remarks on the global sobolev inequalities in the minkowski space rn 1 on demontre des inegalites de sobolev globales sur l espace de minkowski r n 1 le theoreme principal est valable sur des tranches a temps fixe

convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

perspective reformulations of the cta problem with l2 distances any institution that disseminates data in aggregated form has the duty to ensure that individual confidential information is not disclosed either by not releasing data or by perturbing the released data while maintaining data utility controlled tabular adjustment cta is a promising technique of the second type where a protected table that is close to the original one in some chosen distance is constructed the choice of the specific distance shows a trade off although the euclidean distance has been shown and is confirmed here to produce tables with greater utility it gives rise to mixed integer quadratic problems miqps with pairs of linked semi continuous variables that are more difficult to solve than the mixed integer linear problems corresponding to linear norms we provide a novel analysis of perspective reformulations prs for this special structure in particular we devise a projected pr p2r which is piecewise conic but simplifies to a nonseparable miqp when the instance is symmetric we then compare different formulations of the cta problem showing that the ones based on p2r most often obtain better computational results

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

relaxations of factorable functions with convex transformable intermediates we propose to strengthen standard factorable relaxations of global optimization problems through the use of functional transformations of intermedi ate expressions in particular we exploit convex transformability of the component functions of factorable programs as a tool in the generation of bounds we define suit able forms of transforming functions and assess theoretically and computationally the sharpness of the resulting relaxations in comparison to existing schemes

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

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

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

globally optimal algorithms for stratified autocalibration we present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality given a projective reconstruction we first upgrade it to affine by estimating the position of the plane at infinity the plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints in the second stage this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic diac the positive semidefiniteness of the diac is explicitly enforced as part of the optimization process rather than as a post processing step for each stage we construct and minimize tight convex relaxations of the highly non convex objective functions in a branch and bound optimization framework we exploit the inherent problem structure to restrict the search space for the diac and the plane at infinity to a small fixed number of branching dimensions independent of the number of views chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum experimental evidence of the accuracy speed and scalability of our algorithm is presented on synthetic and real data

projected perspective reformulations with applications in design problems the perspective relaxation pr is a general approach for constructing tight approximations to mixed integer nonlinear programs minlp with semicontinuous variables the pr of a minlp can be formulated either as a mixed integer second order cone program mi socp provided that the original objective function is socp representable or as a semi infinite minlp in this paper we show that under some further assumptions rather restrictive but satisfied in several practical applications the pr of a mixed integer quadratic program miqp can also be reformulated as a piecewise quadratic program qp ultimately yielding a qp relaxation of roughly the same size of the standard continuous relaxation furthermore if the original problem has some exploitable structure then this structure is typically preserved in the reformulation thus allowing the construction of specialized approaches for solving the pr we report on implementing these ideas on two miqps with appropriate structure a sensor placement problem and a quadratic cost single commodity network design problem

data separation via a finite number of discriminant functions a global optimization approach abstract this paper presents a mixed 0 1 integer and linear programming milp model for separation of data via a finite number of non linear and non convex discriminant functions the milp model concurrently optimizes the parameters of the user provided individual discriminant functions to implement a decision boundary for an optimal separation of data under analysis the performance of the milp based classification of data is illustrated on randomly generated two dimensional datasets and extensively tested on six well studied datasets in data mining research in comparison with three well established supervised learning methodologies namely the multisurface method the logical analysis of data and the support vector machines numerical results from these experiments show that the new milp based classification of data is an effective and useful methodology for supervised learning

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

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

error bounds for monomial convexification in polynomial optimization convex hulls of monomials have been widely studied in the literature and monomial convexifications are implemented in global optimization software for relaxing polynomials however there has been no study of the error in the global optimum from such approaches we give bounds on the worst case error for convexifying a monomial over subsets of this implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately our main error bounds depend primarily on the degree of the monomial making them easy to compute since monomial convexification studies depend on the bounds on the associated variables in the second part we conduct an error analysis for a multilinear monomial over two different types of box constraints as part of this analysis we also derive the convex hull of a multilinear monomial over

the lifespan of a class of smooth spherically symmetric solutions of the compressible euler equations with variable entropy in three space dimensions we consider smooth three dimensional spherically symmetric eulerian flows of ideal polytropic gases with variable entropy whose initial data are obtained by adding a small smooth perturbation with compact support to a constant state under a natural assumption we obtain precise information on the asymptotic behavior of their lifespan when the size of the initial perturbation tends to 0 this is achieved by the construction and estimate of a suitable approximate flow

existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data on considere le probleme aux valeurs initiales pour certaines equations hyperboliques non lineaires d ordre 2

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

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

is every nonsingular matrix diagonally equivalent to a matrix with all distinct eigenvalues abstract it is shown that a 2 2 complex matrix a is diagonally equivalent to a matrix with two distinct eigenvalues iff a is not strictly triangular it is established in this paper that every 3 3 nonsingular matrix is diagonally equivalent to a matrix with 3 distinct eigenvalues more precisely a 3 3 matrix a is not diagonally equivalent to any matrix with 3 distinct eigenvalues iff det a 0 and each principal minor of a of order 2 is zero it is conjectured that for all n 2 an n n complex matrix is not diagonally equivalent to any matrix with n distinct eigenvalues iff det a 0 and every principal minor of a of order n 1 is zero

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

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

collusive game solutions via optimization a nash based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the nash equilibrium solution in this paper we study the optimization problem of such a collusive game in which the players collectively maximize the nash bargaining objective subject to a set of incentive compatibility constraints we present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem we establish the convexity of the optimization problem in the case where each player s strategy set is unidimensional in the multivariate case we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures computational results with these procedures for solving some test problems are reported

every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues abstract we show that for every invertible n n complex matrix a there is an n n diagonal invertible d such that ad has distinct eigenvalues using this result we affirm a conjecture of feng li and huang that an n n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n 1 are zero

linear programming based converses for finite blocklength lossy joint source channel coding a linear programming lp based framework is presented for obtaining converses for finite blocklength lossy joint source channel coding problems the framework applies for any loss criterion generalizes certain previously known converses and also extends to multi terminal settings the finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift and project like method a close but tractable lp relaxation of this problem is derived lower bounds on the original problem are obtained by the construction of feasible points for the dual of the lp relaxation a particular application of this approach leads to new converses which recover and improve on the converses of kostina and verdu for finite blocklength lossy joint source channel coding and lossy source coding for finite blocklength channel coding the lp relaxation recovers the converse of polyanskiy poor and verdu and leads to a new improvement on the converse of wolfowitz showing thereby that our lp relaxation is asymptotically tight with increasing blocklengths for channel coding lossless source coding and joint source channel coding with the excess distortion probability as the loss criterion using a duality based argument a new converse is derived for finite blocklength joint source channel coding for a class of source channel pairs employing this converse the lp relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise hamming distortion of a q ary uniform source over a q ary symmetric memoryless channel for any q in mathbb n the optimization formulation and the lift and project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of zhou et al for the successive refinement problem for successively refinable source distortion measure triplets

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

efficient and safe global constraints for handling numerical constraint systems numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques local consistencies and interval methods this paper first recalls the principles of tt quad a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints then it introduces a generalization of tt quad to polynomial constraint systems it also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system different linearization techniques are investigated to limit the number of generated constraints tt quadsolver a new branch and prune algorithm that combines tt quad local consistencies and interval methods is introduced tt quadsolver has been evaluated on a variety of benchmarks from kinematics mechanics and robotics on these benchmarks it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state of the art optimization solvers

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

solving mixed integer bilinear problems using milp formulations in this paper we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable this reformulation is obtained by first replacing a general integer variable with its binary expansion and then using mccormick envelopes to linearize the resulting product of continuous and binary variables we present the convex hull of the underlying mixed integer linear set the effectiveness of this reformulation and associated facet defining inequalities are computationally evaluated on five classes of instances

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

jointly constrained semidefinite bilinear programming with an application to dobrushin curves we propose a branch and bound algorithm for minimizing a bilinear functional of the form f x y mathrm tr x otimes y q mathrm tr ax mathrm tr by of pairs of hermitian matrices x y restricted by joint semidefinite programming constraints the functional is parametrized by self adjoint matrices q a and b this problem generalizes that of a bilinear program where x and y belong to polyhedra the algorithm converges to a global optimum and yields upper and lower bounds on its value in every step various problems in quantum information theory can be expressed in this form as an example application we compute dobrushin curves of quantum channels giving upper bounds on classical coding with energy constraints

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

quasiconvex relaxations based on interval arithmetic abstract interval analysis provides a tool for i forward error analysis ii estimating and controlling rounding and approximation errors automatically and iii proving existence and uniqueness of solutions in this context the terms self validating methods inclusion methods orverification methods are in use in this paper we present a new self validating method for solving global constrained optimization problems this method is based on the construction of quasiconvex lower bound and quasiconcave upper bound functions of a given function the latter defined by an arithmetical expression no further assumptions about the nonlinearities of the given function are necessary these lower and upper bound functions are rigorous by using the tools of interval arithmetic in its easiest form they are constructed by taking appropriate linear and or quadratical estimators which yield quasiconvex quasiconcave bound functions we show how these bound functions can be used to define rigorous quasiconvex relaxations for constrained global optimization problems and nonlinear systems these relaxations can be incorporated in a branch and bound framework yielding a self validating method

on separation of eigenvalues by certain matrix subgroups abstract let a be an n n complex matrix with rank r it is shown that there are a monomial matrix m and a unitary matrix u such that each of the matrices ma and ua has r distinct non zero eigenvalues if h is an irreducible subgroup of gl n c and a 0 it is shown that there is an x h such that xa has at least two distinct eigenvalues

perspective cuts for a class of convex 0 1 mixed integer programs we show that the convex envelope of the objective function of mixed integer programming problems with a specific structure is the perspective function of the continuous part of the objective function using a characterization of the subdifferential of the perspective function we derive perspective cuts a family of valid inequalities for the problem perspective cuts can be shown to belong to the general family of disjunctive cuts but they do not require the solution of a potentially costly nonlinear programming problem to be separated using perspective cuts substantially improves the performance of branch cut approaches for at least two models that either naturally or after a proper reformulation have the required structure the unit commitment problem in electrical power production and the mean variance problem in portfolio optimization