a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

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

nonlocal isoperimetric inequalities for kolmogorov fokker planck operators abstract in this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of kolmogorov fokker planck operators which are of interest in physics these operators are very degenerate and do not possess a variational structure the prototypical example was introduced by kolmogorov in his 1938 paper on brownian motion and the theory of gases our work has been influenced by ideas of m ledoux in the local case

a global lower bound for the fundamental solution of kolmogorov fokker planck equations the main result of this paper is a global lower bound for the fundamental solution of the ultraparabolic differential operator

a proximal point algorithm for log determinant optimization with group lasso regularization we consider the covariance selection problem where variables are clustered into groups and the inverse covariance matrix is expected to have a blockwise sparse structure this problem is realized via penalizing the maximum likelihood estimation of the inverse covariance ma trix by group lasso regularization we propose to solve the resulting log determinant optimization problem with the classical proximal point algorithm ppa at each iteration as it is difficult to update the primal variables directly we first solve the dual subproblem by an inexact semismooth newton cg method and then update the primal variables by explicit formulas based on the com puted dual variables we also propose to accelerate the ppa by an inexact generalized newton s method when the iterate is close to the solution theoretically we prove that at the optimal solution the nonsingularity of the generalized hessian matrices of the dual subproblem is equivalent to the constraint nondegeneracy condition for the primal problem global and local convergence results are also presented for the proposed ppa moreover based on the augmented lagrangian function of the dual problem we derive an alternating direction method adm which is easily implementable and is demonstrated to be efficient for random problems numerical results including comparisons with the adm on both synthetic and real data are presented to demonstrate that the proposed newton cg based ppa is stable and efficient and in particular outperforms the adm when high accuracy is required

optimal regularity in the obstacle problem for kolmogorov operators related to american asian options in this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of kolmogorov type we treat smooth obstacles as well as non smooth obstacles all our proofs follow the same line of thought and are based on blow ups compactness barriers and arguments by contradiction the problem considered arises in financial mathematics when considering path dependent derivative contracts with early exercise feature

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

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

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

liouville theorems for non local operators abstract the paper characterizes some classes of pseudo differential operators for which there are or there are not non constant bounded harmonic functions non local perturbations of ornstein uhlenbeck operators and operators with dissipative coefficients are considered the methods used are probabilistic and based on the concept of absorption function and on a new extension of the bismut elworthy li formula the probabilistic interpretation of the liouville theorem by means of absorption functions for general markov processes is given as well

existence uniqueness and invariant measures for stochastic semilinear equations on hilbert spaces a class of stochastic evolution equations with additive noise and weakly continuous drift is considered first regularity properties of the corresponding ornstein uhlenbeck transition semigrouprt are obtained we show thatrt is a compactc0 semigroup in all sobolev spaceswn p which are built on its invariant measure then we show the existence uniqueness compactness and smoothing properties of the transition semigroup for semilinear equations inlp spaces and spacesw1 p as a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to it is shown also that the density of this measure with respect to is inlp for allp 1

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

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

on the ornstein uhlenbeck operator in spaces of continuous functions we study the realization of the ornstein uhlenbeck operator a in the space of the uniformly continuous and bounded functions in rn we prove that it generates a semigroup which is neither strongly continuous nor analytic but enjoys nice smoothing properties such properties are used to prove a schauder type theorem for the elliptic equation u au f

event based optimization of markov systems recent research indicates that markov decision processes mdps and perturbation analysis pa based optimization can be derived easily from two fundamental performance sensitivity formulas with this sensitivity point of view an event based optimization approach including event based sensitivity analysis and event based policy iteration was proposed via an example by x r cao discrete event dyn syst theory appl vol 15 pp 169 197 2005 this approach utilizes the special feature of a system and illustrates how the potentials can be aggregated using the special feature the approach applies to many practical problems that do not fit well the standard mdp formulation this note provides a mathematical formulation and proves the main results for this approach

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming it is known that the karush kuhn tucker kkt conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices we show in this paper that the primal and dual constraint nondegeneracies the strong regularity the nonsingularity of the b subdifferential of this nonsmooth system and the nonsingularity of the corresponding clarke s generalized jacobian at a kkt point are all equivalent moreover we prove the equivalence between each of these conditions and the nonsingularity of clarke s generalized jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing newton methods in particular we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies but without the strict complementarity

completion of squares revisited and extended due to its simplicity the completion of squares technique is quite popular in linear optimal control however this simple technique is limited to linear quadratic gaussian systems in this note by interpreting the completion of squares from a new angle we extend this technique to performance optimization of general markov systems with the long run average criterion leading to a new approach to performance optimization based on direct comparisons of the performance of two policies

pointwise estimates for a class of non homogeneous kolmogorov equations we consider a class of ultraparabolic differential equations that satisfy the hormander s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions the method extends the moser s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption here we remove that assumption by proving some potential estimates and some ad hoc sobolev type inequalities for solutions

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

an algebraic condition equivalent to strong stability of stationary solutions of nonlinear positive semidefinite programs this paper addresses strong stability in the sense of kojima of stationary solutions of nonlinear positive semidefinite programs nsdp first we give a characterization of stability from the po

functional inequalities for a class of nonlocal hypoelliptic equations of hormander type abstract we consider a class of second order partial differential operators a of hormander type which contain as a prototypical example a well studied operator introduced by kolmogorov in the 30s we analyse some properties of the nonlocal operators driven by the fractional powers of a and we introduce some interpolation spaces related to them we also establish sharp pointwise estimates of harnack type for the semigroup associated with the extension operator moreover we prove both global and localised versions of poincare inequalities adapted to the underlying geometry

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

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

backward errors for eigenvalue and singular value decompositions we present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two norm and in the frobenius norm through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition when the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured analogous results hold for the singular vectors we indicate the effect of our error bounds on implementations for eigenvector and singular vector computation in a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal qr factor is not much larger than its distance to the closest orthogonal matrix

the policy iteration algorithm for average reward markov decision processes with general state space the average cost optimal control problem is addressed for markov decision processes with unbounded cost it is found that the policy iteration algorithm generates a sequence of policies which are c regular where c is the cost function under consideration this result only requires the existence of an initial c regular policy and an irreducibility condition on the state space furthermore under these conditions the sequence of relative value functions generated by the algorithm is bounded from below and nearly decreasing from which it follows that the algorithm is always convergent under further conditions it is shown that the algorithm does compute a solution to the optimality equations and hence an optimal average cost policy these results provide elementary criteria for the existence of optimal policies for markov decision processes with unbounded cost and recover known results for the standard linear quadratic gaussian problem in particular in the control of multiclass queueing networks it is found that there is a close connection between optimization of the network and optimal control of a far simpler fluid network model

attaining the optimal gaussian diffusion acceleration sampling from probability distributions in high dimensional spaces is generally impractical diffusion processes with invariant equilibrium distributions can be used as a means to generate approximations an important task in such an endeavor is to design an equilibrium preserving drift to accelerate the convergence starting from a reversible diffusion it is desirable to depart for non reversible dynamics via a perturbed drift so that the convergence rate is maximized with the common equilibrium in the gaussian diffusion acceleration this problem can be cast as perturbing the inverse of a given covariance matrix by skew symmetric matrices so that all resulting eigenvalues have identical real part this paper describes two approaches to obtain the optimal rate of gaussian diffusion the asymptotical approach works universally for arbitrary ornstein uhlenbeck processes whereas the direct approach can be implemented as a fast divide and conquer algorithm a comparison with recently proposed lelievre nier pavliotis algorithm is made

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

semidefinite programs new search directions smoothing type methods and numerical results motivated by some results for linear programs and complementarity problems this paper gives some new characterizations of the central path conditions for semidefinite programs exploiting these characterizations some smoothing type methods for the solution of semidefinite programs are derived the search directions generated by these methods are automatically symmetric and the overall methods are shown to be globally and locally superlinearly convergent under suitable assumptions some numerical results are also included which indicate that the proposed methods are very promising and comparable to several interior point methods moreover the current method seems to be superior to the smoothing method recently proposed by chen and tseng non interior continuation methods for solving semidefinite complementarity problems technical report department of mathematics university of washington seattle 1999

a unified approach to markov decision problems and performance sensitivity analysis with discounted and average criteria multichain cases we propose a unified framework to markov decision problems and performance sensitivity analysis for multichain markov processes with both discounted and average cost performance criteria with the fundamental concept of performance potentials we derive both performance gradient and performance difference formulas which play the central role in performance optimization the standard policy iteration algorithms for both discounted and average reward mdps can be established using the performance difference formulas in a simple and intuitive way and the performance gradient formulas together with stochastic approximation may lead to new optimization schemes this sensitivity based point of view of performance optimization provides some insights that link perturbation analysis markov decision processes and reinforcement learning together the research is an extension of the previous work on ergodic markov chains cao automatica 36 2000 771

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

l1 properties of intrinsic schrodinger semigroups abstract we investigate the l 1 properties of the intrinsic markov semigroup associated with a schrodinger operator on r n which possesses a positive ground state we discover cases for which this semigroup is norm analytic for positive times and others for which the semigroup is norm discontinuous in the strongest possible sense in the case of the harmonic oscillator we show that the generator of the intrinsic semigroup has totally different spectrum depending on whether one works in l 1 r e x 2 dx or l 2 r e x 2 dx in more general cases we show that the equality of the l 1 and l 2 spectrum is closely related to whether the schrodinger semigroup is intrinsically ultracontractive

hardy littlewood sobolev inequalities for a class of non symmetric and non doubling hypoelliptic semigroups in his seminal 1934 paper on brownian motion and the theory of gases kolmogorov introduced a second order evolution equation which displays some challenging features in the opening of his 1967 hypoellipticity paper hormander discussed a general class of degenerate ornstein uhlenbeck operators that includes kolmogorov s as a special case in this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of hardy littlewood sobolev type in the situation when the drift matrix has nonnegative trace our work has been influenced by ideas of e stein and varopoulos in the framework of symmetric semigroups one of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non symmetric setting

on the superlinear local convergence of a penalty free method for nonlinear semidefinite programming abstract this paper is concerned with a sequentially semidefinite programming ssdp algorithm for solving nonlinear semidefinite programming problems nlsdp which does not use a penalty function or a filter this method inspired by the classic sqp method calculates a trial step by a quadratic semidefinite programming subproblem at each iteration the trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced in order to guarantee global convergence the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit we prove the global convergence properties of the algorithm under mild assumptions we also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid maratos effect we prove that under the strict complementarity and the strong second order sufficient conditions with the sigma term the rate of local convergence is superlinear finally some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in c o m p l e i b are given

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

some remarks on the smoluchowski kramers approximation according to the smoluchowski kramers approximation solution qt of the equation mu ddot q t mu b q t mu dot q t mu sigma q t mu dot w t q 0 q dot q p where dot w t is the white noise converges to the solution of equation dot q t b q t sigma q t dot w t q 0 q as 0 many asymptotic problems for the last equation were studied in recent years we consider relations between asymptotics for the first order equation and the original second order equation homogenization large deviations and stochastic resonance approximation of brownian motion wt by a smooth stochastic process stationary distributions are considered

the n th order bias optimality for multichain markov decision processes in this paper we propose a new approach to the theory of finite multichain markov decision processes mdps with different performance optimization criteria we first propose the concept of nth order bias then using the average reward and bias difference formulas derived in this paper we develop an optimization theory for finite mdps that covers a complete spectrum from average optimality bias optimality to all high order bias optimality in a unified way the approach is simple direct natural and intuitive it depends neither on laurent series expansion nor on discounted mdps we also propose one phase policy iteration algorithms for bias and high order bias optimal policies which are more efficient than the two phase algorithms in the literature furthermore we derive high order bias optimality equations this research is a part of our effort in developing sensitivity based learning and optimization theory

generalized ornstein uhlenbeck semigroups littlewood paley stein inequalities and the p a meyer equivalence of norms abstract let be a nondegenerate gaussian measure on a hilbert space h for an arbitrary selfadjoint nonnegative operator a we consider the semigroup e tl e ta on l p where stands for the second quantization operator we provide an explicit characterization of the domains of i l m 2 in l p in terms of gaussian sobolev spaces thus extending the p a meyer result on equivalence of norms the main tools are the littlewood paley stein inequalities which are proved under minimal assumptions by a purely analytic method following e stein 1970 topics in harmonic analysis princeton univ press princeton nj

global l p estimates for degenerate ornstein uhlenbeck operators we consider a class of degenerate ornstein uhlenbeck operators in mathbb r n of the kind mathcal a equiv sum i j 1 p 0 a ij partial x i x j 2 sum i j 1 n b ij x i partial x j where aij bij are constant matrices aij is symmetric positive definite on mathbb r p 0 p0 n and bij is such that mathcal a is hypoelliptic for this class of operators we prove global lp estimates 1 p of the kind left vert partial x i x j 2 u right vert l p left mathbb r n right leq c left left vert mathcal a u right vert l p left mathbb r n right left vert u right vert l p left mathbb r n right right rm for i j 1 2 ldots p 0 and corresponding weak type 1 1 estimates this result seems to be the first case of global estimates in lebesgue lp spaces for complete hormander s operators sum x i 2 x 0 proved in absence of a structure of homogeneous group we obtain the previous estimates as a byproduct of the following one which is of interest in its own left vert partial x i x j 2 u right vert l p left s right leq c left vert lu right vert l p left s right for any u in c 0 infty left s right where s is the strip mathbb r n times left 1 1 right and l is the kolmogorov fokker planck operator mathcal a partial t to get this estimate we use in a crucial way the left invariance of l with respect to a lie group structure in mathbb r n 1 and some results on singular integrals on nonhomogeneous spaces recently proved in bramanti revista matematica iberoamericana 2009 in press

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

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

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

quadratic convergence of a nonsmooth newton type method for semidefinite programs without strict complementarity we consider a newton type method for the solution of semidefinite programs this newton type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations we establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity

level sets of the fundamental solution and harnack inequality for degenerate equations of kolmogorov type in this paper we establish a uniform harnack inequality for a class of degenerate equations whose prototype is kolmogorov s equations in r3 dyyu ydzu dtu o our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution

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

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

harnack inequality for a class of kolmogorov fokker planck equations in non divergence form we prove invariant harnack inequalities for certain classes of non divergence form equations of kolmogorov type the operators we consider exhibit invariance properties with respect to a homogeneous lie group structure the coefficient matrix is assumed either to satisfy a cordes landis condition on the eigenvalues or to admit a uniform modulus of continuity

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

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

a convex matrix optimization for the additive constant problem in multidimensional scaling with application to locally linear embedding the additive constant problem has a long history in multidimensional scaling and it has recently been used to resolve the issue of indefiniteness of the geodesic distance matrix in isomap but it would lead to a large positive constant being added to all eigenvalues of the centered geodesic distance matrix often causing significant distortion of the original distances in this paper we reformulate the problem as a convex optimization of almost negative semidefinite matrix so as to achieve minimal variation of the original distances we then develop a newton cg method and further prove its quadratic convergence finally we include a novel application to the famous lle locally linear embedding in nonlinear dimensionality reduction addressing the issue when the input of lle has missing values we justify the use of the developed method to tackle this issue by establishing that the local gram matrix used in lle can be obtained through a local euclidean distance matrix the effectiveness of our method is