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

approximating the scattering by a material filled 2 d trough in an infinite plane using the impedance boundary condition an application of the impedance boundary condition to 2 d trough scatterers is described in which the approximate field solutions are obtained at a much reduced computational cost compared with more exact numerical approaches the geometries investigated have appreciable subsurface interactions that must be accounted for in the surface impedance after a brief canonical study of the coated circular cylinder the filled 2 d trough in a ground plane which can be used as a model for an antenna cavity mount is analyzed and compared with a more exact numerical approach the results of a parametric study of this geometry demonstrate that excellent far field agreement is obtained when the material and structure satisfy only the requirements for the traditional impedance boundary condition this model is also shown to provide a very good simulation for 2 d troughs with slanted sidewalls which produce inhomogeneous equivalent surface impedance representations

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

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

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

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

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

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

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

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

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

a pivotal method for affine variational inequalities we explain and justify a path following algorithm for solving the equations acx a where a is a linear transformation from rn to rn c is a polyhedral convex subset of rn and ac is the associated normal map when ac is coherently oriented we are able to prove that the path following method terminates at the unique solution of acx a which is a generalization of the well known fact that lemke s method terminates at the unique solution of lcpq m when m is a p matrix otherwise we identity two classes of matrices which are analogues of the class of copositive plus and l matrices in the study of the linear complementarity problem we then prove that our algorithm processes acx a when a is the linear transformation associated with such matrices that is when applied to such a problem the algorithm will find a solution unless the problem is infeasible in a well specified sense

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

a class of exact and higher order surface boundary conditions for layered structures a class of exact and higher order surface impedance boundary conditions hoibcs is derived initially exact impedance boundary conditions ibcs are derived first in the spectral and then in the coordinate domain it is shown that in the coordinate domain they are expressed by a dyadic operator acting on the convolution product between the scalar green s function corresponding to the considered structure and the tangential magnetic field next it is demonstrated that the higher order impedance boundary conditions in the spatial domain correspond to an appropriate expansion of the mentioned convolution product in the space domain finally the accuracy of the hoibcs is estimated by comparing the exact solution with that obtained through the hoibcs for some typical canonical problems the corresponding error curves presented refer to the worst error situation for each one of the chosen cases

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

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

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

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

tm plane wave scattering from finite rectangular grooves in a conducting plane using overlapping t block method transverse magnetic plane wave scattering from finite rectangular grooves in a conducting plane is systematically analyzed with the overlapping t block method multiple rectangular grooves are divided into several overlapping t blocks to obtain the fast cpu time simple applicability and wide versatility the field representations within t blocks are expressed using the green s function relation and mode matching method the scattered fields are obtained in simple closed forms including a fast convergent integral

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

solutions to affine generalized equations using proximal mappings the normal map has proven to be a powerful tool for solving generalized equations of the form find z e c with 0 e f z nc z where c is a convex set and nc z is the normal cone to c at z in this paper we use the t map a generalization of the normal map to solve equations of the more general form find z e dom t with 0 e f z t z where t is a maximal monotone multifunction we present a path following algorithm that determines zeros of coherently oriented piecewise affine functions and we use this algorithm together with the t map to solve the generalized equation for affine coherently oriented functions f and polyhedral multifunctions t the path following algorithm we develop here extends the piecewise linear homotopy framework of eaves to the case where a representation of a subdivided manifold is unknown

higher order impedance boundary conditions applied to scattering by coated bodies of revolution in this paper higher order impedance boundary conditions will be employed in the solution of scattering by coated conducting bodies of revolution the higher order impedance solution reduces the total number of unknowns relative to the exact solution and produces a system matrix which is less dense than that of the exact solution the construction of the solution involves two distinct steps in the first step the body of revolution is replaced by an equivalent set of electric and magnetic currents on its exterior surface which generate the true fields outside the body an integral equation relating these currents through the free space green s function is derived step two employs the higher order impedance boundary condition to relate the electric and magnetic currents on the surface of the body this replaces the rigorous solution of the interior problem the higher order impedance boundary conditions are derived by obtaining an exact impedance boundary condition in the spectral domain for the coated ground plane approximating the impedances as ratios of polynomials in the transform variables and employing the fourier transform the resulting spatial domain differential equations are solved in conjunction with the integral equation using the method of moments several examples of bistatic and monostatic radar cross section for coated bodies of revolution are used to illustrate the accuracy of the higher order impedance boundary condition solution relative to the standard impedance boundary condition solution and the exact solution the effects of coating thickness loss and curvature on the accuracy of the solution are discussed

a gauss newton method for convex composite optimization an extension of the gauss newton method for nonlinear equations to convex composite optimization is described and analyzed local quadratic convergence is established for the minimization ofh f under two conditions namelyh has a set of weak sharp minima c and there is a regular point of the inclusionf x c this result extends a similar convergence result due to womersley this journal 1985 which employs the assumption of a strongly unique solution of the composite functionh f a backtracking line search is proposed as a globalization strategy for this algorithm a global convergence result is established with a quadratic rate under the regularity assumption

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

diffraction of plane waves by a finite array of dielectric loaded cavity backed slots on a common ground plane for oblique incidence and arbitrary polarization plane wave diffraction by a finite array of two dimensional dielectric loaded cavity backed slots on a common ground plane is investigated for oblique incidence and arbitrary polarization the governing system of coupled singular integral integrodifferential equations is discretized using moment method oriented direct singular integral equation methods treating all singular integrals analytically via rapidly converging algorithms leads to numerically stable and efficient analytical expressions for all matrix elements as a result no numerical integration is required to compute these elements several numerical examples are presented to validate the algorithm and illustrate its convergence characteristics results are also presented that reveal the possibility of controlling the absorption efficiency by suitably selecting several geometrical and physical parameters of the structure

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

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

diffraction by a thin dielectric half plane a uniform diffraction coefficient is derived for a thin dielectric half plane illuminated by an h polarized incident plane wave this is accomplished by modeling the half plane using a pair of resistive and modified conductive sheets and then solving the resulting integral equations via the wiener hopf technique in contrast to a recently published solution the one presented here satisfies reciprocity and the numerical results which are included confirm its accuracy

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

derivation and application of a class of generalized boundary conditions electromagnetic scattering boundary conditions involving higher order derivatives are presented for simulating surfaces whose reflection coefficients are known analytically numerically or experimentally procedures for determining the coefficients of the derivatives are discussed along with the effect of displacing the surface where the boundary conditions are applied provided the coefficients satisfy a duality relation equivalent forms of the boundary conditions involving tangential field components are deduced and these provide the natural extension to nonplanar surfaces as an illustration a metal backed uniform dielectric layer is simulated it is shown that fourth order conditions are capable of providing an accurate simulation for layers at least a quarter of a wavelength in thickness

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

equivalence of lcp and pls it is our purpose here to show that two prototype models of complementary pivot and fixed point theory and the corresponding path following solution methods are conceptually equivalent

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

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

application of a conjugate gradient fft method to scattering from thin planar material plates the backscatter cross section is calculated for thin material plates with finite electric permittivity conductivity and magnetic permeability illuminated by a plane wave the plates are assumed to be planar with an arbitrary perimeter the integral equations are formed and solved by a combined conjugate gradient fast fourier transform cg fft method the cg fft method was tested for several geometrics and materials measured and computed backscatter results are compared for a perfectly conducting equilateral triangle plate a square dielectric and magnetic plate and a circular dielectric plate the agreement between measured and computed data is generally good except toward edge on incidence where several factors cause discrepancies accurate approximations to the geometry and far field integrals become critical near edge on incidence and it is postulated that as the incidence angle approaches edge on the sampling interval and tolerance should be decreased

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

pc matrices and the linear complementarity problem abstract we introduce a new matrix class pc which consists of those matrices m for which the solution set of the corresponding linear complementarity problem is connected for every q rn we consider lemke s pivotal method from the perspective of piecewise linear homotopies and normal maps and show that lemke s method processes all matrices in pc q0 we further investigate the relationship of the class pc to other known matrix classes and show that column sufficient matrices are a subclass of pc as are 2 2 p0 matrices

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

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

scattering by superquadric dielectric coated cylinders using higher order impedance boundary conditions a general method for deriving higher order impedance boundary conditions is described it is based on solving an appropriate canonical problem exactly in the spectral domain after approximating the spectral impedance terms as a ratio of polynomials in the transform variable elementary properties of the fourier transform are used to obtain the corresponding boundary condition in the spatial domain the method is applicable to multilayer coatings with arbitrary constitutive relations higher order boundary conditions which neglect the effects of curvature are derived for a dielectric coating using the method the boundary condition equation and the magnetic field integral equation are solved simultaneously using the method of moments yielding the bistatic and monostatic radar cross section for dielectric coated superquadric cylinders the method is also applicable to a combined field integral equation cfie solution which can be used to eliminate the internal resonance problem associated with either the electric field integral equation efie or magnetic field integral equation mfie

te scattering by a two dimensional groove in a ground plane using higher order boundary conditions an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented deficiencies of such higher order boundary conditions are addressed and a correction for the present case is proposed as part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition

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

backscattering from a two dimensional rectangular crack using fie the numerical solution of a rectangular crack in a perfectly conducting surface is appropriate for non destructive testing ndt applications to model faults the paper presents a direct modeling technique for determining the h and e polarized backscattering signatures of a two dimensional crack in a metallic surface that is suitable for inverse scattering problem the governing field integral equations fie with logarithmic and hyper singular kernels is first discretized and solved by a collocation method based on chebyshev polynomials by using ad hoc quadrature rules the integral equation is then reduced to a linear system of algebraic equations this approach does not have the size or frequency limitations of the regular techniques such as modal expansion and quasi static manners the results are in good agreement with the entirely numerical and non reversible solution of a finite element method

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

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

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

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

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