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

long cyclic codes over gf 4 and gf 8 better than bch codes in the high rate region an explicit construction of an infinite family of cyclic codes is presented which over gf 4 resp gf 8 have approximately 8 9 resp 48 49 the redundancy of bch codes of the same minimum distance and length as such the new codes are the best codes currently known in a regime where the minimum distance is fixed and the code length goes to infinity

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

efficient rank reduction of correlation matrices geometric optimisation algorithms are developed that efficiently find the nearest low rank correlation matrix we show in numerical tests that our methods compare favourably with the existing methods in the literature the connection with the lagrange multiplier method is established along with an identification of whether a local minimum is a global minimum an additional benefit of the geometric approach is that any weighted norm can be applied the problem of finding the nearest low rank correlation matrix occurs as part of the calibration of multi factor interest rate market models to correlation

long nonbinary codes exceeding the gilbert varshamov bound for any fixed distance let a q n d denote the maximum size of a q ary code of length n and distance d we study the minimum asymptotic redundancy as n grows while q and d are fixed for any d and q spl ges d 1 long algebraic codes are designed that improve on the bose chaudhuri hocquenghem bch codes and have the lowest asymptotic redundancy known to date prior to this work codes of fixed distance that asymptotically surpass bch codes and the gilbert varshamov bound were designed only for distances 4 5 and 6

continuous selections of trajectories of hybrid systems we present results on existence of continuous selections of trajectories of hybrid systems evolving according to lipschitz nonlinear inclusions first we utilize the skorohod metric to define both a family of pseudo metrics and a metric on the set of trajectories accepted by the hybrid automaton we show that under a non zeno condition the latter metric is complete second the existence of continuous selections with respect to the family of pseudo metrics is proved under a relatively mild assumption of transversality

nonlinear complementary filters on the special orthogonal group this paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units the outputs of such systems are characterized by high noise levels and time varying additive biases we formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group so 3 driven by reconstructed attitude and angular velocity measurements lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error the approach taken leads to an observer that we term the direct complementary filter by exploiting the geometry of the special orthogonal group a related observer termed the passive complementary filter is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs both the direct and passive filters can be extended to estimate gyro bias online the passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude this leads to an observer on so 3 termed the explicit complementary filter that requires only accelerometer and gyro outputs is suitable for implementation on embedded hardware and provides good attitude estimates as well as estimating the gyro biases online the performance of the observers are demonstrated with a set of experiments performed on a robotic test bed and a radio controlled unmanned aerial vehicle

output feedback control for attitude tracking in this paper the problem of attitude tracking control for a rigid spacecraft is addressed it is assumed that only attitude measurements are available and thus spacecraft s angular velocity has to be properly estimated two alternative schemes are proposed in which the unit quaternion is adopted to represent the orientation in the first scheme a second order model based observer is adopted to estimate the angular velocity used in the control law in the second scheme an estimate of the angular velocity error is obtained through a lead filter sufficient conditions ensuring local exponential stability of the two controllers are derived via lyapunov analysis

a framework of constraint preserving update schemes for optimization on stiefel manifold this paper considers optimization problems on the stiefel manifold x mathsf t x i p xtx ip where x in mathbb r n times p x rn p is the variable and i p ip is the p p by p p identity matrix a framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of x x and the null space of x mathsf t xt while this general framework can unify many existing schemes a new update scheme with low complexity cost is also discovered then we study a feasible barzilai borwein like method under the new update scheme the global convergence of the method is established with an adaptive nonmonotone line search the numerical tests on the nearest low rank correlation matrix problem the kohn sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method in particular the new method performs remarkably well for the nearest low rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used scf iteration for the kohn sham total energy minimization

a common framework for complete and incomplete attitude synchronization in networks with switching topology in this paper we study attitude synchronization for elements in the unit sphere in mathbb r scriptscriptstyle scriptscriptstyle 3 and for elements in the three dimensional 3 d rotation group for a network with switching topology the agents angular velocities are assumed to be the control inputs and a switching control law for each agent is devised that guarantees synchronization provided that all elements are initially contained in a region which we identify later in the paper the control law is decentralized and it does not require a common orientation frame among all agents we refer to synchronization of unit vectors in mathbb r scriptscriptstyle 3 as incomplete synchronization and of 3 d rotation matrices as complete synchronization our main contribution lies in showing that these two problems can be analyzed under a common framework where all agents dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

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

on path lifting mechanisms and unwinding in quaternion based attitude control the unit quaternion is a pervasive representation of rigid body attitude used for the design and analysis of feedback control laws because the space of unit quaternions constitutes a double cover of the rigid body attitude space quaternion based control laws are often by design inconsistent i e they do not have a unique value for each rigid body attitude inconsistent quaternion based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation however conversion mechanisms that are memoryless e g selecting the quaternion having positive scalar component have a limited domain where they remain injective and when used globally introduce discontinuities into the closed loop system we show through an explicit construction and lyapunov analysis that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude to remedy this limitation we propose a hybrid dynamic algorithm for smoothly lifting an attitude path to the unit quaternion space we show that this hybrid dynamic mechanism allows us to directly translate quaternion based controllers and their asymptotic stability properties obtained in the unit quaternion space to the actual rigid body attitude space we also show that when quaternion based controllers are not designed to account for the double covering of the rigid body attitude space by a unit quaternion parameterization they can give rise to the unwinding phenomenon which we characterize in terms of the projection of asymptotically stable sets finally we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

computing the nearest low rank correlation matrix by a simplified sqp algorithm in this paper we propose a numerical method for computing the nearest low rank correlation matrix lrcm motivated by the fact that the nearest lrcm problem can be reformulated as a standard nonlinear equality constrained optimization problem with matrix variables via the gramian representation we propose a new algorithm based on the sequential quadratic programming sqp method on each iteration we do not solve the quadratic program qp corresponding to the exact hessian but a modified qp with a simpler hessian this qp subproblem can be solved efficiently by equivalently transforming it to a sparse linear system global convergence is established and preliminary numerical results are presented to demonstrate the proposed method is potentially useful

an inexact primal dual path following algorithm for convex quadratic sdp we propose primal dual path following mehrotra type predictor corrector methods for solving convex quadratic semidefinite programming qsdp problems of the form min x frac 1 2 x bullet mathcal q x c bullet x mathcal a x b x succeq 0 where mathcal q is a self adjoint positive semidefinite linear operator on mathcal s n b r m and mathcal a is a linear map from mathcal s n to r m at each interior point iteration the search direction is computed from a dense symmetric indefinite linear system called the augmented equation of dimension m n n 1 2 such linear systems are typically very large and can only be solved by iterative methods we propose three classes of preconditioners for the augmented equation and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions numerical experiments on a variety of qsdps with n up to 1600 are performed and the computational results show that our methods are efficient and robust

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

brief paper spacecraft relative rotation tracking without angular velocity measurements we present a solution to the problem of tracking relative rotation in a leader follower spacecraft formation using feedback from relative attitude only the controller incorporates an approximate differentiation filter to account for the unmeasured angular velocity we show uniform practical asymptotic stability upas of the closed loop system for simplicity we assume that the leader is controlled and that we know orbital perturbations however this assumption can be easily relaxed to boundedness without degrading the stability property we also assume that angular velocities of spacecraft relative to an inertial frame are bounded simulation results of a leader follower spacecraft formation using the proposed controller structure are also presented

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

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

optimal low rank approximation to a correlation matrix abstract low rank approximation of a correlation matrix is a constrained minimization problem that can be translated into a minimization maximization problem by the method of lagrange multiplier in this paper we solve the inner maximization problems with a single spectral decomposition and the outer minimization problems with gradient based descending methods an in depth analysis is done to characterize the solutions of the inner maximization problem for the case when they are non unique the well posedness of the lagrange multiplier problem and the convergence of the descending methods are rigorously justified numerical results are presented

a majorization algorithm for constrained correlation matrix approximation abstract we desire to find a correlation matrix r of a given rank that is as close as possible to an input matrix r subject to the constraint that specified elements in r must be zero our optimality criterion is the weighted frobenius norm of the approximation error and we use a constrained majorization algorithm to solve the problem although many correlation matrix approximation approaches have been proposed this specific problem with the rank specification and the r ij 0 constraints has not been studied until now we discuss solution feasibility convergence and computational effort we also present several examples

an integral invariance principle for differential inclusions with applications in adaptive control the byrnes martin integral invariance principle for ordinary differential equations is extended to differential inclusions on bbb r n the extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes

robust attitude stabilization of spacecraft using nonlinear quaternion feedback this paper considers the problem of three axis attitude stabilization of a rigid spacecraft a nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability the control law does not require the knowledge of the system parameters and is therefore robust to modeling errors the significance of the control law is that it can be used for large angle maneuvers with guaranteed stability

on the generalized low rank approximation of the correlation matrices arising in the asset portfolio abstract in this paper we consider the generalized low rank approximation of the correlation matrices problem which arises in the asset portfolio we first characterize the feasible set by using the gramian representation together with a special trigonometric function transform and then transform the generalized low rank approximation of the correlation matrices problem into an unconstrained optimization problem finally we use the conjugate gradient algorithm with the strong wolfe line search to solve the unconstrained optimization problem numerical examples show that our new method is feasible and effective

attitude control without angular velocity measurement a passivity approach it is well known that the linear feedback of the quaternion of the attitude error and the angular velocity globally stabilizes the attitude of a rigid body in this note we show that the angular velocity feedback can be replaced by a nonlinear filter of the quaternion thus removing the need for direct angular velocity measurement in contrast to other approaches this design exploits the inherent passivity of the system a model based observer reconstructing the velocity is not needed an application of the proposed scheme is illustrated for the robot control problem simulation results are included to illustrate the theoretical results

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

quaternion based hybrid feedback for robust global attitude synchronization we apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy the proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error through a hysteretic switch of this logic variable the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic the hysteresis eliminates chattering while preventing the unwinding phenomenon apparent in some quaternion based attitude control schemes the results are exercised in a numerical example

unit quaternion based output feedback for the attitude tracking problem in this note we propose a quaternion based dynamic output feedback for the attitude tracking problem of a rigid body without velocity measurement our approach consists of introducing an auxiliary dynamical system whose output which is also a unit quaternion is used in the control law together with the unit quaternion representing the attitude tracking error roughly speaking the necessary damping that would have been achieved by the direct use of the angular velocity can be achieved in our approach by the vector part mathtilde q of the error signal between the output of the auxiliary system and the unit quaternion tracking error the resulting velocity free control scheme guarantees almost global asymptotic stability which is as strong as the topology of the motion space can permit in the regulation case our control law is a pure quaternion feedback i e consisting of two terms that are vector parts of unit quaternion and hence the control torques are naturally bounded by the control gains simulation results are provided to show the effectiveness of the proposed control scheme

passivity based adaptive attitude control of a rigid spacecraft an adaptive control scheme for the attitude control of a rigid spacecraft is derived using a linear parameterization of the equation of motion the tracking error is described with the euler parameter vector global convergence of the tracking error to zero is shown using passivity theory this allows for the use of time varying positive definite feedback gain matrices and the results can easily be extended to other passive parameter update laws

sparse iteration 4d cordic algorithms for multiplying quaternions novel 4d cordic algorithms and hardware architecture for multiplying quaternions are presented aimed at constant coefficient multipliers in our solution microrotations are multiplications by hypercomplex numbers with only one non zero imaginary part such transformations can be described using sparse matrices and implemented using less hardware resources than the iteration of the known quaternion cordic algorithm chip area can be saved because sparse microrotations can be computed using a permutation network and two operand adders without four operand additions which are necessary in the known solution the obtainable area savings depend on the implementation technology and the architectures of adders and bit shifters but can be as large as 45 and 25 percent for asic and fpga circuits respectively the circuitry simplifications come at the cost of slowing down computations our approach improves the area delay product and a single sparse iteration can be executed up to 15 percent faster than the conventional one but more microrotations are usually necessary to achieve the required accuracy on the other hand a sparse 4d iteration can be identified with microrotations of two 2d cordics which work in parallel this makes convergence analysis easier than in the previous 4d solution as it is sufficient to consider only two dimensions

an invariance principle for nonlinear hybrid and impulsive dynamical systems in this paper we develop an invariance principle for dynamical systems possessing left continuous flows specifically we show that left continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems as a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems namely state dependent impulsive dynamical systems these results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

smooth lyapunov functions for hybrid systems part i existence is equivalent to robustness hybrid systems are dynamical systems where the state is allowed to either evolve continuously flow on certain subsets of the state space or evolve discontinuously jump from other subsets of the state space for a broad class of such systems characterized by mild regularity conditions on the data we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth lyapunov function this result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right hand sides furthermore we give a description of forward completeness of a hybrid system in terms of a smooth lyapunov like function

constrained best euclidean distance embedding on a sphere a matrix optimization approach the problem of data representation on a sphere of unknown radius arises from various disciplines such as statistics spatial data representation psychology constrained multidimensional scaling and computer science machine learning and pattern recognition the best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some euclidean distance constraints it is those spherical and euclidean distance constraints that present an enormous challenge to the existing algorithms in this paper we reformulate the problem as an euclidean distance matrix optimization problem with a low rank constraint we then propose an iterative algorithm that uses a quadratically convergent newton cg method at each step we study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized jacobian that ensure the quadratic convergence of the newton method we use some classic examples from the spherical multidimensional scaling to demonstrat

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

nonbinary double error correcting codes designed by means of algebraic varieties linear q ary codes of growing length n spl rarr spl infin and designed distance spl delta are studied at first we examine cyclic codes defined by the sets of code zeros g sup i i q sup s 1 q sup s 1 1 spl middot spl middot spl middot q sup s spl delta 2 1 over a primitive element g of gf q sup m then special cubic varieties are designed and employed in order to attain distances spl delta 5 6 the resulting double error correcting codes of length n q sup m have r spl les 2m m 3 1 parity check symbols and reduce the best known redundancy by 2m 3 symbols a decoding procedure of complexity o rn operations is also considered

an augmented lagrangian approach for sparse principal component analysis principal component analysis pca is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering however the standard pca suffers from the fact that the principal components pcs are usually linear combinations of all the original variables and it is thus often difficult to interpret the pcs to alleviate this drawback various sparse pca approaches were proposed in the literature cadima and jolliffe in j appl stat 22 203 214 1995 d aspremont et al in j mach learn res 9 1269 1294 2008 d aspremont et al siam rev 49 434 448 2007 jolliffe in j appl stat 22 29 35 1995 journee et al in j mach learn res 11 517 553 2010 jolliffe et al in j comput graph stat 12 531 547 2003 moghaddam et al in advances in neural information processing systems 18 915 922 mit press cambridge 2006 shen and huang in j multivar anal 99 6 1015 1034 2008 zou et al in j comput graph stat 15 2 265 286 2006 despite success in achieving sparsity some important properties enjoyed by the standard pca are lost in these methods such as uncorrelation of pcs and orthogonality of loading vectors also the total explained variance that they attempt to maximize can be too optimistic in this paper we propose a new formulation for sparse pca aiming at finding sparse and nearly uncorrelated pcs with orthogonal loading vectors while explaining as much of the total variance as possible we also develop a novel augmented lagrangian method for solving a class of nonsmooth constrained optimization problems which is well suited for our formulation of sparse pca we show that it converges to a feasible point and moreover under some regularity assumptions it converges to a stationary point additionally we propose two nonmonotone gradient methods for solving the augmented lagrangian subproblems and establish their global and local convergence finally we compare our sparse pca approach with several existing methods on synthetic zou et al in j comput graph stat 15 2 265 286 2006 pitprops jeffers in appl stat 16 225 236 1967 and gene expression data chin et al in cancer cell 10 529c 541c 2006 respectively the computational results demonstrate that the sparse pcs produced by our approach substantially outperform those by other methods in terms of total explained variance correlation of pcs and orthogonality of loading vectors moreover the experiments on random data show that our method is capable of solving large scale problems within a reasonable amount of time

on the topological structure of attraction basins for differential inclusions abstract we show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties there exists a smooth differential equation rendering the same compact set globally asymptotically stable the regularity properties assumed in this work stem from the consideration of krasovskii filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation in particular the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions it cannot be robustly globally asymptotically stabilized by discontinuous feedback either the results follow from converse lyapunov theory and parallel what is known for the local stabilization problem

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

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

hybrid feedback control and robust stabilization of nonlinear systems in this paper we show for any nonlinear system that is asymptotically controllable to a compact set that a logic based hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise actuator error and external disturbance the construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system the robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties auxiliary results give uniformity of convergence and of overshoots for hybrid systems and give a characterization of asymptotic stability of compact sets

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

a coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise we propose a new algorithm for estimating constant biases in gyro measurements of angular velocity and demonstrate that the resulting estimates converge to the true bias values exponentially fast the new observer is then combined with a nonlinear attitude tracking control strategy in a certainty equivalence fashion and the combination shown via lyapunov analysis to produce globally stable closed loop dynamics with asymptotically perfect tracking of any commanded attitude sequence the analysis is then extended to consider the effects of stochastic measurement noise in the gyro in addition to the bias a simulation is given for a rigid spacecraft tracking a specified time varying attitude sequence to illustrate the theoretical claims

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

explicit maximally recoverable codes with locality consider a systematic linear code where some local parity symbols depend on few prescribed symbols whereas other heavy parity symbols may depend on all data symbols such codes have been studied recently in the context of erasure coding for data storage where the local parities facilitate fast recovery of any single symbol when it is erased whereas the heavy parities provide tolerance to a large number of simultaneous erasures a code as above is maximally recoverable if it corrects all erasure patterns which are information theoretically correctable given the prescribed dependence relations between data symbols and parity symbols in this paper we present explicit families of maximally recoverable codes with locality we also initiate the general study of the tradeoff between maximal recoverability and alphabet size

a sequential semismooth newton method for the nearest low rank correlation matrix problem based on the well known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem sdp we formulate the nearest low rank correlation matrix problem as a nonconvex sdp and propose a numerical method that solves a sequence of least square problems each of the least square problems can be solved by a specifically designed semismooth newton method which is shown to be quadratically convergent the sequential method is guaranteed to produce a stationary point of the nonconvex sdp our numerical results demonstrate the high efficiency of the proposed method on large scale problems

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