a new primal dual path following interior point algorithm for semidefinite optimization abstract in this paper we present a new primal dual path following interior point algorithm for semidefinite optimization the algorithm is based on a new technique for finding the search direction and the strategy of the central path at each iteration we use only full nesterov todd step moreover we obtain the currently best known iteration bound for the algorithm with small update method namely o n log n which is as good as the linear analogue

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

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

controllability of multi agent systems with periodically switching topologies and switching leaders abstractthis paper considers controllability of multi agent systems with periodically switching topologies and switching leaders the concept of m periodic controllability is proposed and a criterion for m periodic controllability is established the effect of the duration of subsystems on controllability is analysed by utilising a property of analytic functions in addition the influence of switching periods on controllability is investigated and an algorithm is proposed to search for the fewest periods to ensure controllability a necessary condition for m periodic controllability is obtained from the perspective of eigenvectors of the subsystems laplacian matrices for a system with switching leaders it is proved that switching leader controllability is equivalent to multiple leader controllability furthermore both the switching order and the tenure of agents being leaders have no effect on the controllability some examples are provided to illustrate the theoretical results

controllability of heterogeneous multi agent systems under directed and weighted topology abstractthis paper considers the controllability problem for both continuous and discrete time linear heterogeneous multi agent systems with directed and weighted communication topology first two kinds of neighbour based control protocols based on the distributed protocol of first order and second order multi agent systems are proposed under which it is shown that a heterogeneous multi agent system is controllable if the underlying communication topology is controllable then under special leader selection the result shows that the controllability of a heterogeneous multi agent system is solely decided by its communication topology graph furthermore some necessary and or sufficient conditions are derived for controllability of communication topology from algebraic and graphical perspectives finally simulation examples are presented to demonstrate the effectiveness of the theoretical results

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

quadratic stabilisability of multi agent systems under switching topologies this paper addresses the stabilisability of multi agent systems mass under switching topologies necessary and or sufficient conditions are presented in terms of graph topology these conditions explicitly reveal how the intrinsic dynamics of the agents the communication topology and the external control input affect stabilisability jointly with the appropriate selection of some agents to which the external inputs are applied and the suitable design of neighbour interaction rules via a switching topology an mas is proved to be stabilisable even if so is not for each of uncertain subsystem in addition a method is proposed to constructively design a switching rule for mass with norm bounded time varying uncertainties the switching rules designed via this method do not rely on uncertainties and the switched mas is quadratically stabilisable via decentralised external self feedback for all uncertainties with respect to applications of the stabilisability results the formation control and the cooper

leaders in multi agent controllability under consensus algorithm and tree topology abstract by controllability of multi agent systems we mean the property that all agents can be driven to arbitrary configurations only through regulating a few leaders externally while the rest of the agents are subject to the nearest neighbor law it is known that the controllability of multi agent systems depends not only on the communication topologies among agents but also on the selection of leaders the main objective of this paper aims to characterize the virtue that leaders should have from the perspective of algebraic and graphical conditions we begin with a heuristic observation for tree graph to gain more insights necessary and sufficient conditions are then proposed in terms of downer branch and subgraphs respectively in particular it is shown for path topologies that controllability completely depends on the leaders location the obtained results may help us to pave the way towards a new method of formation and reconfiguration control in swarming robots

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

first and second order methods for semidefinite programming in this paper we survey the most recent methods that have been developed for the solution of semidefinite programs we first concentrate on the methods that have been primarily motivated by the interior point ip algorithms for linear programming putting special emphasis in the class of primal dual path following algorithms we also survey methods that have been developed for solving large scale sdp problems these include first order nonlinear programming nlp methods and more specialized path following ip methods which use the preconditioned conjugate gradient or residual scheme to compute the newton direction and the notion of matrix completion to exploit data sparsity

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

a full nesterov todd step feasible primal dual interior point algorithm for convex quadratic semi definite optimization in this paper a short step feasible primal dual path following interior point algorithm is proposed for solving a convex quadratic semidefinite optimization cqsdo problem the algorithm uses at each iteration full nesterov todd nt steps to find an approximated solution of cqsdo the favorable iteration bound namely o n log n is obtained for short step method and which is as good as the linear and semidefinite optimization analogue

strengthened existence and uniqueness conditions for search directions in semidefinite programming abstract primal dual interior point p d i p methods for semidefinite programming sdp are generally based on solving a system of matrix equations for a newton type search direction for a symmetrization of the optimality conditions these search directions followed the derivation of similar p d i p methods for linear programming lp among these a computationally interesting search direction is the aho direction however in contrast to the lp case existence and uniqueness of the aho search direction is not guaranteed under the standard nondegeneracy assumptions two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints the first is given by shida shindoh kojima and is based on the semidefiniteness of the symmetrization of the product sx at the current iterate the second is a centrality condition given first by monteiro zanjacomo and then improved by monteiro todd in this paper we revisit and strengthen both of the above mentioned sufficient conditions we include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions as well we present new results on the relationship between the kronecker product and the symmetric kronecker product that arise from these matrix equations we conclude with a proof that the existence and uniqueness of the aho direction is a generic property for every sdp problem and extend the results to the general monteiro zhang family of search directions

polynomial convergence of a new family of primal dual algorithms for semidefinite programming this paper establishes the polynomial convergence of a new class of primal dual interior point path following feasible algorithms for semidefinite programming sdp whose search directions are obtained by applying newton s method to the symmetric central path equation p x p t 1 2 p t s p 1 p x p t 1 2 mu i 0 where p is a nonsingular matrix specifically we show that the short step path following algorithm based on the frobenius norm neighborhood and the semilong step path following algorithm based on the operator 2 norm neighborhood have o sqrt n l and o nl iteration complexity bounds respectively when p i this yields the first polynomially convergent semilong step algorithm based on a pure newton direction restricting the scaling matrix p at each iteration to a certain subset of nonsingular matrices we are able to establish an o n3 2l iteration complexity for the long step path following method the resulting subclass of search directions contains both the nesterov todd direction and the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction

quenched decay of correlations for slowly mixing systems we study random towers that are suitable to analyse the statistics of slowly mixing random systems we obtain upper bounds on the rate of quenched correlation decay in a general setting we apply our results to the random family of liveranisaussol vaienti maps with parameters in 0 1 0 1 chosen independently with respect to a distribution on 0 1 and show that the quenched decay of correlation is governed by the fastest mixing map in the family in particular we prove that for every 0 for almost every 0 1 z the upper bound n 1 1 0 holds on the rate of decay of correlation for holder observables on the fibre over for three different distributions on 0 1 discrete uniform quadratic we also derive sharp asymptotics on the measure of return time intervals for the quenched dynamics ranging from n 1 0 to log n 1 0 n 1 0 to log n 2 0 n 1 0 respectively

differentiability and analyticity of topological entropy for anosov and geodesic flows in this paper we investigate the regularity of the topological entropyh top forc k perturbations of anosov flows we show that the topological entropy varies almost as smoothly as the perturbation the results in this paper along with several related results have been announced in kkpw

nonsymmetric search directions for semidefinite programming two nonsymmetric search directions for semidefinite programming the xz and zx search directions are proposed they are derived from a nonsymmetric formulation of the semidefinite programming problem the xz direction corresponds to the direct linearization of the central path equation xz nu i while the zx direction corresponds to zx nu i the xz and zx directions are well defined if both x and z are positive definite matrices where x may be nonsymmetric we present an algorithm using the xz and zx directions alternately following the mehrotra predictor corrector framework numerical results show that the xz zx algorithm in many cases requires less cpu time than the xz zx method of alizadeh overton and haeberly siam j optim 8 1998 pp 746 768 while achieving similar accuracy

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

polynomiality of primal dual algorithms for semidefinite linear complementarity problems based on the kojima shindoh hara family of directions kojima shindoh and hara proposed a family of search directions for the semidefinite linear complementarity problem sdlcp and established polynomial convergence of a feasible short step path following algorithm based on a particular direction of their family the question of whether polynomiality could be established for any direction of their family thus remained an open problem this paper answers this question in the affirmative by establishing the polynomiality of primal dual interior point algorithms for sdlcp based on any direction of the kojima shindoh and hara family of search directions we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of sdlcp

flocking of multi agent systems with multiple groups in this paper we consider the flocking problem of multi agent systems with multiple groups first some algorithms using local information are designed to divide the agents into any pre assigned number of groups in fixed and switching heterogeneous networks respectively based on algebraic graph theory and barbalat s lemma convergence criteria are established to ensure velocity alignment and cohesion of each subgroup as well as collision avoidance between any agents in the whole group second an algorithm for homogeneous networks is studied simulation examples are finally presented to verify the effectiveness of our theoretical results

exploiting sparsity in primal dual interior point methods for semidefinite programming the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro and nesterov todd search directions have been used in many primal dual interior point methods for semidefinite programs this paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

whitney holder continuity of the srb measure for transversal families of smooth unimodal maps we consider c 2 families t mapsto f t of c 4 nondegenerate unimodal maps we study the absolutely continuous invariant probability srb measure mu t of f t as a function of t on the set of collet eckmann ce parameters upper bounds assuming existence of a transversal ce parameter we find a positive measure set of ce parameters delta and for each t 0 in delta a set delta 0 subset delta of polynomially recurrent parameters containing t 0 as a lebesgue density point and constants c ge 1 gamma 4 so that for every 1 2 holder function a begin aligned big int a d mu t int a d mu t 0 big le c vert a vert c 1 2 t t 0 1 2 log t t 0 gamma forall t in delta 0 end aligned in addition for all t in delta 0 the renormalisation period p t of f t satisfies p t le p t 0 and there are uniform bounds on the rates of mixing of f t p t for all t with p t p t 0 if f t x tx 1 x the set delta contains almost all ce parameters lower bounds assuming existence of a transversal mixing misiurewicz thurston parameter t 0 we find a set of ce parameters delta mt accumulating at t 0 a constant c ge 1 and a c infty function a 0 so that begin aligned c t t 0 1 2 ge big int a 0 d mu t int a 0 d mu t 0 big ge c 1 t t 0 1 2 forall t in delta mt end aligned

decentralized stabilizability of multi agent systems under fixed and switching topologies abstract the paper studies decentralized stabilizability for multi agent systems with general linear dynamics the stabilizability problem is formulated in a way that the protocol performance can be evaluated by means of the stabilizability region and the feedback gain for fixed topology it is proved that the system is stabilizable if and only if external control inputs are exerted on some indicated agents the result is further shown to be a prerequisite for subsequent design of the corresponding decentralized external self feedback control which is also necessary and sufficient based on this two methods are presented to find the agents under which stabilizability can be reached and the region of stabilizability is given to evaluate the protocol performance for switching interaction topology it is shown that the system is stabilizable even if each of its subsystems is not finally the results are employed to cope with the decentralized set point formation control problem for which some necessary and or sufficient conditions are developed numerical simulations are presented to demonstrate the effectiveness of the proposed results

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

central limit theorem and stable laws for intermittent maps in the setting of abstract markov maps we prove results concerning the convergence of renormalized birkhoff sums to normal laws or stable laws they apply to one dimensional maps with a neutral fixed point at 0 of the form x x1 for 0 1 in particular for 1 2 we show that the birkhoff sums of a holder observable f converge to a normal law or a stable law depending on whether f 0 0 or f 0 0 the proof uses spectral techniques introduced by sarig and wiener s lemma in non commutative banach algebras

discretization method for semi definite programming semi definite programs are convex optimization problems arising in a wide variety of applications and the extension of linear programming most methods for linear programming have been generalized to semi definite programs this paper discusses the discretization method in semi definite programming the convergence and the convergent rate of error between the optimal value of the semi definite programming problems and the optimal value of the discretized problems are obtained an approximately optimal division is given under certain assumptions with the significance of the convergence property the duality result in semi definite programs is proved in a simple way which is different from the other common proofs

linear response for random dynamical systems abstract we study for the first time linear response for random compositions of maps chosen independently according to a distribution p we are interested in the following question how does an absolutely continuous stationary measure acsm of a random system change when p changes smoothly to p e for a wide class of one dimensional random maps we prove differentiability of acsm with respect to e moreover we obtain a linear response formula our results cover random maps whose transfer operator does not necessarily admit a spectral gap we apply our results to iid compositions with respect to various distributions p e of uniformly expanding circle maps gauss renyi maps random continued fractions and pomeau manneville maps our results yield an exact formula for the invariant density of random continued fractions while for pomeau manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

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

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

controllability and observability of grid graphs via reduction and symmetries in this paper we investigate the controllability and observability properties of a family of linear dynamical systems whose structure is induced by the laplacian of a grid graph this analysis is motivated by several applications in network control and estimation quantum computation and discretization of partial differential equations specifically we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph for each eigenvalue based on its multiplicity and on suitable symmetries of the corresponding eigenvectors we provide necessary and sufficient conditions to characterize all and only the nodes from which the induced dynamical system is controllable observable we discuss the proposed criteria and show through suitable examples how such criteria reduce the complexity of the controllability respectively observability analysis of the grid

controllability analysis of multi agent systems with directed and weighted interconnection in this article we investigate the controllability of multi agent systems with leaders as control inputs where the interconnection is directed and weighted we employ weight balanced partition to classify the interconnection graphs and study the controllable subspaces with given nontrivial weight balanced partition we also provide two necessary and sufficient graph conditions on structural controllability and strong structural controllability moreover we consider the effect of the zero row sum restrictions of the system matrices on structural controllability

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

solving some large scale semidefinite programs via the conjugate residual method most current implementations of interior point methods for semidefinite programming use a direct method to solve the schur complement equation sce m delta y h in computing the search direction when the number of constraints is large the problem of having insufficient memory to store m can be avoided if an iterative method is used instead numerical experiments have shown that the conjugate residual cr method typically takes a huge number of steps to generate a high accuracy solution on the other hand it is difficult to incorporate traditional preconditioners into the sce except for block diagonal preconditioners we decompose the sce into a 2 2 block system by decomposing delta y similarly for h into two orthogonal components with one lying in a certain subspace that is determined from the structure of m numerical experiments on semidefinite programming problems arising from the lovasz theta function of graphs and maxcut problems show that high accuracy solutions can be obtained with a moderate number of cr steps using the proposed equation

solving large scale semidefinite programs via an iterative solver on the augmented systems the search directions in an interior point method for large scale semidefinite programming sdp can be computed by applying a krylov iterative method to either the schur complement equation sce or the augmented equation both methods suffer from slow convergence as interior point iterates approach optimality numerical experiments have shown that a diagonally preconditioned conjugate residual method on the sce typically takes a huge number of steps to converge however it is difficult to incorporate cheap and effective preconditioners into the sce this paper proposes to apply the preconditioned symmetric quasi minimal residual psqmr method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior point iterates numerical experiments on sdp problems arising from maximum clique and selected sdplib sdp library problems show that moderately accurate solutions can be obtained with a modest number of psqmr steps using the proposed preconditioned reduced augmented equation an sdp problem with 127600 constraints is solved in about 6 5 hours to an accuracy of 10 6 in relative duality gap

linear response for intermittent maps we consider the one parameter family alpha mapsto t alpha alpha in 0 1 of pomeau manneville type interval maps t alpha x x 1 2 alpha x alpha for x in 0 1 2 and t alpha x 2x 1 for x in 1 2 1 with the associated absolutely continuous invariant probability measure mu alpha for alpha in 0 1 sarig and gouezel proved that the system mixes only polynomially with rate n 1 1 alpha in particular there is no spectral gap we show that for any psi in l q the map alpha to int 0 1 psi d mu alpha is differentiable on 0 1 1 q and we give a linear response formula for the value of the derivative this is the first time that a linear response formula for the srb measure is obtained in the setting of slowly mixing dynamics our argument shows how cone techniques can be used in this context for alpha ge 1 2 we need the n 1 alpha decorrelation obtained by gouezel under additional conditions

primal dual interior point methods for semidefinite programming in finite precision recently a number of primal dual interior point methods for semidefinite programming have been developed to reduce the number of floating point operations each iteration of these methods typically performs block gaussian elimination with block pivots that are close to singular near the optimal solution as a result these methods often exhibit complex numerical properties in practice we consider numerical issues related to some of these methods our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well conditioned but are unstable otherwise with this result we explain why one particular method the one introduced by alizadeh haeberly and overton is in general more stable than others we also explain why the so called least squares variation introduced for some of these methods does not yield more numerical accuracy in general finally we present results from our numerical experiments to support our analysis

on the reachability and observability of path and cycle graphs in this technical note we investigate the reachability and observability properties of a network system running a laplacian based average consensus algorithm when the communication graph is a path or a cycle specifically we provide necessary and sufficient conditions based on simple rules from number theory to characterize all and only the nodes from which the network system is reachable respectively observable interesting immediate corollaries of our results are i a path graph is reachable observable from any single node if and only if the number of nodes of the graph is a power of two n 2i i n and ii a cycle is reachable observable from any pair of nodes if and only if n is a prime number for any set of control observation nodes we provide a closed form expression for the unreachable unobservable eigenvalues and for the eigenvectors of the unreachable unobservable subsystem

an interior point method with a primal dual quadratic barrier penalty function for nonlinear semidefinite programming in this paper we consider an interior point method for nonlinear semidefinite programming yamashita yabe and harada presented a primal dual interior point method in which a nondifferentiable merit function was used by using shifted barrier kkt conditions we propose a differentiable primal dual merit function within the framework of the line search strategy and prove its global convergence property under weaker assumptions than the method of yamashita yabe and harada

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

a primal dual variant of the iri imai algorithm for linear programming a local acceleration method for primal dual potential reduction algorithms is introduced the method developed here uses modified newton search directions to minimize the tanabe todd ye tty potential function and can be regarded as a primal dual variant of the iri imai algorithm based on the multiplicative analogue of karmarkar s potential function when iterates are close to an optimal solution the tty potential function has negative curvature along the generated search directions therefore large reductions in the potential function can be obtained guaranteeing polynomial and quadratic convergence to nondegenerate solutions

search directions in the sdp and the monotone sdlcp generalization and inexact computation we propose a family of directions that generalizes many directions proposed so far in feasible interior point methods for the sdp semidefinite programming and for the monotone sdlcp semidefinite linear complementarity problem we derive the family from the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction by relaxing its centrality equation into a centrality inequality using this family we present a predictor corrector infeasible interior point method to provide a theoretical basis for inexact computation of directions in primal dual interior point methods for the sdp

controllability improvement for multi agent systems leader selection and weight adjustment abstractfor an uncontrollable system adding leaders and adjusting edge weights are two methods to improve controllability in this paper the controllability of multi agent systems under directed topologies is studied especially on the leader selection problem and the weight adjustment problem for a multi agent system necessary and sufficient algebraic conditions for controllability with fewest leaders are proposed to improve controllability by adjusting edge weights the system is supposed to be structurally controllable which holds if and only if the communication topology contains a spanning tree it is also proved that the number of the fewest edges needed to be assigned on new weights equals the rank deficiency of controllability matrix in addition a leader selection algorithm and a weight adjustment algorithm are presented simulation examples are provided to illustrate the theoretical results

primal dual interior point algorithm for convex quadratic semi definite optimization abstract in this paper we present a new primal dual interior point algorithm for solving a special case of convex quadratic semi definite optimization based on a parametric kernel function the proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the center for the algorithm these properties enable us to derive the currently best known iteration bounds for the algorithm with large and small update methods namely o n log n log n e and o n log n e respectively which reduce the gap between the practical behavior of the algorithm and its theoretical performance results

a class of projective transformations for linear programming a class of projective transformations under which potential functions are invariant for linear programming is described as a result a new projective algorithm converging in o l sqrt n iterations is developed the algorithm does not require centering conditions and its convergence speed is improved by a factor sqrt n over karmarkar s projective algorithm and by a constant over the recent affine potential reduction algorithm

a feasible primaldual interior point method for linear semidefinite programming in this paper we consider a feasible primaldual interior point method for linear semidefinite programming problem sdp based on alizadehhaeberlyoverton aho direction monteiro 1997 firstly and by a new and simple technique we establish the existence and uniqueness of optimal solution of the perturbed problem sdp and its convergence to optimal solution of sdp next we present new different alternatives to calculate the displacement step after we establish the convergence of the obtained algorithm and we show that its complexity is o nln 1 x0 s0 finally we present some numerical simulations which show the effectiveness of the algorithm developed in this work

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

sampling of stochastic operators we develop sampling methodology aimed at identifying stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function the data that we use to reconstruct the operator or in some cases only the autocorrelation of the spreading function is based on the response of the unknown operator to a known deterministic test signal

protocols design and uncontrollable topologies construction for multi agent networks this note investigates the controllability issues of multi agent systems where each node contains generic linear dynamics first a neighbor based control protocol is proposed under which it is shown that the controllability of a multi agent system is solely decided by its communication topology structure we then further consider the effects of communication topology on the controllability from a graph theory point of view by explicitly constructing topologies the construction exhibits a partition of the designed graph with identified leaders under which the system is shown to be uncontrollable

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 new o nl iteration predictor corrector algorithm with wide neighborhood for semidefinite programming in this paper we extend the ai zhang predictor corrector method to the class of semidefinite programming first we define a new wide neighborhood n t b another key ingredient of our method is that we treat the classical newton direction as the sum of two other directions corresponding to respectively the negative part and the positive part of the right hand side we prove that besides the predictor steps each corrector step also reduces the duality gap by a rate of 1 1o n then the method enjoys the low iteration bound of o nl which is better than that of usual wide neighborhood algorithm o nl where n is the dimension of the problem and l x 0 t s 0 e with e the required precision and x 0 y 0 s 0 the initial interior solution