elastic mode algorithms for mathematical programs with equilibrium constraints global convergence and stationarity properties the elastic mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that for a finite value of the penalty parameter local solutions satisfying first and second order necessary optimality conditions for the original problem are also first and second order points of the elastic mode formulation here we study global convergence properties of methods based on this formulation which involve generating an exact or inexact first or second order point of the formulation for nondecreasing values of the penalty parameter under certain regularity conditions on the active constraints we establish finite or asymptotic convergence to points having a certain stationarity property such as strong stationarity m stationarity or c stationarity numerical experience with these approaches is discussed in particular our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic mode formulation is solved inexactly

quasi polynomials linear diophantine equations and semi linear sets we investigate the family of semi linear sets of n t and z t we study the growth function of semi linear sets and we prove that such a function is a piecewise quasi polynomial on a polyhedral partition of n t moreover we give a new proof of combinatorial character of a famous theorem by dahmen and micchelli on the partition function of a system of diophantine linear equations

on inertia and schur complement in optimization partitioned symmetric matrices in particular the hessian of the lagrangian play a fundamental role in nonlinear optimization for this type of matrices s p han and o fujiwara recently presented an inertia theorem under a certain regularity assumption we prove that this theorem is true without any regularity assumption then we consider matrix extensions preserving the sign of the determinant such extensions are shown to be related with the positive definiteness of some schur complement under a regularity assumption this shows from the viewpoint of linear algebra the equivalence of strong stability in the sense of m kojima and strong regularity in the sense of s m robinson finally we discuss the inertia of a typical one parameter family of symmetric matrices occurring in various places in optimization augmented lagrangians focal point theory etc

on iterated minimization in nonconvex optimization in dynamic programming and decomposition methods one often applies an iterated minimization procedure the problem variables are partitioned into several blocks say x and y treating y as a parameter the first phase consists of minimization with respect to the variable x in a second phase the minimization of the resulting optimal value function depending on y is considered in this paper we treat this basic idea on a local level it turns out that strong stability in the sense of kojima in the first phase is a natural assumption in order to show that the iterated local minima of the parametric problem lead to a local minimum for the whole problem we use a generalized version of a positive definiteness criterion of fujiwara han mangasarian

on optimality conditions for structured families of nonlinear programming problems optimality conditions for families of nonlinear programming problems inrn are studied from a generic point of view the objective function and some of the constraints are assumed to depend on a parameter while others are held fixed techniques of differential topology are used to show that under suitable conditions certain strong second order conditions are necessary for optimality except possibly for parameter values lying in a negligible set

primal dual properties of sequential gradient restoration algorithms for optimal control problems 2 general problem abstract the problem of minimizing a functional subject to differential constraints nondifferential constraints initial constraints and final constraints is considered in connection with sequential gradient restoration algorithms sgra for optimal control problems both the primal formulation and the dual formulation are presented depending on whether the primal formulation is used or the dual formulation is used one obtains a primal sequential gradient restoration algorithm psgra or a dual sequential gradient restoration algorithm dsgra for the problem under consideration it is found convenient to split the control vector into an independent control vector and a dependent control vector the latter having the same dimension as the nondifferential constraint vector this modification enhances the computational efficiency of both the primal formulation and the dual formulation the system of lagrange multipliers associated with i the gradient phase of sgra and ii the restoration phase of sgra is examined for each phase it is shown that the lagrange multipliers are endowed with a duality property they minimize a special functional quadratic in the multipliers subject to the multiplier differential equations and boundary conditions for given state control and parameter these duality properties have considerable computational implications they allow one to reduce the auxiliary optimal control problems associated with i and ii to mathematical programming problems involving a finite number of parameters as unknowns

global stabilization of nonlinear systems with inputs subject to magnitude and rate bounds a parametric optimization approach a bounded feedback control design approach is proposed for the global asymptotic stabilization of a class of nonlinear systems with stable free dynamics the control inputs and their derivatives are constrained to take values on sets defined by a cartesian product of eta dimensional closed balls mathcal b mathbf r eta p which are defined by means of a p norm and a radius vector parameter mathbf r in order to derive the bounded control stabilizer the resulting procedure implies that gains as state functions are obtained from the solution to a set of c parameterized nonlinear programming problems in general the resulting closed loop system could be implicitly defined i e consisting of a system of differential equations plus a set of nonlinear algebraic equations required to compute the control special interest is focused on an important class of homogeneous systems that includes a class of globally asymptotically stabilizable systems by linear feedback and bilinear systems for those systems the problem of inputs subject to globally bounded rates is also addressed

on structure and computation of generalized nash equilibria we consider generalized nash equilibrium problems gnep from a structural and computational point of view in gnep the players feasible sets may depend on the other players strategies moreover the players may share common constraints in particular the latter leads to the stable appearance of nash equilibria which are fritz john fj points but not karush kuhn tucker kkt points basic in our approach is the representation of fj points as zeros of an appropriate underdetermined system of nonsmooth equations here additional nonsmooth variables are taken into account we prove that the set of fj points together with corresponding active lagrange multipliers generically constitutes a lipschitz manifold its dimension is n 1 j 0 where n is the number of players and j 0 is the number of active common constraints in a structural analysis of nash equilibria the number n 1 j 0 plays a crucial role in fact the latter number encodes both the possible degeneracies for the players parametric subproblems and the dimension of the set of nash equilibria in particular in the nondegenerate case the dimension of the set of nash equilibria equals locally n 1 j 0 for the computation of fj points we propose a nonsmooth projection method npm which aims at nding solutions of an underdetermined system of nonsmooth equations npm is shown to be well dened for gnep local convergence of npm is conjectured for gnep under generic assumptions and its proof is challenging however we indicate special cases known from the literature in which convergence holds

a conjugate decomposition of the euclidean space abstract given a closed convex cone k in the n dimensional real euclidean space rn and an n n real matrix a that is positive definite on k we show that each vector in rn can be decomposed into a component that lies in k and another that lies in the conjugate cone induced by a and such that the two vectors are conjugate to each other with respect to a at as a consequence of this decomposition we establish the following characterization of positive definite matrices an n n real matrix a is positive definite if and only if it is positive definite on some closed convex cone k in rn and a at 1 exists and is positive semidefinite on the polar cone k0 if k is a subspace of rn then k0 is its orthogonal complement k unk other applications include local duality results for nonlinear programs and other characterizations of positive definite and semidefinite matrices

a polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed we prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d dimensional euclidean space previously such algorithms were known for dimensions d 1 2 3 and 4 only

a restricted signature normal form for hermitian matrices quasi spectral decompositions and applications abstract in recent years a number of results on the relationships between the inertias of hermitian matrices and the inertias of their principal submatrices have appeared in the literature in this paper we study restricted congruence transformations of hermitian matrices m that at the same time induce a congruence transformation of a given principal submatrix a of m such transformations lead to the concept of the restricted signature normal form of m in particular by means of this normal form we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type for some applications a special class of almost unitary restricted congruence transformations turns out to be useful we show that with such transformations m can be reduced to a quasi diagonal form which in particular displays the eigenvalues of a moreover this quasi spectral decomposition is used to derive a generalized signature formula and to study hermitian matrix pencils

structure of the first order solution set for a class of nonlinear programs wit h parameters this paper studies the set of first order solutions of a family of nonlinear programs in which the inequality constraints are fixed but the right hand side of the equality constraints varies it is shown that under certain conditions this set is a topological manifold the results are applied to the problem of describing the set of first order pareto optima in a pure exchange economy with monotone utility functions

a double threshold image binarization method based on edge detector this paper presents a new double threshold image binarization method based on the edge and intensity information we first find seeds near the image edges and present an edge connection method to close the image edges then we use closed image edges to partition the binarized image that is generated using a high threshold and obtain a primary binarization result by filling the partitioned high threshold binary image with the seeds finally the final binarization result is obtained by remedying the primary binarization result with the low threshold binary image compared with the classical binarization methods and the similar binarization methods our method is effective on the binarization of images with low contrast noise and non uniform illumination

a note on weakly active constraints in connection with nonconvex quadratic programming abstract in general it is very difficult to determine a true local minimizer in nonconvex quadratic programming the main problems arise if we have so called weakly active constraints we discuss an efficient method for checking the local optimality or determining a direction of descent

review v i arnol d s m gusein zade and a n varchenko singularities of differentiable maps vol 1 the classification of critical sets caustics and wave fronts

on generic one parametric semi infinite optimization we consider differentiable semi infinite optimization problems depending on a real parameter for generic one parametric families we classify the corresponding set of generalized critical points into eight types five of these types also occur in problems with a finite number of inequality constraints whereas the other three types are typical for the semi infinite case we discuss types 7 and 8 in detail while at points of type 6 the singularity is due to the fact that in the associated lower level problem a lagrange multiplier corresponding to an active inequality constraint vanishes at points of type 7 and 8 the gradients of the active constraints in the lower level problem are linearly dependent if the total number of active constraints in the lower level problem does not exceed the lower level dimension the point is of type 7 otherwise it is of type 8 moreover we distinguish between points of type 8a and 8b where a point is of type 8a if the mangasarian fromovitz constraint qualification holds in the lower level problem and of type 8b otherwise at points of type 8a the set of generalized critical points is not smooth but it does not exhibit a turning point the linear and quadratic indices remain constant when passing along a point of type 8a points of type 7 and type 8b are relative boundary points of the set of generalized critical points

bilevel optimization on the structure of the feasible set we consider bilevel optimization from the optimistic point of view let the pair x y denote the variables the main difficulty in studying such problems lies in the fact that the lower level contains a global constraint in fact a point x y is feasible if y solves a parametric optimization problem l x in this paper we restrict ourselves to the special case that the variable x is one dimensional we describe the generic structure of the feasible set m moreover we discuss local reductions of the bilevel problem as well as corresponding optimality criteria finally we point out typical problems that appear when trying to extend the ideas to higher dimensional x dimensions this will clarify the high intrinsic complexity of the general generic structure of the feasible set m and corresponding optimality conditions for the bilevel problem u

determining subspaces on which a matrix is nonnegative definite abstract we give methods for finding a subspace l of r n of high dimension on which a given real symmetric n n matrix a is nonnegative positive definite the subspace l is constructed as the intersection of some hyperplanes the first method assumes that a has few negative nonpositive eigenvalues and selects the hyperplanes from a given collection which includes the coordinate planes the second method finds a subspace l of maximal dimension it proceeds by selecting some hyperplanes from the given collection and then successively constructing additional hyperplanes until a maximal l is determined both methods rely on inertia theory some new inertia results are derived numerical examples are given

on the commutative equivalence of semi linear sets of n k given two subsets s 1 s 2 of n k we say that s 1 is commutatively equivalent to s 2 if there exists a bijection f s 1 s s 2 from s 1 onto s 2 such that for every v s 1 v f v where v denotes the sum of the components of v we prove that every semi linear set of n k is commutatively equivalent to a recognizable subset of n k

a note on differentiability of global optimal values the differentiability of global optimal values for smooth constrained programs with respect to general perturbations is considered the strong whitney c2 topology is used to the space of objective functions and constraints it will be shown generically the global optimal value is twice continuously differentiable with respect to the function space of objective functions and constraints an application of this genericity argument to a nonsmooth objective function a function of maximum type is also discussed

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

critical sets in parametric optimization we deal with one parameter families of optimization problems in finite dimensions the constraints are both of equality and inequality type the concept of a generalized critical point g c point is introduced in particular every local minimum kuhn tucker point and point of fritz john type is a g c point under fairly weak even generic conditions we study the set consisting of all g c points due to the parameter the set is pieced together from one dimensional manifolds the points of can be divided into five characteristic types the subset of nondegenerate critical points first type is open and dense in nondegenerate means strict complementarity nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints a nondegenerate critical point is completely characterized by means of four indices the change of these indices along is presented finally the kuhn tucker subset of is studied in more detail in particular in connection with the failure of the mangasarian fromowitz constraint qualification

on the commutative equivalence of bounded context free and regular languages the semi linear case this is the third paper of a group of three where we prove the following result let a be an alphabet of t letters and let a nt be the corresponding parikh morphism given two languages l1 l2 a we say that l1 is commutatively equivalent to l2 if there exists a bijection f l1 l2 from l1 onto l2 such that for every u l1 u f u then every bounded context free language is commutatively equivalent to a regular language

on the structure of the counting function of sparse context free languages we give an exact description of the counting function of a sparse context free language let l be a sparse context free language and let fl be its counting function then there exist polynomials p0 p1 pk 1 with rational coefficients and an integer constant k0 such that for any n k0 one has fl n pj n where j is such that j n mod k as a consequence one can easily show the decidability of some questions concerning sparse context free languages finally we show that for any sparse context free language l there exists a regular language l such that for any n 0 one has fl n fl n and therefore fl is rational

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

the parikh counting functions of sparse context free languages are quasi polynomials let l be a sparse context free language over an alphabet of t letters and let f l n t n be its parikh counting function we prove the following two results 1 there exists a partition of n t into a finite family of polyhedra such that the function f l is a quasi polynomial on each polyhedron of the partition 2 there exists a partition of n t into a finite family of rational subsets such that the function f l is a polynomial on each set of the partition

local duality of nonlinear programs it is shown that the second order sufficient necessary optimality condition for the dual of a nonlinear program is equivalent to the inverse of the hessian of the lagrangian being positive definite semidefinite on the normal cone to the local primal constraint surface this compares with the hessian itself being positive definite semidefinite on the tangent cone on the local primal constraint surface for the corresponding second order condition for the primal problem we also show that primal second order sufficiency necessity and dual second order necessity sufficiency is essentially equivalent to the hessian of the lagrangian being positive definite this follows from the following interesting linear algebra result a necessary and sufficient condition for a nonsingular symmetric n times n matrix to be positive definite is that for some subspace of r n the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace

restricted quadratic forms inertia theorems and the schur complement the starting point of this investigation is the properties of restricted quadratic forms x ax x s rm where a is an m m real symmetric matrix and s is a subspace the index theory of hestenes 1951 and maddocks 1985 that treats the more general hilbert space version of this problem is first specialized to the finite dimensional context and appropriate extensions valid only in finite dimensions are made the theory is then applied to obtain various inertia theorems for matrices and positivity tests for quadratic forms expressions for the inertias of divers symmetrically partitioned matrices are described in particular an inertia theorem for the generalized schur complement is given the investigation recovers links and extends several formerly disparate results in the general area of inertia theorems

a parametric view on the mangasarian fromovitz constraint qualification in this paper we consider the behaviour of the multipliers for one parametric semi infinite programs near critical parameters where the mangasarian fromovitz constraint qualification can be violated the growth of the multipliers when the parameter approaches such a critical parameter is characterized by a parametric constraint qualification which is introduced here it is equivalent to a bound on the growth of the multipliers

a technique of three level thresholding based on probability partition and fuzzy 3 partition thresholding is a commonly used technique in image segmentation selecting the correct thresholds is a critical issue in this paper the relationship between a probability partition pp and a fuzzy c partition fp in thresholding is given this relationship and the entropy approach are used to derive a thresholding technique to select the best fuzzy c partition the measure of the selection quality is the compatibility between the fp and the pp generated by the problem an entropy function defined by the pp and fp is used to measure the compatibility a necessary condition of the entropy function arriving at a maximum is derived based on this condition an efficient algorithm for three level thresholding is deduced experiments to verify the efficiency of the proposed method and comparison to some existing techniques are also presented the experiment results show that our proposed method gives the best performance in three level thresholding using fuzzy c partition

on the commutative equivalence of bounded context free and regular languages this is the first paper of a group of three where we prove the following result let a be an alphabet of t letters and let a s n t be the corresponding parikh morphism given two languages l 1 l 2 a we say that l 1 is commutatively equivalent to l 2 if there exists a bijection f l 1 s l 2 from l 1 onto l 2 such that for every u l 1 u f u then every bounded context free language is commutatively equivalent to a regular language

linear rate convergence of the alternating direction method of multipliers for convex composite programming in this paper we aim to prove the linear rate convergence of the alternating direction method of multipliers admm for solving linearly constrained convex composite optimization problems under a mild calmness condition which holds automatically for convex composite piecewise linear quadratic programming we establish the global q linear rate of convergence for a general semi proximal admm with the dual step length being taken in 0 1 51 2 2 this semi proximal admm which covers the classic one has the advantage to resolve the potentially nonsolvability issue of the subproblems in the classic admm and possesses the abilities of handling the multi block cases efficiently we demonstrate the usefulness of the obtained results when applied to two and multi block convex quadratic semidefinite programming

the c index a new stability concept for quadratic programs with complementarity constraints we introduce nondegeneracy and the c index for c stationary points of a qpcc that is for a mathematical program with a quadratic objective function and linear complementarity constraints the c index characterizes the qualitative local behavior of a qpcc around a nondegenerate c stationary point the article focuses on the structure of the c stationary set of qpccs depending on a real parameter we show that for generic qpcc data the c index changes exactly at turning points of the c stationary set and that it changes by exactly one to illustrate this concept we introduce and analyze two homotopy methods for finding c stationary points numerical results illustrate that for randomly generated test problems the two homotopy methods very often identify b stationary points

characterization of strong stability for c stationary points in mpcc we study mathematical programs with complementarity constraints mpcc special focus will be on c stationary points under the linear independence constraint qualification we characterize strong stability of c stationary points in the sense of kojima by means of first and second order information of the defining functions it turns out that strong stability of c stationary points allows a possible degeneracy of bi active lagrange multipliers some relations to other stationarity concepts such as a m s and b stationarity are shortly discussed

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

moreau s decomposition in banach spaces moreau s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics in this paper it is extended to reflexive banach spaces and in the context of generalized proximity measures this extension unifies and significantly improves upon existing results

on the hessian of lagrangian and second order optimality conditions for a constrained minimization problem the restriction of a hessian of lagrangian to a tangent space of the feasible set can be used to detect whether a karush kuhn tucker point is a local minimum maximum or saddle point of the problem it is shown in this paper that the restriction of the hessian to a normal space with respect to the indefinite inner product induced by the hessian can be used to characterize a karush kuhn tucker point for the wolfe dual from this result and by an inertia theorem in s p han and o fujiwara an inertia theorem and its application to nonlinear programs manuscript march 1984 we deduce that under a regularity condition the hessian is positive semidefinite if and only if the considered karush kuhn tucker point satisfies the second order necessary condition for a local minimum point of the primal problem and at the same time satisfies the second order necessary condition for a local maximum point of thee dual similar results on the positive definiteness of the hes

shape stress trade off design of membrane structures for specified sequence of boundary shapes a new method is presented for parametrically generating optimal shapes and stresses of a membrane structure supported by a frame the problem considered here is a so called trade off design problem where the weighted sum of the deviations of the initial stress and the shape from the desired distribution is to be minimized the membrane is discretized by using the finite element method and optimal shapes are expanded with respect to a parameter defining the shape of the boundary frame an approximate method is presented without exact calculation of the hessian of the stresses with respect to the nodal coordinates the explicit developability conditions derived by the authors are used to guarantee the curved surface to be reduced to a set of plane cutting patterns after removing the pretensioning force in the examples trade off designs are found for an hp type membrane structure and the efficiency of the proposed method is discussed

an inertia theorem for symmetric matrices and its application to nonlinear programming abstract the relationship between the inertia of a symmetric real matrix and the inertia of its projection to a subspace is studied the result is then used to characterize a karush kuhn tucker point of a nonlinear programming problem by the inertia of a hessian of lagrangian with respect to both primal and dual variables a theorem of kojima on the determinant of a matrix in nonlinear programming is also derived as a direct consequence of the result

a bifurcation analysis of the nonlinear parametric programming problem the structure of solutions to the nonlinear parametric programming problem with a one dimensional parameter is analyzed in terms of the bifurcation behavior of the curves of critical points and the persistence of minima along these curves changes in the structure of the solution occur at singularities of a nonlinear system of equations motivated by the fritz john first order necessary conditions it has been shown that these singularities may be completely partitioned into seven distinct classes based upon the violation of one or more of the following a complementarity condition a constraint qualification and the nonsingularity of the hessian of the lagrangian on a tangent space to apply classical bifurcation techniques to these singularities a further subdivision of each case is necessary the structure of curves of critical points near singularities of lowest zero codimension within each case is analyzed as well as the persistence of minima along curves emanating from these singularities bifurcation behavior is also investigated or discussed for many of the subcases giving rise to a codimension one singularity

lifting mathematical programs with complementarity constraints we present a new smoothing approach for mathematical programs with complementarity constraints based on the orthogonal projection of a smooth manifold we study regularity of the lifted feasible set and since the corresponding optimality conditions are inherently degenerate introduce a regularization approach involving a novel concept of tilting stability a correspondence between the c index in the original problem and the quadratic index in the lifted problem is shown in particular a local minimizer of the mathematical program with complementarity constraints may numerically be found by minimization of the lifted smooth problem we report preliminary computational experience with the lifting approach

change of generalized indices of stationary solutions to multiparametric optimization we study the local change of the generalized index which is a modification of the morse index and the stationary index for the multiparametric optimization under the regular value condition the change of the generalized index around a triplet x v t is locally bounded by the dimension of the parameter vector t where x is a variable vector and v a vector of the lagrange multiplier space we also discuss the local change of the generalized index around a pair x t

local duality of nonlinear semidefinite programming recently chan and sun chan z x d sun constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming siam j optim 19 370 376 reported for semidefinite programming sdp that the primal dual constraint nondegeneracy is equivalent to the dual primal strong second order sufficient condition ssosc this result is responsible for a number of important results in stability analysis of sdp in this paper we study duality of this type in nonlinear semidefinite programming nsdp we introduce the dual ssosc at a karush kuhn tucker kkt triple of nsdp and study its various characterizations and relationships to the primal nondegeneracy although the dual ssosc is nothing but the ssosc for the wolfe dual of the nsdp it suggests new information for the primal nsdp for example it ensures that the inverse of the hessian of the lagrangian function exists at the kkt triple and the inverse is positive definite on some normal space it also ensures the primal nondegeneracy some of our results generalize the corresponding classical duality results in nonlinear programming studied by fujiwara et al fujiwara o s p han o l mangasarian 1984 local duality of nonlinear programs siam j control optim 22 162 169 for the convex quadratic sdp qsdp we have complete characterizations for the primal and dual ssosc our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual ssosc but also the primal and dual nondegeneracy at its solution suggesting that it is a well conditioned qsdp

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

an adaptive logical method for binarization of degraded document images abstract this paper describes a modified logical thresholding method for binarization of seriously degraded and very poor quality gray scale document images this method can deal with complex signal dependent noise variable background intensity caused by nonuniform illumination shadow smear or smudge and very low contrast the output image has no obvious loss of useful information firstly we analyse the clustering and connection characteristics of the character stroke from the run length histogram for selected image regions and various inhomogeneous gray scale backgrounds then we propose a modified logical thresholding method to extract the binary image adaptively from the degraded gray scale document image with complex and inhomogeneous background it can adjust the size of the local area and logical thresholding level adaptively according to the local run length histogram and the local gray scale inhomogeneity our method can threshold various poor quality gray scale document images automatically without need of any prior knowledge of the document image and manual fine tuning of parameters it keeps useful information more accurately without overconnected and broken strokes of the characters and thus has a wider range of applications compared with other methods

a modified standard embedding with jumps in nonlinear optimization the paper deals with a combination of pathfollowing methods embedding approach and feasible descent direction methods so called jumps for solving a nonlinear optimization problem with equality and inequality constraints since the method that we propose here uses jumps from one connected component to another one more than one connected component of the solution set of the corresponding one parametric problem can be followed numerically it is assumed that the problem under consideration belongs to a generic subset which was introduced by jongen jonker and twilt there already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers in this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set the topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed furthermore this new method is applied to a so called modified standard embedding which is a particular construction without equality constraints finally an algorithmic version of this new method as well as computational results are presented

relative eigenvalues of hermitian matrices abstract motivated by the interest in a more thorough understanding of the relationship between the eigenvalues of a hermitian matrix m and those of a matrix of the form a ma we study the relative eigenvalues of a hermitian matrix with respect to a subspace a geometrically meaningful result and some of its useful corollaries are established as a consequence some inertia theorems given in earlier work are also strengthened

bifurcation problems in nonlinear parametric programming the nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters this system which is motivated by the fritz john first order necessary conditions contains all fritz john and all karush kuhn tucker points as well as local minima and maxima saddle points feasible and nonfeasible critical points necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition the linear dependence of the gradients of the active constraints and the singularity of the hessian of the lagrangian on a tangent space any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution for problems with one parameter we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity

conjugate cone characterization of positive definite and semidefinite matrices abstract positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in rn these results are obtained by generalizing moreau s polar decomposition to a conjugate decomposition some typical results are the matrix a is positive definite if and only if for some closed convex cone k a is positive definite on k and a at 1 exists and is semidefinite on the polar cone k the matrix a is positive semidefinite if and only if for some closed convex cone k such that either k is polyhedral or a at k is closed a is positive semidefinite on both k and the conjugate cone ka s xt a at s 0 x k and a at x 0 for all x in k such that xtax 0

morse programs a topological approach to smooth constrained optimization the purpose of this paper is to give a geometrical answer to the question to the strong second order sufficiency conditions hold at any local minimum point for almost all nonlinear programs our idea is to reduce the nonlinear programming problem to a finite family of well behaved nonlinear programs by perturbing the objective function in a linear fashion and perturbing the right hand side of the constraints by adding a constant each of the well behaved nonlinear programs will consist of minimizing a morse function on a manifold with boundary where the morse function has no critical points on the boundary

on the augmented lagrangian it is well known that for a nonlinear program any karush kuhn tucker point that satisfies the second order sufficient condition is also a strict local minimum of the augmented lagrangian with an underlying penalty parameter sufficiently large a main theme of the paper is to strengthen the theorem and bring to light an interesting connection among general critical points of a nonlinear program and its augmented lagrangian when a penalty parameter is sufficiently large it will be shown that a karush kuhn tucker point of a nonlinear program is also a critical point of its augmented lagrangian with the same index and the same nullity the aforementioned theorem is just the special case when the index and the nullity are zeros this result is established through an extension of a lemma due to finsler and debreu which is interesting in its own right consequently under a constraint qualification any karush kuhn tucker point can also be shown to satisfy a second order necessary condition for a local maximum of the generalized wolfe dual induced from the augmented lagrangian interestingly this relation holds without requiring the karush kuhn tucker point itself to satisfy any second order optimality condition