strong convergence theorem for nonexpansive mappings on star shaped sets in hilbert spaces abstract in this paper using the concept of attractive points of a nonlinear mapping we obtain a strong convergence theorem of halpern s type 7 for nonexpansive mappings in a star shaped subset of a hilbert space using this result we obtain a new fixed point theorem and a strong convergence theorem which generalizes wittmann s theorem 26

irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

copositive relaxation beats lagrangian dual bounds in quadratically and linearly constrained quadratic optimization problems we study nonconvex quadratic minimization problems under possibly nonconvex quadratic and linear constraints characterizing both lagrangian and semi lagrangian dual bounds in terms of conic optimization while the lagrangian dual is equivalent to the sdp relaxation which has been known for quite a while although the presented form incorporating explicitly linear constraints seems to be novel we show that the semi lagrangian dual is equivalent to a natural copositive relaxation and this has apparently not been observed before this way we arrive at conic bounds tighter than the usual lagrangian dual and thus than the sdp bounds any of the known tractable inner approximations of the copositive cone can be used for this tightening but in particular the above mentioned characterization with explicit linear constraints is a natural much cheaper relaxation than the usual zero order approximation by doubly nonnegative dnn matrices and still improves upon the lagrangian dual bounds these app

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

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

semidefinite relaxations for quadratically constrained quadratic programming a review and comparisons at the intersection of nonlinear and combinatorial optimization quadratic programming has attracted significant interest over the past several decades a variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps the primary purpose of this paper is to present a systematic comparison of sdp relaxations for qcqp using theoretical analysis it is shown that the recently developed doubly nonnegative relaxation is equivalent to the shor relaxation when the latter is enhanced with a partial first order relaxation linearization technique these two relaxations are shown to theoretically dominate six other sdp relaxations a computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations while providing relaxation gaps averaging 3 as opposed to gaps of up to 19 for weaker relaxations on 700 randomly generated problems with up to 60 variables an sdp relaxation derived from lagrangian relaxation after the addition of redundant nonlinear constraints to the primal achieves gaps averaging 13 in a few cpu seconds

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

positive energy representations holomorphic discrete series and finite dimensional irreps let g be a semi simple non compact lie group with unitary lowest highest weight representations we consider explicitly the relation among three types of representations of g positive energy unitary lowest weight representations holomorphic discrete series representations and non unitary finite dimensional irreps we consider mainly the conformal groups soo n 2 treating in full detail the cases n 1 3 4

generalized bilinear programming part i models applications and linear programming relaxation abstract the evolution of the bilinear programming problem is reviewed and a new more general model is discussed the model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed the class of problems under consideration contains traditional bilinear programs general quadratic programs and bilinearly constrained and quadratically constrained extensions of these problems we describe how several important applications from the literature including the multiple modular design problem can be modeled as generalized bilinear programs finally we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound

constrained optimal input signal design for data centric estimation methods this technical note examines the design of constrained input signals for data centric estimation methods which systematically generate a local function approximation from a database of regressors at a current operating point the proposed method addresses the optimal distribution of regressor vectors under constraints for a linear time invariant lti system the resulting nonconvex optimization problems are solved using semidefinite relaxation methods numerical examples illustrate the benefits and usefulness of the proposed input signal design formulations

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

new and old bounds for standard quadratic optimization dominance equivalence and incomparability a standard quadratic optimization problem stqp consists in minimizing a quadratic form over a simplex among the problems which can be transformed into a stqp are the general quadratic problem over a polytope and the maximum clique problem in a graph in this paper we present several new polynomial time bounds for stqp ranging from very simple and cheap ones to more complex and tight constructions the main tools employed in the conception and analysis of most bounds are semidefinite programming and decomposition of the objective function into a sum of two quadratic functions each of which is easy to minimize we provide a complete diagram of the dominance incomparability or equivalence relations among the bounds proposed in this and in previous works in particular we show that one of our new bounds dominates all the others furthermore a specialization of such bound dominates schrijver s improvement of lovasz s function bound for the maximum size of a clique in a graph

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

renormalisation of noncommutative ϕ 4 theory by multi scale analysis in this paper we give a much more efficient proof that the real euclidean 4 model on the four dimensional moyal plane is renormalisable to all orders we prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non local matrix models on the other hand our bounds permit a powerful multi scale analysis of the resulting ribbon graphs here the dual graphs play a particular role because the angular momentum conservation is conveniently represented in the dual picture choosing a spanning tree in the dual graph according to the scale attribution we prove that the summation over the loop angular momenta can be performed at no cost so that the power counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

lie group convolution algebras as deformation quantizations of linear poisson structures introduction let l be a finite dimensional lie algebra over the real numbers r and let l be its dual vector space it is well known 24 that the lie algebra structure on l defines a natural poisson structure on l in fact this was already known to lie 24 and these poisson structures are exactly what are now called the linear poisson structures given a manifold m equipped with a poisson structure one can seek deformation quantizations in the direction of as first studied in 3 these are loosely speaking one parameter families h ler of deformations of the pointwise multiplication on c m or an appropriate subalgebra such that

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

a finite branch and bound algorithm for nonconvex quadratic programming via semidefinite relaxations existing global optimization techniques for nonconvex quadratic programming qp branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch and bound nodes an open question of theoretical interest is how to develop a finite branch and bound algorithm for nonconvex qp one idea which guarantees a finite number of branching decisions is to enforce the first order karush kuhn tucker kkt conditions through branching in addition such an approach naturally yields linear programming lp relaxations at each node however the lp relaxations are unbounded a fact that precludes their use in this paper we propose and study semidefinite programming relaxations which are bounded and hence suitable for use with finite kkt branching computational results demonstrate the practical effectiveness of the method with a particular highlight being that only a small number of nodes are required

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

how to convexify the intersection of a second order cone and a nonconvex quadratic a recent series of papers has examined the extension of disjunctive programming techniques to mixed integer second order cone programming for example it has been shown by several authors using different techniques that the convex hull of the intersection of an ellipsoid mathcal e e and a split disjunction l x j x j u le 0 l xj xj u 0 with l u l

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

quantum field theory on quantized bergman domain we present an oscillator realization of discrete series representations of group su 2 2 we give formulas for the coherent state star product quantization of a bergman domain d a formulation of a regularized noncommutative scalar field on a quantized d is given

asymptotically non expansive self maps and global stability with ultimate boundedness of dynamic systems this paper investigates self maps t x x which satisfy a distance constraint in a metric space with mixed point dependent non expansive properties or in particular contractive ones and potentially expansive properties related to some distance threshold the above mentioned constraint is feasible in certain real world problems of usefulness for instance when discussing ultimate boundedness in dynamic systems which guarantees lyapunov stability this fact makes the proposed analysis to be potentially useful to investigate global stability properties in dynamic systems in the potential presence of some locally unstable equilibrium points the results can be applied to stability problems of dynamics systems and circuit theory as the given examples suggest

an iterative algorithm for approximating convex minimization problem we suggest and analyze an iterative algorithm without the assumption of any type of commutativity on an infinite family of nonexpansive mappings we show that the proposed iterative algorithm converges to the unique minimizer of some quadratic function over the common fixed point sets of an infinite family of nonexpansive mappings our result extend and improve many results announced by many authors

a gentle geometric introduction to copositive optimization this paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way we focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive style problem which itself can be recast in terms of linear second order cone and semidefinite optimization a particular highlight is the role played by the geometry of the feasible set

strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive banach spaces in this paper by using some new analysis techniques we study the approximation problems of common fixed points of halpern s iterative sequence for a class of finite nonexpansive mappings in strictly convex and reflexive banach spaces by using banach s limit the main results presented in this paper generalize extend and improve the corresponding results of bauschke the approximation of fixed points of compositions of nonexpansive mappings in hilbert spaces j math anal appl 202 1996 150 159 halpern fixed points of nonexpansive maps bull am math soc 73 1967 957 961 shioji and takahashi strong convergence of approximated sequences for nonexpansive mappings in banach spaces proc am math soc 125 1997 3641 3645 takahashi et al approximation of common fixed points of a family of finite nonexpansive mappings in banach spaces sci math jpn 56 2002 475 480 wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 xu another control condition in an iterative method for nonexpansive mappings bull austral math soc 65 2002 109 113 remarks on an iterative method for nonexpansive mappings commun appl nonlinear anal 10 2003 67 75 and others

a note on lack of strong duality for quadratic problems with orthogonal constraints abstract the general quadratically constrained quadratic program qqp is an important modelling tool for many diverse problems the qqp is in general np hard and numerically intractable lagrangian relaxations often provide good approximate solutions to these hard problems such relaxations are equivalent to semidefinite programming relaxations and can be solved efficiently for several special cases of qqp the lagrangian relaxation provides the exact optimal value this means that there is a zero duality gap and the problem is tractable it is important to know for which cases this is true since they can then be used as subproblems to improve lagrangian relaxation for intractable qqps in this paper we study the special qqp with orthogonal matrix constraints xx t i if c 0 the zero duality gap result holds if the redundant orthogonal constraints x t x i are added we show that this is not true in the general case however we show how to close the duality gap in the pure linear case by adding variables in addition to constraints

on the convergence of han s method for convex programming with quadratic objective we present an iterative method for minimizing strictly convex quadratic functions over the intersection of a finite number of convex sets the method consists in computing projections onto the individual sets simultaneously and the new iterate is a convex combination of those projections we give convergence proofs even for the inconsistent case i e when the intersection of the sets is empty

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

power counting theorem for non local matrix models and renormalisation solving the exact renormalisation group equation a la wilson polchinski perturbatively we derive a power counting theorem for general matrix models with arbitrarily non local propagators the power counting degree is determined by two scaling dimensions of the cut off propagator and various topological data of ribbon graphs as a necessary condition for the renormalisability of a model the two scaling dimensions have to be large enough relative to the dimension of the underlying space in order to have a renormalisable model one needs additional locality properties typically arising from orthogonal polynomials which relate the relevant and marginal interaction coefficients to a finite number of base couplings the main application of our power counting theorem is the renormalisation of field theories on noncommutative d in matrix formulation