hidden conic quadratic representation of some nonconvex quadratic optimization problems the problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property even when the quadratic forms are indefinite the equivalent convex problem is a semidefinite one and the equivalence is based on the celebrated s lemma in this paper we show that when the quadratic forms are simultaneously diagonalizable sd it is possible to derive an equivalent convex problem which is a conic quadratic cq one and as such is significantly more tractable than a semidefinite problem the sd condition holds for free for many problems arising in applications in particular when deriving robust counterparts of quadratic or conic quadratic constraints affected by implementation error the proof of the hidden cq property is constructive and does not rely on the s lemma this fact may be significant in discovering hidden convexity in some nonquadratic problems

grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

optimal designs for bivariate logistic regression summary in drug testing experiments the primary responses of interest are efficacy and toxicity these can be modeled as a bivariate quantal response using the gumbel model for bivariate logistic regression d optimal and q optimal experimental designs are developed for this model the q optimal design minimizes the average asymptotic prediction variance of p l 0 d the probability of efficacy without toxicity at dose d over a desired range of doses the robustness of these designs to parameter misspecification is discussed in addition d efficiencies of q optimal designs and q efficiencies of doptimal designs are presented an extension of the general equivalence theorem to the multivariate case is applied to these designs

convergent semidefinite programming relaxations for global bilevel polynomial optimization problems in this paper we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials we present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming sdp relaxations and provide convergence results for the methods our scheme for problems with a convex lower level problem involves solving a transformed equivalent single level problem by a sequence of sdp relaxations whereas our approach for general problems involving a nonconvex polynomial lower level problem solves a sequence of approximation problems via another sequence of sdp relaxations

a note on polynomial solvability of the cdt problem recent research has focused on the complexity of extensions of the classical trust region subproblem which addresses the minimization of a quadratic function over a unit ball in mathbb r n

efficient and exact mesh deformation using multiscale rbf interpolation abstract radial basis function rbf interpolation is popular for mesh deformation due to robustness and generality but the cost scales with the number of surface points sourcing the deformation as o n s 3 hence there have been numerous works investigating efficient methods using reduced datasets however although reduced data methods are efficient they require a secondary method to treat an error vector field to ensure surface points not included in the primary deformation are moved to the correct location and the volume mesh moved accordingly a new method is presented which captures global and local motions at multiple scales using all the surface points and so no correction stage is required all surface points are used and a single interpolation built but the cost and conditioning issues associated with rbf methods are eliminated moreover the sparsity introduced is exploited using a wall distance function to further reduce the cost the method is compared to an efficient greedy method and it is shown mesh quality is always comparable with or better than with the greedy method and cost is comparable or cheaper at all stages surface mesh preprocessing is the dominant cost for reduced data methods and this cost is reduced significantly here greedy methods select points to minimise interpolation error requiring repeated system solution and cost o n r e d 4 to select n r e d points the multiscale method has no error and the problem is transferred to a geometric search with cost o n s log n s resulting in an eight orders of magnitude cost reduction for three dimensional meshes furthermore since the method is dependent on geometry not deformation it only needs to be applied once prior to simulation as the mesh deformation is decoupled from the point selection process

a combination conformal transfinite mapping method for grids about fin afterbody combinations an algebraic procedure is presented for the generation of a smooth computational grid about an afterbody fin configuration the method makes use of a sequence of conformal transformations to unwrap the geometry and remove the corner singularities at the fin trailing edge and tail of the afterbody a 3 d grid is generated by stacking a sequence of 2 d grids of the c type on predetermined smooth tubular surfaces clustering is accomplished by a sequence of one dimensional stretching functions in physical space examples are presented to show the character of the resulting grid

weighted least squares method of grid generation we propose the creation of a grid smoother to improve the local grid distorsion nonsmoothness and overlapping of the transfinite interpolated grid points the second purpose of this study is to smooth the resulting grid finding through the application of the jeng and liou modified multiple one dimensional adaptive grid method

finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains a finite difference time dependent numerical method for the wave equation supported by recently derived novel elliptic grids is analyzed the method is successfully applied to single and multiple two dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries the new grids have nearly uniform cell area j grids and nearly uniform grid line spacing email protected numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state the restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing as a consequence moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost also numerical solutions for wave problems inside annular regions of complex shapes are obtained the use of the new grids results in late time stability in contrast with other classical finite difference time dependent methods

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

generation of boundary conforming grids around wing body configurations using transfinite interpolation a computational procedure for generating three dimensional nonorthogonal surface fitted mesh systems around wing fuselage configurations is presented the method is based on the concept of transfinite interpolation which has been extended to handle very general mapping function specifications at the boundaries thereby making it possible to generate single block mappings with geometry data specified only at the outer boundaries of the computational domain since it is a direct algebraic mapping technique the method is very inexpensive in terms of computer cost different types of possible mappings are compared with respect to resolution and economy of nodal points a procedure for a novel type of mapping designated type o o is described and several plots of generated grids demonstrate the capabilities of the method the singular lines inherent in every three dimensional mesh for this type of surface geometry are also discussed

on the classical necessary second order optimality conditions in the presence of equality and inequality constraints for nonconvex optimization problems in rn a counterexample was recently given by anitescu siam j optim 10 2000 pp 1116 1135 that showed that the mangasarian fromovitz constraint qualification mfcq is not sufficient for the classical necessary second order optimality conditions to hold we prove these optimality conditions with the assumptions that the set of lagrange multipliers is a bounded line segment and a relaxed strict complementary slackness scs condition holds a new constraint qualification is presented in this paper we assume that the mfcq holds and the active constraint rank deficiency is 1 this assumption relaxes the linear independence constraint qualification and enforces the mfcq in addition the set of lagrange multipliers is a bounded line segment

mesh generation for the computation of flowfields over complex aerodynamic shapes methods are presented for generating both structured and unstructured meshes about three dimensional shapes results for both approaches are shown and their strengths and weaknesses are compared for relatively simple configurations such as wing body combinations a structured mesh is the preferred approach for a complete aircraft however structured meshes lack the necessary flexibility but unstructured meshes do offer the opportunity to treat completely general configurations with relative ease

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

narrowing the difficulty gap for the celis dennis tapia problem we study the celis dennis tapia cdt problem minimize a non convex quadratic function over the intersection of two ellipsoids in contrast to the well studied trust region problem where the feasible set is just one ellipsoid the cdt problem is not yet fully understood our main objective in this paper is to narrow the difficulty gap that occurs when the hessian of the lagrangian is indefinite at all karush kuhn tucker points we prove new sufficient and necessary conditions both for local and global optimality based on copositivity giving a complete characterization in the degenerate case

s lemma with equality and its applications let f x x tax 2a tx c f x xtax 2atx c and h x x tbx 2b tx d h x xtbx 2btx d be two quadratic functions having symmetric matrices a a and b b the s lemma with equality asks when the unsolvability of the system f x 0 h x 0 f x 0 h x 0 implies the existence of a real number mu such that f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn the problem is much harder than the inequality version which asserts that under slater condition f x 0 h x le 0 f x 0 h x 0 is unsolvable if and only if f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn for some mu ge 0 0 in this paper we show that the s lemma with equality does not hold only when the matrix a a has exactly one negative eigenvalue and h x h x is a non constant linear function b 0 b not 0 b 0 b 0 as an application we can globally solve inf f x h x 0 inf f x h x 0 as well as the two sided generalized trust region subproblem inf f x l le h x le u inf f x l h x u without any condition moreover the convexity of the joint numerical range f x h 1 x ldots h p x x in mathbb r n f x h1 x hp x x rn where f f is a possibly non convex quadratic function and h 1 x ldots h p x h1 x hp x are affine functions can be characterized using the newly developed s lemma with equality

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

non convex quadratic minimization problems with quadratic constraints global optimality conditions in this paper we first examine how global optimality of non convex constrained optimization problems is related to lagrange multiplier conditions we then establish lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints we also obtain necessary global optimality conditions which are different from the lagrange multiplier conditions for special classes of quadratic optimization problems these classes include weighted least squares with ellipsoidal constraints and quadratic minimization with binary constraints we discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi extremal non convex quadratic optimization problems

trust region problems with linear inequality constraints exact sdp relaxation global optimality and robust optimization the trust region problem which minimizes a nonconvex quadratic function over a ball is a key subproblem in trust region methods for solving nonlinear optimization problems it enjoys many attractive properties such as an exact semi definite linear programming relaxation sdp relaxation and strong duality unfortunately such properties do not in general hold for an extended trust region problem having extra linear constraints this paper shows that two useful and powerful features of the classical trust region problem continue to hold for an extended trust region problem with linear inequality constraints under a new dimension condition first we establish that the class of extended trust region problems has an exact sdp relaxation which holds without the slater constraint qualification this is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range convexity whenever the dimension condition is fulfilled second we show that the dimension condition together with the slater condition ensures that a set of combined first and second order lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust region problem and consequently for strong duality through simple examples we also provide an insightful account of our development from sdp relaxation to strong duality finally we show that the dimension condition is easily satisfied for the extended trust region model that arises from the reformulation of a robust least squares problem lsp as well as a robust second order cone programming model problem socp as an equivalent semi definite linear programming problem this leads us to conclude that under mild assumptions solving a robust lsp or socp under matrix norm uncertainty or polyhedral uncertainty is equivalent to solving a semi definite linear programming problem and so their solutions can be validated in polynomial time

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

damped euler equation method to compute transonic flow around wing body combinations inviscid transonic flows containing either strong shock waves or complex vortex structure call for the euler equations as a realistic model presented here is a computational procedure for solving the euler equations for transonic flow around a wing and fuselage upon an o o mesh generated by transfinite interpolation an explicit time marching finite volume procedure solves the flow equations and features a nonreflecting far field boundary condition an internal mechanism for temporal damping and use of the local time step all of which improve the convergence of the computation converged after several hundred iterations results computed on the cyber 203 vector processor are compared with experimental data and potential flow computations the euler equation model is found to predict the existence of a tip vortex created by inviscid flow separation in the downstream region of the tip of the m6 wing where the radius of curvature approaches zero ast year a workshop 1 was held in order to assess the currently used computational procedures one against the other in several carefully specified two dimensional transonic flow problems while reasonable agreement among the results given by the various full potential and euler equation methods was obtained in the subcritical cases a disparity between the results of these two models was found to grow with cases of increasing shock strength for the naca 0012 airfoil and conditions m 0 80 and a 1 25 deg for example the lift coefficients cl given by all the euler methods ranged between 0 30 and 0 38 whereas the range of cl for the potential method was 0 28 1 1 this discrepancy has led to a reconsideration of the validity of the potential model when strong shocks are present in the flow not only locally in terms of the isentropic shock jumps but also and perhaps even more importantly with regard to the correct modeling of vortex phenomena throughout the entire flowfield in the potential representation the vorticity that is bound to the airfoil is accounted for by a jump in the potential across a line originating at the trailing edge and lying a priori along some chosen coordinate direction downstream the so called kutta condition the euler equations in contrast admit vorticity in the solution and the equivalent to the kutta condition evidently does not need to be enforced explicitly this situation is currently under study by researchers using a variety of numerical methods in three dimensions the flow past a finite wing is even more complex and less is known vortices for example are shed continuously from the wing tips and the entire trailing edge whether a kutta condition or its equivalent is necessary in this case has not been in vestigated but because vorticity is so crucial to the realism achieved by inviscid flow models interest has been aroused in questions like this and in numerical methods that solve the euler equations a number of methods 2 4 exist to solve the euler equations for three dimensional flow but they have been developed exclusively for and applied only to internal flows apparently the solution of the euler equations for

robust duality for generalized convex programming problems under data uncertainty abstract in this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization we establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain lagrangian dual by showing strong duality between the deterministic counterparts robust counterpart of the primal model and the optimistic counterpart of its dual problem a robust strong duality theorem is given whenever the lagrangian function is convex we provide classes of uncertain non convex programming problems for which robust strong duality holds under a constraint qualification in particular we show that robust strong duality is guaranteed for non convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized slater condition numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems we further show that robust duality continues to hold under a weakened convexity condition

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

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

a hybrid differential algebraic method for three dimensional grid generation an effective method is presented for generating three dimensional boundary fitted grids the technique is based on a two step procedure involving the solution of differential equations and algebraic interpolations a coarse grid with an arbitrarily given point distribution at all the boundaries is first generated by solving the poisson equations with the forcing functions being automatically adjusted during the grid generation to achieve orthogonality at the boundaries a fine grid with a desired concentration of grid lines is then obtained by fitting tricubic spline functions on the coarse grid a primary feature of the method is that the mesh spacing can be quickly and explicitly changed while retaining the smoothness over the entire field and the boundary orthogonality applications to a variety of geometries demonstrate the performance of the method

an alternative theorem for quadratic forms and extensions abstract this note concerns an alternative theorem for quadratic forms established recently by y yuan we explore some directions in which yuan s result can be extended and we indicate also some important situations where an alternative theorem a la yuan does not hold

alternative theorems for quadratic inequality systems and global quadratic optimization we establish alternative theorems for quadratic inequality systems consequently we obtain lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems we present a generalization of dine s theorem to a system of two homogeneous quadratic functions with a regular cone the class of regular cones are cones k for which k cup k is a subspace as a consequence we establish a generalization of the powerful s lemma which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint we then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint range of three homogeneous quadratic functions using a regular cone this yields lagrange multiplier characterizations of global optimality for classes of trust region type problems with two inequality constraints finally we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving z matrices which are matrices with nonpositive off diagonal elements and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems

maximum likelihood regression methods for paired binary data we discuss maximum likelihood methods for analysing binary responses measured at two times such as in a cross over design we construct a 2 2 table for each individual with cell probabilities corresponding to the cross classification of the responses at the two times the underlying likelihood for each individual is multinomial with four cells the three dimensional parameter space of the multinomial distribution is completely specified by the two marginal probabilities of success of the 2 2 table and an association parameter between the binary responses at the two times we examine a logistic model for the marginal probabilities of the 2 2 table for individual i the association parameters we consider are either the correlation coefficient the odds ratio or the relative risk simulations show that the parameter estimates for the logistic regression model for the marginal probabilities are not very sensitive to the parameters used to describe the association between the binary responses at the two times thus we suggest choosing the measure of association for ease of interpretation

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

three dimensional full navier stokes solvers for incompressible flows past arbitrary geometries the computation of the three dimensional viscous flow past several geometries is investigated an iterative technique resting on the fully elliptic mode is applied to the reynolds averaged navier stokes equations written in a non orthogonal curvilinear body fitted co ordinate system results of the computation are compared with available experiments

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

a bivariate logistic regression model based on latent variables bivariate observations of binary and ordinal data arise frequently and require a bivariate modeling approach in cases where one is interested in aspects of the marginal distributions as separate outcomes along with the association between the two we consider methods for constructing such bivariate models based on latent variables with logistic marginals and propose a model based on the ali mikhail haq bivariate logistic distribution we motivate the model as an extension of that based on the gumbel type 2 distribution as considered by other authors and as a bivariate extension of the logistic distribution which preserves certain natural characteristics basic properties of the obtained model are studied and the proposed methods are illustrated through analysis of two data sets a basic science cognitive experiment of visual recognition and awareness and a clinical data set describing assessments of walking disability among multiple sclerosis patients

convexifiability of continuous and discrete nonnegative quadratic programs for gap free duality abstract in this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi lagrangian duals more importantly we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs such as mixed integer quadratic programs revealing zero duality gaps as an application we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions various sufficient conditions for convexifiability are also given

marginal modeling of correlated ordinal data using a multivariate plackett distribution abstract an extension of the bivariate model suggested by dale is proposed for the analysis of dependent ordinal categorical data the so called multivariate dale model is constructed by first generalizing the bivariate plackett distribution to any dimensions because the approach is likelihood based it satisfies properties that are not fulfilled by other popular methods such as the generalized estimating equations approach the proposed method models both the marginal and the association structure in a flexible way the attractiveness of the multivariate dale model is illustrated in three key examples covering areas such as crossover trials longitudinal studies with patients dropping out from the study and discriminant analysis applications the differences and similarities with the generalized estimating approach are highlighted

a geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints we first establish a relaxed version of dines theorem associated to quadratic minimization problems with finitely many linear equality and a single nonconvex quadratic inequality constraints the case of unbounded optimal valued is also discussed then we characterize geometrically the strong duality and some relationships with the conditions employed in finsler theorem are established furthermore necessary and sufficient optimality conditions with or without the slater assumption are derived our results can be used to situations where none of the results appearing elsewhere are applicable in addition a revisited theorem due to frank and wolfe along with that due to eaves is established for asymptotically linear sets

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

algebraic grid generation for an afterbody with finite span tapered fins an algebraic technique is presented for the generation of a 3 d grid about a pointed afterbody with four tapered finite span fins the grid topology is largely determined by a polar type singularity which passes through the tip parallel to the body centerline a 3 d grid is generated by stacking a sequence of 2 d grids on predetermined blank surfaces in the fin and tip regions on each blank surface a grid of the c type is computed by a combination of conformal mapping and transfinite interpolation the method is an extension of previous work for the generation of a 3 d grid about an afterbody with four constant chord infinite span fins an example grid is presented to illustrate the method and its characteristics are discussed