on the approximate augmented lagrangian for nonlinear symmetric cone programming abstract this paper studies the approximate augmented lagrangian for nonlinear symmetric cone programming the analysis is based on some results under the framework of euclidean jordan algebras we formulate the approximate lagrangian dual problem and study conditions for approximate strong duality results and an approximate exact penalty representation we also show under robinson s constraint qualification that the sequence of stationary points of the approximate augmented lagrangian problems converges to a stationary point of the original nonlinear symmetric cone programming

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

monotonicity of lowner operators and its applications to symmetric cone complementarity problems we prove necessary and sufficient conditions for locally lipschitz lowner operators to be monotone strictly monotone and strongly monotone utilizing our characterization of the strict monotonicity of lowner operators we generalize mangasarian class of nonlinear complementarity problem ncp functions to the setting of symmetric cone complementarity problem this affirmatively answers a question of tseng math program 83 159 185 1998

a continuation method for nonlinear complementarity problems over symmetric cones in this paper we introduce a new p type property for nonlinear functions defined over euclidean jordan algebras and study a continuation method for nonlinear complementarity problems over symmetric cones this new p type property represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically

associative and jordan algebras and polynomial time interior point algorithms for symmetric cones we present a general framework whereby analysis of interior point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras in particular such analyses are extendible to the cone of positive semidefinite hermitian matrices with complex and quaternion entries and to the lorentz cone we prove the case of the lorentz cone by using the embedding of its associated jordan algebra in the clifford algebra as an example of such extensions we take monterio s polynomial time complexity analysis of the family of similarly scaled directions introduced by monteiro and zhang 1998 and generalize it to cone lp over all representable symmetric cones

project scheduling problem with mixed uncertainty of randomness and fuzziness abstract project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time this paper considers project scheduling problem with mixed uncertainty of randomness and fuzziness where activity duration times are assumed to be random fuzzy variables three types of random fuzzy models as expected cost minimization model cost minimization model and chance maximization model are built to meet different management requirements random fuzzy simulations for some uncertain functions are given and embedded into genetic algorithm to design a hybrid intelligent algorithm finally some numerical experiments are given for the sake of illustration of the effectiveness of the algorithm

a primal dual interior point algorithm for second order cone optimization with full nesterov todd step in this paper we propose a primal dual path following interior point algorithm for second order cone optimization the algorithm is based on a new technique for finding the search directions and the strategy of the central path at each iteration we use only full nesterov todd step moreover we derive the currently best known iteration bound for the algorithm with small update method namely email protected where n denotes the number of second order cones in the problem formulation and e the desired accuracy

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems for any function f from mathbb r to mathbb r one can define a corresponding function on the space of n n block diagonal real symmetric matrices by applying f to the eigenvalues of the spectral decomposition we show that this matrix valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as rho order semismoothness our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices we also apply our results to the semidefinite complementarity problem addressing some basic issues in the analysis of smoothing semismooth newton methods for solving this problem

on max k sums the max k sum of a set of real scalars is the maximum sum of a subset of size k or alternatively the sum of the k largest elements we study two extensions first we show how to obtain smooth approximations to functions that are pointwise max k sums of smooth functions second we discuss how the max k sum can be defined on vectors in a finite dimensional real vector space ordered by a closed convex cone

on the coerciveness of some merit functions for complementarity problems over symmetric cones abstract one of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function an important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded the coerciveness of the merit function in this paper we introduce the generalized weak coerciveness of a continuous transformation under this condition we prove the coerciveness of some merit functions such as the natural residual function the normal map and the fukushima yamashita function for complementarity problems over symmetric cones we note that this is a much milder condition than strong monotonicity used in the current literature

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

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

loss of superlinear convergence for an sqp type method with conic constraints in this short note we consider a sequential quadratic programming sqp type method with conic subproblems and compare this method with a standard sqp method in which the conic constraint is linearized at each step for both approaches we restrict our attention to convex subproblems since these are easy to solve and guarantee a certain global descent property using the example of a simple nonlinear program nlp and its conic reformulation we show that the sqp method with conic subproblems displays a slower rate of convergence than standard sqp methods we then explain why an sqp subproblem that is based on a better approximation of the feasible set of the nlp results in a much slower algorithm

interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones we study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones while the analysis in l faybusovich positivity 1 1997 pp 331 357 depends on the optimization theory of convex log barrier functions our approach is based on the paper of monteiro and pang math oper res 23 1998 pp 39 60 where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices as an application of the results we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects consequently we show the existence of a path having the following properties a the path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions b any accumulation point of the path is a solution of the homogeneous model c if the original problem is solvable then every accumulation point of the path gives us a finite solution d if the original problem is strongly infeasible then under the assumption of lipschitz continuity any accumulation point of the path gives us a finite certificate proving infeasibility

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

the rate of convergence of the augmented lagrangian method for nonlinear semidefinite programming we analyze the rate of local convergence of the augmented lagrangian method in nonlinear semidefinite optimization the presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices an implicit function theorem for semismooth functions and variational analysis on the projection operator in the symmetric matrix space without requiring strict complementarity we prove that under the constraint nondegeneracy condition and the strong second order sufficient condition the rate of convergence is linear and the ratio constant is proportional to 1 c where c is the penalty parameter that exceeds a threshold overline c 0

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

traveling salesmen in the presence of competition we propose the competing salesmen problem csp a two player competitive version of the classical traveling salesman problem this problem arises when considering two competing salesmen instead of just one the concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does in particular we consider the situation where players take turns moving along one edge at a time within a graph g v e the set of customers is given by a subset vc v of the vertices at any given time both players know of their opponent s position a player wins if he is able to reach a majority of the vertices in vc before the opponent does we prove that the csp is pspace complete even if the graph is bipartite and both players start at distance 2 from each other furthermore we show that the starting player may not be able to avoid losing the game even if both players start from the same vertex however for the case of bipartite graphs we show that the starting player always can avoid a loss on the other hand we show that the second player can avoid to lose by more than one customer when play takes place on a graph that is a tree t and vc consists of leaves of t it is unclear whether a polynomial strategy exists for any of the two players to force this outcome for the case where t is a star i e a tree with only one vertex of degree higher than two and vc consists of n leaves of t we give a simple and fast strategy which is optimal for both players if vc consists not only of leaves we point out that the situation is more involved

some majorization inequalities induced by schur products in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n an element x is said to be majorized by an element y symbolically x y if the corresponding eigenvalue vector x is majorized by y in r n in this article we describe pointwise majorization inequalities of the form t x s x where t and s are linear transformations induced by schur products specializing we recover analogs of majorization inequalities of schur hadamard and oppenheimer stated in the setting of euclidean jordan algebras as well as majorization inequalities connecting quadratic and lyapunov transformations on v we also show how schur products induced by certain scalar means such as arithmetic geometric harmonic and logarithmic means naturally lead to majorization inequalities

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

complexity measures from interaction structures we evaluate information theoretic quantities that quantify complexity in terms of k text th order statistical dependences that cannot be reduced to interactions among k ensuremath 1 random variables using symbolic dynamics of coupled maps and cellular automata as model systems we demonstrate that these measures are able to identify complex dynamical regimes

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

an augmented lagrangian method for optimization problems in banach spaces we propose a variant of the classical augmented lagrangian method for constrained optimization problems in banach spaces our theoretical framework does not require any convexity or second order assumptions and it allows the treatment of inequality constraints with infinite dimensional image space moreover we discuss the convergence properties of our algorithm with regard to feasibility global optimality and kkt conditions some numerical results are given to illustrate the practical viability of the method

convergence analysis of the augmented lagrangian method for nonlinear second order cone optimization problems abstract the paper focuses on the convergence rate of the augmented lagrangian method for nonlinear second order cone optimization problems under a set of assumptions of sufficient conditions including the componentwise strict complementarity condition the constraint nondegeneracy condition and the second order sufficient condition we first study some properties of the augmented lagrangian and then show that the rate of local convergence of the augmented lagrangian method is proportional to 1 where the penalty parameter is not less than a threshold 0

production scheduling in a price competition abstract in this paper we study the production scheduling problem in a price competition two firms produce the same product and compete with each other the production capacity of each firm is random due to random breakdowns the demand process and its allocation are random and affected by the pricing strategies of both firms we consider a finite planning horizon the scheduling problem is formulated as a finite dynamic game algorithms are developed to determine the control policies of interest numerical cases are studied to observe the system behavior under those policies we also construct a single firm optimization model and demonstrate that the policy from the single firm model may not perform well in a competitive environment

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

local convergence of exact and inexact augmented lagrangian methods under the second order sufficient optimality condition we establish local convergence and rate of convergence of the classical augmented lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second order sufficient optimality condition in particular no constraint qualifications of any kind are needed previous literature on the subject required in addition the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second order sufficient condition that said the classical results allow the initial multiplier estimate to be far from the optimal one at the expense of proportionally increasing the threshold value for the penalty parameters although our primary goal is to avoid constraint qualifications if the stronger assumptions are introduced then starting points far from the optimal multiplier are allowed within our analysis as well using only the second order sufficient optimality condition for penalty parameters large

on the complexity of project scheduling to minimize exposed time we consider project scheduling where the project manager s objective is to minimize the time from when an adversary discovers the project until the completion of the project we analyze the complexity of the problem identifying both polynomially solvable and np hard versions of the problem the complexity of the problem is seen to be dependent on the nature of renewable resource constraints precedence constraints and the ability to crash activities in the project

the dynamics of complex amplitude norm preserving lattices of coupled oscillators we introduce a class of models composed by lattices of coupled complex amplitude oscillators which preserve the norm these models are particularly well adapted to investigate phenomena described by the nonlinear schr6 odinger equation the coupling between oscillators is parameterized by the mass while their local dynamics is illustrated for two area preserving maps one obtained from the exact local solution of the schr6 odinger equation the other obtained from its crank nicholson discretization in both cases we determine all periodic orbits and show how to detect artifacts introduced by the discretization c

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

information capacity and pattern formation in a tent map network featuring statistical periodicity we provide quantitative support to the observation that lattices of coupled maps are efficient information coding devices it has been suggested recently that lattices of coupled maps may provide a model of information coding in the nervous system because of their ability to create structured and stimulus dependent activity patterns which have the potential to be used for storing information in this paper we give an upper bound to the effective number of patterns that can be used to store information in the lattice by evaluating numerically its information capacity or information rate as a function of the coupling strength between the maps we also estimate the time taken by the lattice to establish a limiting activity pattern

log sigmoid nonlinear lagrange method for nonlinear optimization problems over second order cones this paper analyzes the rate of local convergence of the log sigmoid nonlinear lagrange method for nonconvex nonlinear second order cone programming under the componentwise strict complementarity condition the constraint nondegeneracy condition and the second order sufficient condition we show that the sequence of iteration points generated by the proposed method locally converges to a local solution when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter finally we report numerical results to show the efficiency of the method

optimality conditions and global convergence for nonlinear semidefinite programming sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization in this paper we extend theses concepts for nonlinear semidefinite programming we define two sequential optimality conditions for nonlinear semidefinite programming the first is a natural extension of the so called approximate karush kuhn tucker akkt well known in nonlinear optimization the second one called trace akkt is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided we propose an augmented lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed weaker than previously considered ones which are sufficient for the global convergence of the algorithm to a stationary point

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

nash equilibria in competitive project scheduling we consider the problem of scheduling activities of a project by a firm that competes with another firm that has to perform the same project the profit that a firm gets from each activity depends on whether the firm finishes the activity before or after its competitor it is required to find a nash equilibrium solution or show that no such solutions exist we present a structural characterization of nash equilibrium solutions and a low order polynomial algorithm for the problem

a c0 finite element method for an inverse problem a c 0 finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem the unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error error estimates in a weighted h 1 norm and l 2 are given numerical examples are presented to show features of the method

the augmented lagrangian method for equality and inequality constraints in hilbert spaces in this paper we consider an augmented lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a hilbert space an inequality constraint whose image space is finite dimensional and an affine inequality constraint whose image space is in an infinite dimensional hilbert space we obtain local convergence of this method without imposing strict complementarity conditions when the equality as well as the inequality constraint with finite dimensional image space are augmented to the author s knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

some p properties for nonlinear transformations on euclidean jordan algebras in this article we introduce the concepts of p and p0 properties for a nonlinear transformation defined on a euclidean jordan algebra and study existence of solution in the associated complementarity problems in particular we show in this general setting that if a transformation has the p0 and r0 properties then all associated complementarity problems have solutions we also describe a necessary condition for a transformation to have the global uniqueness of solution property

lagrange multipliers and optimality lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first order optimality conditions formally as a system of equations modern applications with their emphasis on numerical methods and more complicated side conditions than equations have demanded deeper understanding of the concept and how it fits into a larger theoretical picture a major line of research has been the nonsmooth geometry of one sided tangent and normal vectors to the set of points satisfying the given constraints another has been the game theoretic role of multiplier vectors as solutions to a dual problem interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black and white constraints to be replaced by penalty expressions this paper traces su

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

uzawa block relaxation domain decomposition method for the two body contact problem with tresca friction we propose a uzawa block relaxation domain decomposition method for a two body contact problem with tresca friction we introduce auxiliary interface unknowns to transform the variational problem into a saddle point problem applying a uzawa block relaxation algorithm to the corresponding augmented lagrangian functional we obtain a domain decomposition algorithm in which we have to solve two uncoupled linear elasticity subproblems in each iteration the auxiliary unknowns are computed explicitly using kuhn tucker conditions and fenchel duality theory numerical experiments show the scalability of the domain decomposition algorithm on matching or nonmatching meshes for two or three dimensional contact problems

pattern formation in diffusive advective coupled map lattices we investigate pattern formation and evolution in coupled map lattices when advection is incorporated inaddition to the usual diffusive term all patterns may be suitably grouped into ve classes three periodic supporting static patterns and traveling waves and two nonperiodic relative frequencies are determined as afunction of all model parameters diffusion advection local nonlinearity and lattice size advection plays animportant role in coupled map lattices being capable of considerably altering pattern evolution for instance advection may induce synchronization making chaotic patterns evolve periodically as a byproduct we de scribe a practical algorithm for classifying generic pattern evolutions and for measuring velocities of travelingwaves doi 10 1103 physreve 69 066206 pacs number s 89 75 kd 05 45 xt 05 45 rai introduction

estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented lagrangian technique in this paper we apply a hybrid method to estimate a temporally and spatially varying diffusion coefficient in a parabolic system this technique combines the output least squares and the equation error method the resulting optimization problem is solved by an augmented lagrangian approach and convergence as well as rate of convergence proofs are provided the stability of the estimated coefficient with respect to perturbations in the observation is guaranteed

analysis of nonsmooth vector valued functions associated with second order cones let open image in new window be the lorentz second order cone in open image in new window for any function f from open image in new window to open image in new window one can define a corresponding function fsoc x on open image in new window by applying f to the spectral values of the spectral decomposition of x open image in new window with respect to open image in new window we show that this vector valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as order semismoothness these results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second order cone programs and complementarity problems

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

sequential and parallel splitting methods for bilinear control problems in hilbert spaces an augmented lagrangian method with second order update is developed and its relationship to the sequential quadratic programming method is described the rate of convergence proof depends on a second order sufficient optimality condition which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type numerical examples are included which demonstrate the globalizing effect of the augmented lagrangian method

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

mesh independence for an augmented lagrangian sqp method in hilbert spaces an augmented lagrangian sqp algorithm for optimal control of differential equations in hilbert spaces is analyzed this algorithm has second order convergence rate provided that a second order sufficient optimality condition is satisfied the internal approximation of this method is investigated and convergence results are presented a mesh independence principle for the augmented lagrangian sqp method is proved which asserts that asymptotically the infinite dimensional algorithm and finite dimensional discretizations have the same convergence property more precisely for sufficiently small mesh size there is at most a difference of one iteration step between the number of steps required by the infinite dimensional method and its discretization to converge within a given tolerance varepsilon 0 the theoretical results are demonstrated by two optimal control problems for the burgers equation

some properties of the augmented lagrangian in cone constrained optimization a large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions in this paper we discuss augmented lagrangian duality for such optimization problems we formulate the augmented lagrangian dual problems and study conditions ensuring existence of the corresponding augmented lagrange multipliers we also discuss sensitivity of optimal solutions to small perturbations of augmented lagrange multipliers

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

improved local convergence results for augmented lagrangian methods in varvec c mathbf 2 c2 cone reducible constrained optimization this paper deals with a class of cone reducible constrained optimization problems which encompasses nonlinear programming semidefinite programming second order cone programming and any combination thereof using the second order sufficient condition and a strict version of the robinson constraint qualification we provide a semi local error bound which generalizes known results from the literature moreover under the same assumptions we prove that an augmented lagrangian method is locally convergent with rate proportional to 1 rho k where rho k is the penalty parameter and that rho k remains bounded