primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

an adaptive linear approximation algorithm for copositive programs we study linear optimization problems over the cone of copositive matrices these problems appear in nonconvex quadratic and binary optimization for instance the maximum clique problem and other combinatorial problems can be reformulated as such problems we present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit in contrast to previous approximation schemes our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not using these approximations we derive an adaptive linear approximation algorithm for copositive programs numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

balanced line bundles on fano varieties a conjecture of batyrev and manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants in particular counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones this framework leads to the notion of balanced line bundles whose counting functions conjecturally capture generic distributions of rational points we investigate balanced line bundles in the context of the minimal model program with special regard to the classification of fano threefolds

equivalent unconstrained minimization and global error bounds for variational inequality problems new merit functions for variational inequality problems are constructed through the moreau yosida regularization of some gap functions the proposed merit functions constitute unconstrained optimization problems equivalent to the original variational inequality problem under suitable assumptions conditions are studied for those merit functions to be differentiable and for any stationary point of those functions to be a solution of the original variational inequality problem moreover those functions are shown to provide global error bounds for general variational inequality problems under the strong monotonicity assumption only

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

solvability of semidefinite complementarity problems abstract the concept of exceptional family has been introduced to study the existence theorem for nonlinear complementarity problems and variational inequality problems we describe extensions of such concepts to complementarity problems defined over the cone of block diagonal symmetric positive semidefinite real matrices using the concept of exceptional family we propose a very general existence theorem for the semidefinite complementarity problem extensions of isac carbone s condition karamardian s condition properness and coercivity are also introduced several applications of the main results are presented and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone semidefinite complementarity problems

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

a solution condition for complementarity problems with an application to spatial price equilibrium a sufficient condition is developed for the existence of solutions to complementarity problems which is applicable to all continuous functions in addition it is shown to imply a number of simpler conditions including the well known coerciveness condition finally this condition is applied to an existence problem in spatial price equilibrium where it is shown to yield not only a more general result but also a simpler method of proof

application of degree theory in stability of the complementarity problem in this paper we study the behavior of a solution of the complementarity problem when the problem is perturbed the basic result is that if the degree of a certain function is nonzero then the problem is stable we show that several well known conditions for the existence of a solution to the problem also imply stability of the problem we also show that in the nondegenerate case there are simple conditions that ensure stability

a class of gap functions for variational inequalities recently auchmuty 1989 has introduced a new class of merit functions or optimization formulations for variational inequalities in finite dimensional space we develop and generalize auchmuty s results and relate his class of merit functions to other works done in this field especially we investigate differentiability and convexity properties and present characterizations of the set of solutions to variational inequalities we then present new descent algorithms for variational inequalities within this framework including approximate solutions of the direction finding and line search problems the new class of merit functions include the primal and dual gap functions introduced by zuhovickii et al 1969a 1969b and the differentiable merit function recently presented by fukushima 1992 also the descent algorithm proposed by fukushima is a special case from the class of descent methods developed in this paper through a generalization of auchmuty s class of merit functions we extend those inherent in the works of dafermos 1983 cohen 1988 and wu et al 1991 new algorithmic equivalence results relating these algorithm classes to each other and to auchmuty s framework are also given

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

matrix model approach to minimal liouville gravity revisited using the connection with the frobenius manifold fm structure we study the matrix model description of minimal liouville gravity mlg based on the douglas string equation our goal is to find an exact discrete formulation of the mlg model that intrinsically contains information about the conformal selection rules we discuss how to modify the fm structure appropriately for this purposes we propose a modification of the construction for lee yang series involving the algebra instead of the previously used a1 algebra with the new prescription we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations

equivalence of variational inequality problems to unconstrained minimization in this paper we propose a class of merit functions for variational inequality problems vi through these merit functions the variational inequality problem is cast as unconstrained minimization problem we estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of vi

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

generalized tensor equations with leading structured tensors abstract systems of tensor equations tes have received much considerable attention in the recent literature in this paper we consider a class of generalized tensor equations gtes an important difference between gtes and tes is that gtes can be regarded as a system of non homogenous polynomial equations whereas tes is a homogenous one such a difference usually make the theoretical and algorithmic results tailored for tes not necessarily applicable to gtes to study properties of the solution set of gtes we first introduce a new class of so named z tensors which includes the set of all p tensors as a proper subset with the help of degree theory we prove that the system of gtes with a leading coefficient z tensor has at least one solution for any right hand side vector moreover we study the local error bounds under some appropriate conditions finally we employ a levenberg marquardt algorithm to find a solution to gtes and report some preliminary numerical results

analysis for a homotopy path of complementarity problems based on µ exceptional family this paper is concerned with several structural properties of a homotopy path for nonlinear continuous semi monotone complementarity problems especially several sufficient conditions are established for the existence and boundedness of this homotopy path under the p 0 property all these sufficient conditions guarantee the uniqueness and continuity of this path the concept of m exceptional family plays a key role in the analysis throughout this paper

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

d orientation sequences for continuous functions and nonlinear complementarity problems we introduce the new concept of d orientation sequence for continuous functions it is shown that if there does not exist a d orientation sequence for a continuous function then the corresponding complementarity problem cp has a solution we believe that such a result characterizes an intrinsic property of cps as the concept of exceptional family of elements the notion of d orientation sequence of a function is also a powerful tool for investigating the existence theorems of cps we use this new tool to establish among other things a new existence result for a class of p mapping cps

global convergence properties of some iterative methods for linear complementarity problems the subject of this work is a class of iterative methods for solving the linear complementarily problem lcp these methods are based on a reformulation of the lcp consisting of a usually differentiable system of nonlinear equations to which newton s method is applied thus the algorithms are locally q quadratically convergent furthermore global convergence results for these methods are proved for lcps associated with r 0 nondegenerate and p matrices finally some promising numerical results are reported for both constructed and randomly generated problems

alternative theorems for nonlinear projection equations and applications to generalized complementarity problems let k be a closed convex subset in rn and let f g and h be three functions from rn into itself we consider the following nonlinear projection equation npe h x k g x f x x r 1 where k is the orthogonal projection operator onto the set k this equation provides a uni ed formulation of several interesting and important special cases in the case when h x g x for any x rn then 1 reduces to the equation studied by pang and yao 17 under the name of generalized normal equation which is equivalent to the following generalized variational inequality gvi g x k y g x f x 0 for all y k 2

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

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

a note on the existence of traffic equilibria this note presents an alternative proof of the existence of a solution to the network equilibrium problem which is based on the recent work in complementarity theory by t smith 11

an existence condition for open set nonlinear complementarity problems with an application to a closed economic system a sufficient condition for the existence of solutions to open set nonlinear complementarity problems oncp is developed this condition is applied to an existence problem in a closed general economic equilibrium system

on the geometry of thin exceptional sets in manin s conjecture manin s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set we study whether the exceptional set in manin s conjecture is a thin set using the minimal model program and boundedness of log fano varieties

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

exceptional family and solvability of copositive complementarity problems abstract in this paper by extending the concept of exceptional family to complementarity problems over the cone of symmetric copositive real matrices we propose an existence theorem of a solution to the copositive complementarity problem extensions of isac carbone s condition karamardian s condition weakly properness and coercivity are also introduced several applications of these results are presented and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone copositive complementarity problems

exceptional families and finite dimensional variational inequalities over polyhedral convex sets a new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper it generalizes the concepts for complementarity problem introduced by smith and isac we applied the new analytical tool to the study of existence problem for variational inequalities it is shown that our existence condition is weaker than most of the sufficient conditions which have been known

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

on correlation numbers in 2d minimal gravity and matrix models we test recent results for the four point correlation numbers in minimal liouville gravity against calculations in the one matrix models and find full agreement in the process we construct the resonance transformation which relates coupling parameters k of the liouville gravity to the couplings tk of the matrix models up to the terms of the order 4 we also conjecture the general form of this transformation

neron severi groups under specialization andre used hodge theoretic methods to show that in a smooth proper family x b of varieties over an algebraically closed field k of characteristic 0 there exists a closed fiber having the same picard number as the geometric generic fiber even if k is countable we give a completely different approach to andre s theorem which also proves the following refinement in a family of varieties with good reduction at p the locus on the base where the picard number jumps is p adically nowhere dense our proof uses the p adic lefschetz 1 1 theorem of berthelot and ogus combined with an analysis of p adic power series we prove analogous statements for cycles of higher codimension assuming a p adic analogue of the variational hodge conjecture and prove that this analogue implies the usual variational hodge conjecture applications are given to abelian schemes and to proper families of projective varieties

a copositive programming approach to graph partitioning we consider 3 partitioning the vertices of a graph into sets s 1 s 2 and s 3 of specified cardinalities such that the total weight of all edges joining s 1 and s 2 is minimized this problem is closely related to several np hard problems like determining the bandwidth or finding a vertex separator in a graph we show that this problem can be formulated as a linear program over the cone of completely positive matrices leading in a natural way to semidefinite relaxations of the problem we show in particular that the spectral relaxation introduced by helmberg e 1995 can equivalently be formulated as a semidefinite program finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound

exceptional family and solvability of the second order cone complementarity problems abstract in this paper we introduce a concept of exceptional family for second order cone complementarity problems denoted by soccp which is the particular case of the concept of exceptional family of elements introduced by isac and carbone 1999 25 for the nonlinear complementarity problems and the exceptional family for second order cone complementarity problems has some pretty unique properties furthermore we propose a new sufficient existence condition of a solution to soccp and give a particular example to show that the new condition is not stronger than known isac carbone s condition

explicit solution for p q duality in two dimensional quantum gravity by using sato s infinite dimensional grassmannian theory of the kp hierarchy we study the global structure of the theory space of 2d quantum gravity coupled to various minimal conformal fields labeled by a pair of integers p q after giving a rigorous proof of the equivalence of douglas s equation and the schwinger dyson equation w constrainton a function we establish thep q duality of the p q quantum gravity at green s function level as an application we discuss the metamorphosis of operators under unitarity preserving renormalization group flows