under a regularity condition numerical examples are given to illustrate the significance of the results

patterns in parabolic problems with nonlinear boundary conditions we obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels we prove the convergence of eigenvalues and eigenfunctions of the laplace operator in such domains this information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively weakly coupled the main tools employed are the invariant manifold theory and a uniform trace theorem

on some properties and an application of the logarithmic barrier method we analyze the logarithmic barrier method for nonsmooth convex optimization in the setting of point to set theory this general framework allows us to both extend and include classical results we also propose an application for finding efficient points of nonsmooth constrained convex vector valued problems

the narrow escape problem the narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain the problem is equivalent to solving the mixed dirichlet neumann boundary value problem for the poisson equation with small dirichlet and large neumann parts the mixed boundary value problem which goes back to lord rayleigh originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the neumann problem in domains with bottlenecks we review here recent developments in the non standard asymptotics of the problem which are based on several ingredients a better resolution of the singularity of neumann s function resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks and the breakup of composite domains into simpler components the new methodology applies to two and higher dimensional problems selected applications are r

interior proximal and multiplier methods based on second order homogeneous kernels we study a class of interior proximal algorithms and nonquadratic multiplier methods for solving convex programs where the usual proximal quadratic term is replaced by an homogeneous functional of order two defined in terms of a convex function we prove under mild assumptions several new convergence results in both the primal and dual framework allowing also for approximate minimization in particular we introduce a new class of interior proximal methods which is globally convergent and allows for generating c multiplier methods with bounded hessians which exhibit strong convergence properties we also consider linearly constrained convex problems and establish global quadratic convergence rates results for linear programs we then study in detail a particular realization of these algorithms leading to a new class of logarithmic quadratic interior point algorithms which are shown to enjoy several attractive properties for solving constrained convex optimization problems

convex composite multi objective nonsmooth programming this paper examines nonsmooth constrained multi objective optimization problems where the objective function and the constraints are compositions of convex functions and locally lipschitz and g teaux differentiable functions lagrangian necessary conditions and new sufficient optimality conditions for efficient and properly efficient solutions are presented multi objective duality results are given for convex composite problems which are not necessarily convex programming problems applications of the results to new and some special classes of nonlinear programming problems are discussed a scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions

tubularity and asymptotic convergence of penalty trajectories in convex programming in this paper we give a sufficient condition for the asymptotic convergence of penalty trajectories in convex programming with multiple solutions we show that for a wide class of penalty methods the associated optimal trajectory converges to a particular solution of the original problem characterized through a minimization selection principle our main assumption for this convergence result is that all the functions involved in the convex program are tubular this new notion of regularity weaker than that of quasianalyticity is defined and studied in detail

asymptotic analysis for penalty and barrier methods in convex and linear programming we consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature we provide a systematic way to generate penalty and barrier functions in this class and we analyze the existence of primal and dual optimal paths generated by these penalty methods as well as their convergence to the primal and dual optimal sets for linear programming we prove that these optimal paths converge to single points

coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm we study the asymptotic behavior of the continuous flow generated by coupling the steepest descent method with a general class of penalty schemes for convex programming we establish convergence of the trajectories towards primal optimal solutions as well as convergence of some naturally associated dual paths the results are extended to the sequences generated by an implicit discretization scheme which corresponds to the coupling of an inexact proximal point iteration with the penalty schemes

penalty and smoothing methods for convex semi infinite programming in this paper we consider min max convex semi infinite programming to solve these problems we introduce a unified framework concerning remez type algorithms and integral methods coupled with penalty and smoothing methods this framework subsumes well known classical algorithms but also provides some new methods with interesting properties convergence of the primal and dual sequences are proved under minimal assumptions

asymptotic analysis of congested communication networks this paper is devoted to the mathematical study of a routing problem in telecommunication networks when the cost function is the average delay of communications we establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem the study is strongly related to the one of a partial inverse barrier method for linear programming

complete characterizations of stable farkas lemma and cone convex programming duality we establish necessary and sufficient conditions for a stable farkas lemma we then derive necessary and sufficient conditions for a stable duality of a cone convex optimization problem where strong duality holds for each linear perturbation of a given convex objective function as an application we obtain stable duality results for convex semi definite programs and convex second order cone programs

on dual convergence of the generalized proximal point method with bregman distances the use of generalized distances e g bregman distances instead of the euclidean one in the proximal point method for convex optimization allows for elimination of the inequality constraints from the subproblems in this paper we consider the proximal point method with bregman distances applied to linearly constrained convex optimization problems and study the behavior of the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem under rather general assumptions which cover most bregman distances of interest we obtain an ergodic convergence result namely that a sequence of weighted averages of the dual sequence converges to a specific point of the dual optimal set as an intermediate result we prove under the same assumptions that the dual central path generated by a large class of barriers including the generalized bregman distances converges to the same point

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

further study on the levitin polyak well posedness of constrained convex vector optimization problems abstract in this paper we first establish characterizations of the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with a general ordering cone with or without a cone constraint defined in a finite dimensional space using one of the characterizations we further establish for a convex vector optimization problem with a general ordering cone and a cone constraint defined in a finite dimensional space the equivalence between the nonemptiness and compactness of its weakly efficient solution set and the generalized type i levitin polyak well posednesses finally for a cone constrained convex vector optimization problem defined in a banach space we derive sufficient conditions for guaranteeing the generalized type i levitin polyak well posedness of the problem

properties of the principal eigenvalues of a general class of non classical mixed boundary value problems in this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions our theory is a substantial extension of the classical theory by p hess and t kato 1980 comm partial differential equations5 999 1030 in obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself these auxiliary results complement and in some sense complete the theory of d daners and e n dancer 1997 j differential equations138 86 132 the main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to h amann and j lopez gomez 1998 j differential equations146 336 374