alternating projections on manifolds we prove that if two smooth manifolds intersect transversally then the method of alternating projections converges locally at a linear rate we bound the speed of convergence in terms of the angle between the manifolds which in turn we relate to the modulus of metric regularity for the intersection problem a natural measure of conditioning we discuss a variety of problem classes where the projections are computationally tractable and we illustrate the method numerically on a problem of finding a low rank solution of a matrix equation

existence of solutions for variational inequalities on riemannian manifolds abstract we establish the existence and uniqueness results for variational inequality problems on riemannian manifolds and solve completely the open problem proposed in s z nemeth variational inequalities on hadamard manifolds nonlinear anal 52 2003 1491 1498 also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on riemannian manifolds are studied

invex sets and preinvex functions on riemannian manifolds the concept of a geodesic invex subset of a riemannian manifold is introduced geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined the relation between geodesic invexity and preinvexity of functions on manifolds is studied using proximal subdifferential certain results concerning extremum points of a non smooth geodesic preinvex function on a geodesic invex set are obtained the main value inequality and the mean value theorem in invexity analysis are extended to cartan hadamard manifolds

optimality conditions and algorithms for integral global minimization with the integral approach of global optimization new optimality conditions for global minimization are proposed and studied using m mean value condition one can design algorithms for finding global minimizers with n variance condition one can set the stopping criterion

steepest descent method with a generalized armijo search for quasiconvex functions on riemannian manifolds this paper extends the full convergence of the steepest descent method with a generalized armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete riemannian manifolds previous convergence results are obtained as particular cases and some examples in non euclidian spaces are given in particular our approach can be used to solve constrained minimization problems with nonconvex objective functions in euclidian spaces if the set of constraints is a riemannian manifold and the objective function is quasiconvex in this manifold

linear convergence of subgradient algorithm for convex feasibility on riemannian manifolds we study the convergence issue of the subgradient algorithm for solving the convex feasibility problems in riemannian manifolds which was first proposed and analyzed by bento and melo j optim theory appl 152 2012 pp 773 785 the linear convergence property about the subgradient algorithm for solving the convex feasibility problems with the slater condition in riemannian manifolds are established and some step sizes rules are suggested for finite convergence purposes which are motivated by the work due to de pierro iusem appl math optim 17 1988 pp 225 235 as a by product the convergence result of this algorithm is obtained for the convex feasibility problem without the slater condition assumption these results extend and or improve the corresponding known ones in both the euclidean space and riemannian manifolds

line search algorithms for locally lipschitz functions on riemannian manifolds this paper presents line search algorithms for finding extrema of locally lipschitz functions defined on riemannian manifolds to this end we generalize the so called wolfe conditions for nonsmooth functions on riemannian manifolds using varepsilon subgradient oriented descent directions and the wolfe conditions we propose a nonsmooth riemannian line search algorithm and establish the convergence of our algorithm to a stationary point moreover we extend the classical bfgs algorithm to nonsmooth functions on riemannian manifolds numerical experiments illustrate the effectiveness and efficiency of the proposed algorithm

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

set valued robust mappings and approximatable mappings abstract the concepts of robustness of sets and mappings were proposed for the theory of integral global optimization this paper discusses the robustness of a set valued mapping associating with the stability problem we extend the equivalence between the robustness and the approximatability to the case of a set valued mapping we deal also with bi robustness and in robustness

discontinuity and measurability of robust functions in the integral global minimization abstract robustness and measurability of a set and of a function are the foundation of the integral approach to global minimization however a robust function may be discontinuous and nonmeasurable in this paper we show that the set of points of discontinuity of a robust function has empty interior and is of the first category a robust function on the interval 0 1 is constructed such that the lebesque measure of its set of points of discontinuity approaches 1 we also show that there is a robust set on 0 1 which is lebesque nonmeasurable and then a lebesque nonmeasurable robust function is constructed

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

a maximum principle for evolution hamilton jacobi equations on riemannian manifolds abstract we establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t f t d x u 0 u 0 x u 0 x where u 0 m r is a bounded uniformly continuous function m is a riemannian manifold and f 0 t m r this yields uniqueness of the viscosity solutions of such hamilton jacobi equations

equilibrium problems on riemannian manifolds with applications abstract we study equilibrium problem on general riemannian manifolds focusing on existence of solutions and the convexity properties of the solution set our approach consists in relating the equilibrium problem to a suitable variational inequality problem on riemannian manifolds and is completely different from previous ones on this topic in the literature we apply our results to the mixed variational inequality and the nash equilibrium problem moreover we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem in particular correct proofs are provided for the results claimed in j math anal appl 388 2012 61 77 i e theorems 3 5 and 4 9 there regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on hadamard manifolds

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

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

on generalized semi infinite optimization and bilevel optimization the paper studies the connections and differences between bilevel problems bl and generalized semi infinite problems gsip under natural assumptions gsip can be seen as a special case of a bl we consider the so called reduction approach for bl and gsip leading to optimality conditions and newton type methods for solving the problems we show by a structural analysis that for gsip problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general bl problems not the genericity behavior of bl and gsip is in particular studied for linear problems

quasi tangent vectors in flow invariance and optimization problems on banach manifolds abstract the aim of this paper is threefold 1 1 to introduce the notion of quasi tangent vector to a subset s of a banach manifold m and to extend the definitions and some properties of a generalized gradients and tangent cones from a banach space e to m 2 2 to apply some of the above notions to the problem of flow invariance of a closed subset of the manifold m generalizing nagumo brezis result 3 3 to extend some fundamental results in optimization e g clarke s general lagrange multipliers rule in terms of generalized gradient from e to m points 1 and 3 are based on the results of clarke rockafellar and ursescu on tangent cones

discontinuous robust mappings are approximatable the concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization a set a of a topological space x is said to be robust iff cl a cl int a a mapping f x y is said tobe robust iffforeach open set uy of y f i uy is robust we prove that if x is a baire space and y satisfies the second axiom of countability then a mapping f x y is robust iff it is approximatable in the sense that the set of points of continuity of f is dense in x and that for any other point x e x x f x is the limit of x f x where for all a x is a continuous point of f this result justifies the notion of robustness

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

porosity results on fixed points for nonexpansive set valued maps in hyperbolic spaces abstract we establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set valued maps in hyperbolic spaces which in particular extend and or improve some known results in this direction

upper robust mappings and vector minimization an integral approach abstract a study of upper robust mapping from a topological space to r n and development of optimality conditions for vector minimization of upper robust mappings are presented in the framework of integral based optimization theory under some general assumptions optimality conditions are established for several well developed scalarization techniques such as weighting constraint and reference point these optimality conditions are applied to design integral algorithms for finding the set of efficient solutions of a vector optimization problem a numerical example is presented to illustrate the effectiveness of the algorithm

invariant monotone vector fields on riemannian manifolds abstract various concepts of invariant monotone vector fields on riemannian manifolds are introduced some examples of invariant monotone vector fields are given several notions of invexities for functions on riemannian manifolds are defined and their relations with invariant monotone vector fields are studied

existence of equilibria via ekeland s principle abstract in the literature when dealing with equilibrium problems and the existence of their solutions the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity together with some weak continuity assumptions of the function in this paper we focus on conditions that do not involve any convexity concept neither for the domain nor for the function involved starting from the well known ekeland s theorem for minimization problems we find a suitable set of conditions on the function f that lead to an ekeland s variational principle for equilibrium problems via the existence of e solutions we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems

a proof of a general maximum principle for optimal controls via a multiplier rule on metric space a unified proof is given of the maximum principle for optimal control with various kinds of constraints by using a multiplier rule on metric spaces

global optimum shape design abstract an optimum shape design problem can be formulated as a minimization problem of a functional subject to certain constraints usually it is nonlinear and nonconvex conventional optimization techniques are gradient based they highly depend on the initial design and are difficult to be applied to find a global solution integral global optimization algorithm is proposed to solve optimum shape design problems three design examples are given to illustrate the effectiveness of the algorithm

an iterative method for finding common solutions of equilibrium and fixed point problems we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem

alternating projections in cat 0 spaces abstract by using recently developed theory which extends the idea of weak convergence into cat 0 space we prove the convergence of the alternating projection method for convex closed subsets of a cat 0 space given the right notion of weak convergence it turns out that the generalization of the well known results in hilbert spaces is straightforward and allows the use of the method in a nonlinear setting as an application we use the alternating projection method to minimize convex functionals on a cat 0 space

variational inequalities for set valued vector fields on riemannian manifolds convexity of the solution set and the proximal point algorithm we consider variational inequality problems for set valued vector fields on general riemannian manifolds the existence results of the solution convexity of the solution set and the convergence property of the proximal point algorithm for the variational inequality problems for set valued mappings on riemannian manifolds are established applications to convex optimization problems on riemannian manifolds are provided

proximal subgradient and a characterization of lipschitz function on riemannian manifolds a characterization of lipschitz behavior of functions defined on riemannian manifolds is given in this paper first it is extended the concept of proximal subgradient and some results of proximal analysis from hilbert space to riemannian manifold setting a technique introduced by clarke stern and wolenski f h clarke r j stern p r wolenski subgradient criteria for monotonicity the lipschitz condition and convexity canad j math 45 1993 1167 1183 for generating proximal subgradients of functions defined on a hilbert spaces is also extended to riemannian manifolds in order to provide that characterization a number of examples of lipschitz functions are presented so as to show that the lipschitz behavior of functions defined on riemannian manifolds depends on the riemannian metric

a conceptual method for solving generalized semi infinite programming problems via global optimization by exact discontinuous penalization abstract we consider a generalized semi infinite programming problem gsip with one semi infinite constraint where the index set depends on the variable to be minimized keeping in mind the integral global optimization method of zheng and chew and its modifications we would like to outline theoretical considerations for determining coarse approximations of a solution of gsip via global optimization of an exact discontinuous penalty approach we consider an auxiliary parametric semi infinite programming problem and the behavior of its marginal functional in so doing we extend the theory of robust analysis to study robustness of marginal functions and robustness of set valued mappings with given structures

scalarization in vector optimization in this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem this will be done for weakly minimal and minimal solutions respectively finally the results will be applied to a certain class of approximation problems

characterization of compact geodesic spaces abstract driven by klee s well known work about the topological properties of convex sets in locally convex linear spaces we give a complete characterization of compact geodesic spaces with curvature bounded below in terms of the fixed point property for continuous functions furthermore we provide an example which highlights the role of the sectional curvature in our result

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

the schauder fixed point theorem in geodesic spaces abstract we give a direct proof of schauder s fixed point theorem in the setting of geodesic metric spaces generalizing the classical schauder s theorem and improving a recent version of this theorem in cat spaces as an application we prove an existence result for a variational inequality in the setting of cat spaces

discontinuous optimization by smoothing we present a simple tool for solving many discontinuous optimization problems the basic idea is to express discontinuities by means of a step function and then to approximate the step function by a smooth one this way a smooth once or twice continuously differentiable approximate problem is obtained this problem can be solved by any gradient technique the approximations introduced contain a single parameter which controls their accuracy so that the original problem is replaced only in some neighborhoods of the points of discontinuity some convergence properties are established and numerical experiments with some test problems are reported

variational inequalities on hadamard manifolds the notion of variational inequalities is extended to hadamard manifolds and related to geodesic convex optimization problems existence and uniqueness theorems for variational inequalities on hadamard manifolds are proved a convexity property of the solution set of a variational inequality on a hadamard manifold is presented

existence and stability of solutions for maximal element theorem on hadamard manifolds with applications abstract in this paper maximal element theorem on hadamard manifolds is established first we prove the existence of solutions for maximal element theorem on hadamard manifolds further we prove that most of problems in maximal element theorem on hadamard manifolds in the sense of baire category are essential and that for any problem in maximal element theorem on hadamard manifolds there exists at least one essential component of its solution set as applications we study existence and stability of solutions for variational relation problems on hadamard manifolds and existence and stability of weakly pareto nash equilibrium points for n person multi objective games on hadamard manifolds

monotonicity resolvents and yosida approximations of operators on hilbert manifolds abstract monotonicity resolvents and yosida approximations of operators on manifolds of class c 3 are considered strong convergence of difference approximations to differential equations is shown by using resolvents our motivation is to give an elementary step toward constructing nonlinear semigroup theory on infinite dimensional manifolds

newton methods for nonsmooth convex minimization connections among lagrangian riemannian newton and sqp methods this paper studies newton type methods for minimization of partly smooth convex functions sequential newton methods are provided using local parameterizations obtained from lagrangian theory and from riemannian geometry the hessian based on the lagrangian depends on the selection of a dual parameter g by revealing the connection to riemannian geometry a natural choice of g emerges for which the two newton directions coincide this choice of g is also shown to be related to the least squares multiplier estimate from a sequential quadratic programming sqp approach and with this multiplier sqp gives the same search direction as the newton methods

local convergence of the proximal point method for a special class of nonconvex functions on hadamard manifolds abstract local convergence analysis of the proximal point method for a special class of nonconvex functions on hadamard manifold is presented in this paper the well definedness of the sequence generated by the proximal point method is guaranteed moreover it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and under additional assumptions its convergence for a minimizer is obtained

global minimization of constrained problems with discontinuous penalty functions abstract with the integral approach to global optimization a class of discontinuous penalty functions is proposed to solve constrained minimization problems optimality conditions of a penalized minimization problem are generalized to a discontinuous case necessary and sufficient conditions for an exact penalty function are examined a nonsequential algorithm is proposed numerical examples are given to illustrate the effectiveness of the algorithm

generalized semi infinite programming theory and methods generalized semi infinite optimization problems gsip are considered the difference between gsip and standard semi infinite problems sip is illustrated by examples by applying the reduction ansatz optimality conditions for gsip are derived numerical methods for solving gsip are considered in comparison with methods for sip from a theoretical and a practical point of view it is investigated under which assumptions a gsip can be transformed into an sip

on certain conditions for the existence of solutions of equilibrium problems the main purpose of this paper is the study of sufficient and or necessary conditions for existence of solutions of equilibrium problems we discuss some of the assumptions of the problem under which the introduced conditions are sufficient and or necessary and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem

optimality condition and algorithm with deviation integral for global optimization to study the integral global minimization a general form of deviation integral is introduced and its properties are examined in this work in terms of the deviation integral optimality condition and algorithms are given algorithms are implemented by a properly designed monte carlo simulation numerical tests are given to show the effectiveness of the method

some generalizations of ekeland type variational principle with applications to equilibrium problems and fixed point theory abstract in this paper we introduce the concept of a q function defined on a quasi metric space which generalizes the notion of a function and a w distance we establish ekeland type variational principles in the setting of quasi metric spaces with a q function we also present an equilibrium version of the ekeland type variational principle in the setting of quasi metric spaces with a q function we prove some equivalences of our variational principles with caristi kirk type fixed point theorems for multivalued maps the takahashi minimization theorem and some other related results as applications of our results we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps we also extend the nadler s fixed point theorem for multivalued maps to a q function and in the setting of complete quasi metric spaces as a consequence we prove the banach contraction theorem for a q function and in the setting of complete quasi metric spaces the results of this paper extend and generalize many results appearing recently in the literature

proximal point method for minimizing quasiconvex locally lipschitz functions on hadamard manifolds abstract in this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex locally lipschitz objective functions on hadamard manifolds to reach this goal we use the concept of clarke subdifferential on hadamard manifolds and assuming that the function is bounded from below we prove the global convergence of the sequence generated by the method to a critical point of the function

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions

the regular module problem i in the study of induced representations the following problem arises let h ag be a finite solvable group and k a field with chark a let v be an irreducible faithful primitive k 4g module suppose h contains a normal extraspecial r subgroup r with z r z h and that a acts faith fully on r under what conditions does a have a regular direct summand in vi in this paper we consider this question under the hypotheses that g mr where 1 m is normal abelian in am a is nilpotent a mr m r 1 and r z r is a faithful irreducible am modu e we show that a has at least three regular direct summands in v unless a exp m and r satisfy certain very restrictive conditions let h ag be a finite solvable group where a is nilpotent g is normal and y4 g 1 let k be a field of characteristic not dividing a and v a zc module assume that v is faithful and irreducible the study of induced representations often leads to the following two questions 1 regular orbit problem under what conditions does a have a regular orbit on f

equilibrium problems in hadamard manifolds abstract an equilibrium theory is developed in hadamard manifolds the existence of equilibrium points for a bifunction is proved under suitable conditions and applications to variational inequality fixed point and nash equilibrium problems are provided the convergence of picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points

firmly nonexpansive mappings in classes of geodesic spaces firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators in this paper we do a thorough study of fixed point theory and the asymptotic behaviour of picard iterates of these mappings in different classes of geodesic spaces such as uniformly convex w hyperbolic spaces busemann spaces and cat 0 spaces furthermore we apply methods of proof mining to obtain effective rates of asymptotic regularity for the picard iterations msc primary 47h09 47h10 53c22 secondary 03f10 47h05 90c25 52a41