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

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

copositivity detection by difference of convex decomposition and ω subdivision we present three new copositivity tests based upon difference of convex d c decompositions and combine them to a branch and bound algorithm of subdivision type the tests employ lp or convex qp techniques but also can be used heuristically using appropriate test points we also discuss the selection of efficient d c decompositions and propose some preprocessing ideas based on the spectral d c decomposition we report on first numerical experience with this procedure which are very promising

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

the directional instability problem in systems with frictional contacts the research summarized in this paper addresses the directional instability of finite dimensional systems with unilateral frictional contacts conditions for the occurrence of this divergence type instability are discussed complementarity formulations are developed and numerical procedures are proposed for the solution of the corresponding non smooth stability eigenproblems various examples are analytically or numerically solved and discussed namely some finite element examples that have instability modes involving evolution towards slip or stick in different portions of the contact surface

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

symmetric tensor approximation hierarchies for the completely positive cone in this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors we show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone as an application we consider a class of bounds on the stability number of a graph obtained from the polyhedral approximation hierarchy and we construct a primal optimal solution with its tensor lifting for each of such linear programs

copositive programming motivated bounds on the stability and the chromatic numbers the lovasz theta number of a graph g can be viewed as a semidefinite programming relaxation of the stability number of g it has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of g we introduce a related strengthening of the lovasz theta number toward the chromatic number of g which is shown to be equal to the fractional chromatic number of g solving copositive programs is np hard this motivates the study of tractable approximations of the copositive cone we investigate the parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs we provide some computational results indicating that the lovasz theta number can be strengthened significantly toward the fractional chromatic number of g on some hamming graphs

conjugate points in length spaces in this paper we extend the concept of a conjugate point in a riemannian manifold to geodesic spaces in particular we introduce symmetric conjugate points and ultimate conjugate points and relate these notions to prior notions developed for more restricted classes of spaces we generalize the long homotopy lemma of klingenberg to this setting as well as the injectivity radius estimate also due to klingenberg which was used to produce closed geodesics or conjugate points on riemannian manifolds we close with applications of these new kinds of conjugate points to cba spaces proving both known and new theorems in particular we prove a rauch comparison theorem a relative rauch comparison theorem the fact that there are no ultimate conjugate points less than p apart in a cba 1 space and a few facts concerning closed geodesics this paper is written to be accessible to students and includes open problems

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

higher kronecker limit formulas for real quadratic fields for every integer kf 2 we introduce an analytic function of a positive real variable and give a universal formula expressing the values z b k of the zeta func tions of narrow ideal classes in real quadratic fields in terms of this function and its deriva tives up to order k 1 evaluated at reduced real quadratic irrationalities associated to b we show that our functions satisfy functional equations and use these to deduce explicit formulas for the rational numbers z b 1 k we also give an interpretation of our for mula for z b k in terms of cohomology groups of sl 2 z with analytic coe cients and describe a twisted extension of the main formula that allows one to treat zeta values of zeta functions of ray classes rather than just ideal classes finally we use our formulas to compute some zeta values numerically and test that they are expressible as combinations of higher polylogarithm functions evaluated at algebraic arguments

copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints the problem of finding the minimum value of a quadratic function on a set defined by nonnegativity and quadratic equality constraints is analyzed the difficulty in finding the solution to this problem is primarily due to the fact that the feasible region is nonconvex an algorithm that requires the hessian of the quadratic constraint function be strictly copositive is developed for finding the minimal value of the quadratic objective function the problem of finding this global minima can be mapped into the problem of determining whether or not a particular matrix is copositive this result is equivalent to earlier results characterizing the solutions to a large class of fractional programming problems a more efficient algorithm for finding solutions that satisfy the kuhn tucker necessary conditions is developed and its convergence behavior is analyzed this algorithm requires that the hessians of the quadratic constraint and objective functions be both positive semidefinite and strictly copositive

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

values of zeta functions at negative integers dedekind sums and toric geometry in the present paper we study relations among special values of zeta functions of real quadratic fields properties of generalized dedekind sums and todd classes of toric varieties the main theme of the paper is the use of toric geometry to explain in a conceptual way properties of the values of zeta functions and dedekind sums as well as to provide explicit computations both toric varieties and zeta functions associate numerical invariants to cones in lattices with different motivations and applications though we will focus on the case of two dimensional cones in the present paper we introduce notation and definitions that are valid for cones of arbitrary dimension the reason for this added generality is clarity as well as preparation for the results of a subsequent publication

three manifolds with many flat planes we discuss the rigidity or lack thereof imposed by dierent no tions of having an abundance of zero curvature planes on a complete riemann ian 3 manifold we prove a rank rigidity theorem for complete 3 manifolds showing that having higher rank is equivalent to having reducible universal covering we also study 3 manifolds such that every tangent vector is con tained in a at plane including examples with irreducible universal covering and discuss the eect of nite volume and real analiticity assumptions

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

copositivity and constrained fractional quadratic problems we provide completely positive and copositive optimization formulations for the constrained fractional quadratic problem cfqp and standard fractional quadratic problem stfqp based on these formulations semidefinite programming relaxations are derived for finding good lower bounds to these fractional programs which can be used in a global optimization branch and bound approach applications of the cfqp and stfqp related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

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

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

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

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

a computational study for bilevel quadratic programs using semidefinite relaxations abstract in this paper we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem for this purpose we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent karush kuhn tucker kkt conditions then we use the single level formulations to obtain mixed integer linear programming milp models and semidefinite programming sdp relaxations thus we compute optimal solutions and upper bounds using linear programming lp and sdp relaxations our numerical results indicate that the sdp relaxations are considerably tighter than the lp ones consequently the sdp relaxations allow finding tight feasible solutions for the problem especially when the number of variables in the leader problem is larger than in the follower problem moreover they are solved at a significantly lower computational cost for large scale instances

the operator psi for the chromatic number of a graph we investigate hierarchies of semidefinite approximations for the chromatic number chi g of a graph g we introduce an operator psi mapping any graph parameter beta g nested between the stability number alpha g and chi overline g to a new graph parameter psi beta g nested between alpha overline g and chi g psi beta g is polynomial time computable if beta g is as an application there is no polynomial time computable graph parameter nested between the fractional chromatic number chi cdot and chi cdot unless text p text np moreover based on the motzkin straus formulation for alpha g we give quadratically constrained quadratic and copositive programming formulations for chi g under some mild assumptions n beta g le psi beta g but while n beta g remains below chi g psi beta g can reach chi g e g for beta cdot alpha cdot we also define new polynomial time computable lower bounds for chi g improving the classic lovasz theta number and its strengthenings obtained by adding nonnegativity and triangle inequalities experimental results on hamming graphs kneser graphs and dimacs benchmark graphs will be given in the follow up paper n gvozdenovic and m laurent siam j optim 19 2008 pp 592 615

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

topological polylogarithms and p adic interpolation of l values of totally real fields we develop the topological polylogarithm which provides an integral version of nori s eisenstein cohomology classes for mathrm gl n mathbb z and yields classes with values in an iwasawa algebra this implies directly the integrality properties of special values of l functions of totally real fields and a construction of the associated p adic l function using a result of graf we also apply this to prove some integrality and p adic interpolation results for the eisenstein cohomology of hilbert modular varieties

asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi complementary sequence sets a quasi complementary sequence set qcss refers to a set of two dimensional matrices with low nontrivial aperiodic auto and cross correlation sums for multicarrier code division multiple access applications the availability of large qcsss with low correlation sums is desirable the generalized levenshtein bound glb is a lower bound on the maximum aperiodic correlation sum of qcsss the bounding expression of glb is a fractional quadratic function of a weight vector mathbf w and is expressed in terms of three additional parameters associated with qcss the set size k the number of channels m and the sequence length n it is known that a tighter glb compared to the welch bound is possible only if the condition m geq 2 and k geq overline k 1 where overline k is a certain function of m and n is satisfied a challenging research problem is to determine if there exists a weight vector that gives rise to a tighter glb for all not just some k geq overline k 1 and m geq 2 especially for large n i e the condition is asymptotically both necessary and sufficient to achieve this we analytically optimize the glb which is in general nonconvex as the numerator term is an indefinite quadratic function of the weight vector our key idea is to apply the frequency domain decomposition of the circulant matrix in the numerator term to convert the nonconvex problem into a convex one following this optimization approach we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the glb under certain conditions

jointly constrained bilinear programs and related problems an overview abstract linear complementarity quadratic programming and separably constrained bilinear programming problems are all special cases of jointly constrained bilinear programs in this paper we review current theory and procedures for bilinear programs and related problems the key notion of convex envelopes that drives the general procedure is explored and some minor extensions to known results are presented extensions and directions for future research are also discussed

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

a block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem in this paper we address the solution of the symmetric eigenvalue complementarity problem eicp by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex the spectral projected gradient spg method has been recommended to this optimization problem when the dimension of the symmetric eicp is large and the accuracy of the solution is not a very important issue we suggest a new algorithm which combines elements from the spg method and the block active set method where the latter was originally designed for box constrained quadratic programs in the new algorithm the projection onto the unit simplex in the spg method is replaced by the much cheaper projection onto a box this can be of particular advantage for large and sparse symmetric eicps global convergence to a solution of the symmetric eicp is established computational experience with medium and large symmetric eicps is reported to illustrate the efficacy and efficiency of the new algorithm

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

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

computing the stability number of a graph via linear and semidefinite programming we study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph our work is based on and refines de klerk and pasechnik s approach to approximating the stability number via copositive programming siam j optim 12 2002 pp 875 892 we provide a closed form expression for the values computed by the linear programming approximations we also show that the exact value of the stability number alpha g is attained by the semidefinite approximation of order alpha g 1 as long as alpha g leq 6 our results reveal some sharp differences between the linear and the semidefinite approximations for instance the value of the linear programming approximation of any order is strictly larger than alpha g whenever alpha g 1

index parity of closed geodesics and rigidity of hopf fibrations the diameter rigidity theorem of gromoll and grove 1987 states that a riemannian manifold with sectional curvature 1 and diameter 2 is either homeomorphic to a sphere locally isometric to a rank one symmetric space or it has the cohomology ring of the cayley plane ca the reason that they were only able to recognize the cohomology ring of ca is due to an exceptional case in another theorem gromoll and grove 1988 a riemannian submersion m b b with connected fibers that is defined on the euclidean sphere m is metrically congruent to a hopf fibration unless possibly m b 15 8 we will rule out the exceptional cases in both theorems our argument relies on a rather unusal application of morse theory for that purpose we give a general criterion which allows to decide whether the morse index of a closed geodesic is even or odd

focal radius rigidity and lower curvature bounds this is the accepted version of the following article luis guijarro and frederick wilhelm focal radius rigidity and lower curvature bounds which has been published in final form at https doi org 10 1112 plms 12113

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

shintani zeta functions and gross stark units for totally real fields let f be a totally real number field and let p be a finite prime of f such that p splits completely in the finite abelian extension h of f tate has proposed a conjecture 22 conjecture 5 4 stating the existence of a p unit u in h with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to h f this conjecture is an analogue of stark s conjecture which tate called the brumer stark conjecture gross 12 conjecture 7 6 proposed a refinement of the brumer stark conjecture that gives a conjectural formula for the image of u in fp e where fp denotes the completion of f at p and e denotes the topological closure of the group of totally positive units e of f we present a further refinement of gross s conjecture by proposing a conjectural formula for the exact value of u in fp

a convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid we consider the nonconvex problem rq of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid this formulation is motivated by the so called regularized total least squares problem rtls which is a special case of the problem s class we study we prove that under a certain mild assumption on the problem s data problem rq admits an exact semidefinite programming relaxation we then study a simple iterative procedure which is proven to converge superlinearly to a global solution of rq and show that the dependency of the number of iterations on the optimality tolerance varepsilon grows as o sqrt ln varepsilon 1

connections between the total least squares and the correction of an infeasible system of linear inequalities given an infeasible system of linear inequalities axb we address the problem of correcting both the matrix of coefficients a by a h and vector b by b p to minimize the frobenius norm of h p for a system of linear equations this problem can be solved by an algebraic and well studied method known as the total least squares for inequalities vatolin seminarber humboldt univ berlin sekt math 81 1986 145 154 was the first to approach this problem presenting a result with necessary and sufficient conditions for local minimizers unfortunately the direct application of these results is impracticable for large problems since the sufficient conditions are not necessary in case of their failure one is unable to draw conclusions on a search path for a local minimizer we have analyzed the problem using the kkt conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima in terms of the solution of the total least squares and the set of active constraints establishing the common features between these two problems is not only important from a theoretical point of view but it opens the possibility of using theoretical developments related with the total least squares to solve the problem with inequalities

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

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time