comments on stochastic stability of jump linear systems for original paper see ibid vol 47 p 1204 8 2002 the purpose of the article is to show that the conjecture presented in remark 3 of the above paper by y fang and k loparo 2002 is actually true

discrete time lq optimal control problems for infinite markov jump parameter systems optimal control problems for discrete time linear systems subject to markovian jumps in the parameters are considered for the case in which the markov chain takes values in a countably infinite set two situations are considered the noiseless case and the case in which an additive noise is appended to the model the solution for these problems relies in part on the study of a countably infinite set of coupled algebraic riccati equations icare conditions for existence and uniqueness of a positive semidefinite solution to the icare are obtained via the extended concepts of stochastic stabilizability ss and stochastic detectability sd which turn out to be equivalent to the spectral radius of certain infinite dimensional linear operators in a banach space being less than one for the long run average cost ss and sd guarantee existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy furthermore an extension of a lyapunov equation result is derived for the countably infinite markov state space case

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

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

stability analysis for stochastic differential equations with infinite markovian switchings abstract this paper is concerned with stability analysis of linear ito stochastic differential equations with countably infinite markovian switchings a spectral criterion is proposed for exponential stability of the considered models by means of the established spectral criterion the relationship between exponential stability and stochastic l 2 stability is clarified moreover under the disturbance of random signals with finite energy a sufficient condition is presented for l 2 input output stability of the perturbed stochastic differential equations

a maximum likelihood approach to density estimation with semidefinite programming density estimation plays an important and fundamental role in pattern recognition machine learning and statistics in this article we develop a parametric approach to univariate or low dimensional density estimation based on semidefinite programming sdp our density model is expressed as the product of a nonnegative polynomial and a base density such as normal distribution exponential distribution and uniform distribution when the base density is specified the maximum likelihood estimation of the polynomial is formulated as a variant of sdp that is solved in polynomial timewith the interior point methods since the base density typically contains just one or two parameters computation of the maximum likelihood estimate reduces to a one or two dimensional easy optimization problem with this use of sdp thus the rigorous maximum likelihood estimate can be computed in our approach furthermore such conditions as symmetry and unimodality of the density function can be easily handled within this framework aic is used to choose the best model through applications to several instances we demonstrate flexibility of the model and performance of the proposed procedure combination with amixture approach is also presented the proposed approach has possible other applications beyond density estimation this point is clarified through an application to the maximum likelihood estimation of the intensity function of a nonstationary poisson process

stochastic stability of jump linear systems in this note some testable conditions for mean square i e second moment stability for discrete time jump linear systems with time homogenous and time inhomogenous finite state markov chain form processes are presented

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

subgame perfection in stochastic games with perfect information and recursive payoffs we consider a class of n player stochastic games with the following properties 1 in every state the transitions are controlled by one player 2 the payoffs are equal to zero in every non absorbing state 3 the payoffs are non negative in every absorbing state with respect to the expected average reward we provide a constructive proof that a subgame perfect equilibrium exists in pure strategies for every 0 moreover if all transitions of a game in our class are deterministic then the game has a subgame perfect 0 equilibrium in pure strategies

bandgap optimization of two dimensional photonic crystals using semidefinite programming and subspace methods in this paper we consider the optimal design of photonic crystal structures for two dimensional square lattices the mathematical formulation of the bandgap optimization problem leads to an infinite dimensional hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector to make the problem tractable the original eigenvalue problem is discretized using the finite element method into a series of finite dimensional eigenvalue problems for multiple values of the wave vector parameter the resulting optimization problem is large scale and non convex with low regularity and non differentiable objective by restricting to appropriate eigenspaces we reduce the large scale non convex optimization problem via reparametrization to a sequence of small scale convex semidefinite programs sdps for which modern sdp solvers can be efficiently applied numerical results are presented for both transverse magnetic tm and transverse electric te polarizations at several frequency bands the optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design

scalable large margin mahalanobis distance metric learning for many machine learning algorithms such as k nearest neighbor k nn classifiers and k means clustering often their success heavily depends on the metric used to calculate distances between different data points an effective solution for defining such a metric is to learn it from a set of labeled training samples in this work we propose a fast and scalable algorithm to learn a mahalanobis distance metric the mahalanobis metric can be viewed as the euclidean distance metric on the input data that have been linearly transformed by employing the principle of margin maximization to achieve better generalization performances this algorithm formulates the metric learning as a convex optimization problem and a positive semidefinite p s d matrix is the unknown variable based on an important theorem that a p s d trace one matrix can always be represented as a convex combination of multiple rank one matrices our algorithm accommodates any differentiable loss function and solves the resulting optimization problem using a specialized gradient descent procedure during the course of optimization the proposed algorithm maintains the positive semidefiniteness of the matrix variable that is essential for a mahalanobis metric compared with conventional methods like standard interior point algorithms or the special solver used in large margin nearest neighbor our algorithm is much more efficient and has a better performance in scalability experiments on benchmark data sets suggest that compared with state of the art metric learning algorithms our algorithm can achieve a comparable classification accuracy with reduced computational complexity

subgame perfection in quitting games with perfect information and differential equations we introduce a new approach to studying subgame perfect equilibrium payoffs in stochastic games the differential equations approach we apply our approach to quitting games with perfect information those are sequential games in which at every stage one ofn players is chosen each player is chosen with probability 1 n the chosen playeri decides whether to quit in which case the game terminates and the terminal payoff is some vectora i rn or whether to continue in which case the game continues to the next stage if no player ever quits the payoff is some vectora rn we define a certain differential inclusion prove that it has at least one solution and prove that every vector on a solution of this differential inclusion is a subgame perfect equilibrium payoff

mean square stability for discrete bounded linear systems in hilbert space the asymptotic behaviour for infinite dimensional discrete linear systems driven by white noise is considered in this paper both the evolution and convergence of the state correlation operators sequence are analysed mean square stability conditions are investigated including a comparison with the deterministic stability problem the particular case of compact operators is considered in some detail

compact convex projections we study the usefulness of conditional gradient like methods for determining projections onto convex sets in particular projections onto naturally arising convex sets in reproducing kernel hilbert spaces our work is motivated by the recently introduced kernel herding algorithm which is closely related to the conditional gradient method cgm it is known that the herding algorithm converges with a rate of 1 t where t counts the number of iterations when a point in the interior of a convex set is approximated we generalize this result and we provide a necessary and sufficient condition for the algorithm to approximate projections with a rate of 1 t the cgm which is in general vastly superior to the herding algorithm achieves only an inferior rate of 1 t in this setting we study the usefulness of such projection algorithms further by exploring ways to use these for solving concrete machine learning problems in particular we derive non parametric regression algorithms which use at their core a slightly modified kernel herding algorithm to determine projections we derive bounds to control approximation errors of these methods and we demonstrate via experiments that the developed regressors are en par with state of the art regression algorithms for large scale problems

subgame perfection in free transition games abstract we prove the existence of a subgame perfect e equilibrium for every e 0 in a class of multi player games with perfect information which we call free transition games the novelty is that a non trivial class of perfect information games is solved for subgame perfection with multiple non terminating actions in which the payoff structure is generally not upper or lower semi continuous due to the lack of semi continuity there is no general rule of comparison between the payoffs that a player can obtain by deviating a large but finite number of times or respectively infinitely many times we introduce new techniques to overcome this difficulty our construction relies on an iterative scheme which is independent of e and terminates in polynomial time with the following output for all possible histories h a pure action a h 1 or in some cases two pure actions a h 2 and b h 2 for the active player at h the subgame perfect e equilibrium then prescribes for every history h that the active player plays a h 1 with probability 1 or respectively plays a h 2 with probability 1 e and b h 2 with probability e here e is arbitrary as long as it is positive and small compared to e so the strategies can be made almost pure

connections between mutually unbiased bases and quantum random access codes we present a new quantum communication complexity protocol the promise quantum random access code which allows us to introduce a new measure of unbiasedness for bases of hilbert spaces the proposed measure possesses a clear operational meaning and can be used to investigate whether a specific number of mutually unbiased bases exist in a given dimension by employing semidefinite programming techniques

a new general sufficient condition for almost sure stability of jump linear systems in this paper we study the almost sure stability of discrete time jump linear systems with a finite state markov form process a general condition for almost sure stability which is a necessary and sufficient condition for scalar one dimensional systems is derived many simpler testable sufficient conditions for almost sure stability are derived from this sufficient condition

product form cholesky factorization in interior point methods for second order cone programming second order cone programming socp problems are typically solved by interior point methods as in linear programming lp interior point methods can in theory solve socps in polynomial time and can in practice exploit sparsity in the problem data specifically when cones of large dimension are present the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an lp here we propose a product form cholesky factorization pfcf approach and show that it is more numerically stable than the alternative sherman morrison woodbury approach we derive several pfcf variants and compare their theoretical perfomance finally we prove that the elements of l in the cholesky factorizations ldlt that arise in interior point methods for socp are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path

recursive repeated games with absorbing states we show the existence of stationary limiting average e equilibria e 0 for two person recursive repeated games with absorbing states these are stochastic games where all states but one are absorbing and in the nonabsorbing state all payoffs are equal to zero a state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions for the purpose of our proof we introduce properness for stationary strategy pairs our result is sharp since it extends neither to the case with more nonabsorbing states nor to the n person case with n 2 moreover it is well known that the result cannot be strengthened to the existence of 0 equilibria and that repeated games with absorbing states generally do not admit stationary e equilibria

higher order moment stability region for markov jump systems based on cumulant generating function this paper is concerned with the solution of higher order moment stability region for markov jump systems with respect to cases of both known and unknown transition probabilities by exploring the cumulant generating function the original stochastic system with markov jumping modes is transformed to a deterministic system then based on the estimation of matrix eigenvalues the explicit solution for higher order moment stability region is expressed in terms of transition probabilities operation modes state transition matrix and its dimension and the moment order several examples are presented to illustrate the effectiveness of the proposed method and its associated algorithm

perfect information games with lower semicontinuous payoffs we prove that every multiplayer perfect information game with bounded and lower semicontinuous payoffs admits a subgame perfect e equilibrium in pure strategies this result complements example 3 in solan and vieille solan e n vieille 2003 deterministic multi player dynkin games j math econom 39 911 929 which shows that a subgame perfect e equilibrium in pure strategies need not exist when the payoffs are not lower semicontinuous in addition if the range of payoffs is finite we characterize in the form of a folk theorem the set of all plays and payoffs that are induced by subgame perfect 0 equilibria in pure strategies

compensatability and optimal compensation of systems with white parameters the optimal compensation problem is considered in the case of linear discrete time systems with stationary white parameters and quadratic criteria a generalization of the notion of mean square stabilizability namely mean square compensatability is introduced it is shown that suitable mean square compensatability and detectability conditions are sufficient and necessary in general for the existence of a unique optimal mean square stabilizing compensator tests are given to determine whether or not a system is mean square compensatable it is indicated how to calculate numerically the tests and the optimal mean square stabilizing compensator the results are illustrated with examples

almost sure and moments stability of jump linear systems abstract the stability of piecewise deterministic linear systems driven by an underlying finite markov chain is analyzed necessary and sufficient conditions for moment stability are obtained by means of an explicit formula for the corresponding liapunov exponent the relationship between almost sure and moment stability is elucidated revealing the possible occurrence of high order moment instability and large deviations from a stable sample path behavior

robust stability and stabilization of uncertain discrete time markovian jump linear systems this note deals with robust stability and control of uncertain discrete time linear systems with markovian jumping parameters systems with polytopic type parameter uncertainty in either the state space model matrices or in the transition probability matrix of the markov process are considered this note develops methods of robust stability analysis and robust stabilization in the mean square sense which are dependent on the system uncertainty the design of both mode dependent and mode independent control laws is addressed the proposed methods are given in terms of linear matrix inequalities numerical examples are provided to demonstrate the effectiveness of the derived results

stability analysis of discrete time semi markov jump linear systems with partly unknown semi markov kernel abstract this paper is concerned with the problems of stability analysis and stabilization synthesis for a class of discrete time semi markov jump linear systems the novel sufficient stability conditions for such systems are obtained in which the sojourn times are allowed to be unbounded and thus these conditions are applicable to discrete time markov jump linear systems moreover by truncating the sojourn times of switching process to be bounded the numerically testable existence conditions of stabilizing controller for the resulting closed loop systems with partly unknown semi markov kernel are obtained an example is given to illustrate the effectiveness of our results

zames falb multipliers for convergence rate motivating example and convex searches the bound of convergence rates for discrete time lurye sometimes written lur e systems has recently attracted much attention the contributions of this technical note are twofold firstly we sho

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

three player absorbing games an n player absorbing game is an n player stochastic game where all the states but one are absorbing a state is absorbing if once it is reached the probability to leave it is zero whatever the players play we prove that every three player absorbing game has an undiscounted equilibrium payoff

linearization of expectation based inequality conditions in control for discrete time linear systems represented with random polytopes abstract a technique of linearizing what we call expectation based inequality conditions is proposed for control of discrete time linear systems with stochastic dynamics in particular the coefficient random matrices of the systems are assumed to be represented with random polytopes and the linearization technique is discussed so as to appropriately deal with associated uncertainties our expectation based inequality is an inequality that involves decision variables contained in the expectation operation and has a unique difficulty in direct linearization in general hence two key lemmas are provided so that the decision variables can be taken out from the expectation operation the combinational use of such lemmas and the conventional linear matrix inequality lmi techniques which is nothing but our linearization technique is expected to be useful for transforming various kinds of expectation based nonlinear inequality into numerically solvable standard lmis as a demonstration new robust stability conditions are derived with the technique whose effectiveness is confirmed numerically

quadratic control with partial information for discrete time jump systems with the markov chain in a general borel space this paper deals with the finite horizon quadratic optimal control problem of discrete time markov jump linear systems mjls considering the case in which the markov chain takes values in a general borel space s it is assumed that the controller has access to an output variable as well as the jump parameter the goal is to design a dynamic markov jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time it is shown that an optimal controller can be obtained from two s coupled difference riccati equations one associated to a filtering problem and the other one associated to a control problem in which the state variable is fully available by s coupled it is meant that the difference riccati equations are coupled via an integral over s this result which can be seen as a separation principle for discrete time mjls generalizes previous ones that were restricted to the markov chain taking values in a finite set

controllability observability and discrete time markovian jump linear quadratic control this paper is concerned with the controllability and observability of discrete time linear systems that possess randomly jumping parameters described by finite stale markov processes and the relationship between these properties and the solution of the infinite time jump linear quadratic jlq optimal control problem the solution of the markovian jlq problem with finite or infinite time horizons is known necessary and sufficient conditions for the existence of optimal constant control laws that lead to finite optimal expected costs as the time horizon becomes infinite are also known sufficient conditions for these steady state control laws to stabilize the controlled system are also available chizeck et al 1986 these conditions are not easy to test however various definitions of controllability and observability for stochastic systems exist in the literature these definitions are unfortunately not related to the steady state jlq control problem in a manner that is analogous to the role of deter

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

benjamin ono equation with unbounded data abstract the initial value problem ivp for the benjamin ono equation with unbounded initial data is considered we show existence and uniqueness of global solutions for sublinear growth data the method of proof relies in an appropriate splitting the study of an ivp associated with a benjamin ono type equation with variable coefficients and the persistence of decay for solutions of the burgers equation

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

socp relaxation bounds for the optimal subset selection problem applied to robust linear regression abstract this paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion with an special emphasis on applications to computing the least trimmed squares estimator ltse for robust regression the computation of the ltse is a challenging subset selection problem involving a nonlinear program with continuous and binary variables linked in a highly nonlinear fashion the selection of a globally optimal subset using the branch and bound bb algorithm is limited to problems in very low dimension typically d 5 as the complexity of the problem increases exponentially with d we introduce a bold pruning strategy in the bb algorithm that results in a significant reduction in computing time at the price of a negligeable accuracy lost the novelty of our algorithm is that the bounds at nodes of the bb tree come from pseudo convexifications derived using a linearization technique with approximate bounds for the nonlinear terms the approximate bounds are computed solving an auxiliary semidefinite optimization problem we show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the ltse problem

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

rigorous error bounds for the optimal value in semidefinite programming a wide variety of problems in global optimization combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming sometimes due to the use of floating point arithmetic in combination with ill conditioning and degeneracy erroneous results may be produced the purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver it turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver numerical results are presented including problems from the sdplib and the netlib lp library these libraries contain many ill conditioned and real life problems

primal dual first order methods with mathcal o 1 epsilon iteration complexity for cone programming in this paper we consider the general cone programming problem and propose primal dual convex smooth and or nonsmooth minimization reformulations for it we then discuss first order methods suitable for solving these reformulations namely nesterov s optimal method nesterov in doklady an sssr 269 543 547 1983 math program 103 127 152 2005 nesterov s smooth approximation scheme nesterov in math program 103 127 152 2005 and nemirovski s prox method nemirovski in siam j opt 15 229 251 2005 and propose a variant of nesterov s optimal method which has outperformed the latter one in our computational experiments we also derive iteration complexity bounds for these first order methods applied to the proposed primal dual reformulations of the cone programming problem the performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances we also compare the approach based on the variant of nesterov s optimal method with the low rank method proposed by burer and monteiro math program ser b 95 329 357 2003 math program 103 427 444 2005 for solving a set of randomly generated sdp instances

sparse semi parametric estimation of harmonic chirp signals in this paper we present a method for estimating the parameters detailing an unknown number of linear possibly harmonically related chirp signals using an iterative sparse reconstruction framework the proposed method is initiated by a re weighted group sparsity approach followed by an iterative relaxation based refining step to allow for high resolution estimates numerical simulations illustrate the achievable performance offering a notable improvement as compared to other recent approaches the resulting estimates are found to be statistically efficient achieving the corresponding cramer rao lower bound

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

equivalent stability notions lyapunov inequality and its application in discrete time linear systems with stochastic dynamics determined by an i i d process this paper is concerned with stability analysis and synthesis for discrete time linear systems with stochastic dynamics equivalence is first proved for three stability notions under some key assumptions on the randomness behind the systems in particular we use the assumption that the stochastic process determining the system dynamics is independent and identically distributed with respect to the discrete time then a lyapunov inequality condition is derived for stability in a necessary and sufficient sense although our lyapunov inequality will involve decision variables contained in the expectation operation an idea is provided to solve it as a standard linear matrix inequality the idea also plays an important role in state feedback synthesis based on the lyapunov inequality motivating numerical examples are further discussed as an application of our approach

on the existence of easy initial states for undiscounted stochastic games this paper deals with undiscounted infinite stage two person zero sum stochastic games with finite state and action spaces it was recently shown that such games possess a value but in general there are no optimal strategies we prove that for each player there exists a nonempty set of easy initial states i e starting states for which the player possesses an optimal stationary strategy this result is proved with the aid of facts derived by bewley and kohlberg for the limit discount equation for stochastic games

borel stochastic games with lim sup payoff we consider two person zero sum stochastic games with limit superior payoff function and borel measurable state and action spaces the games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic the paper extends results of our earlier article 17 in which the same class of games was considered for countable state spaces and finite action sets

some new search directions for primal dual interior point methods in semidefinite programming search directions for primal dual path following methods for semidefinite programming sdp are proposed these directions have the properties that 1 under certain nondegeneracy and strict complementarity assumptions the jacobian matrix of the associated symmetrized newton equation has a bounded condition number along the central path in the limit as the barrier parameter mu tends to zero and 2 the schur complement matrix of the symmetrized newton equation is symmetric and the cost for computing this matrix is 2mn3 0 5m2n2 flops where n and m are the dimension of the matrix and vector variables of the semidefinite program respectively these two properties imply that a path following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by alizadeh haeberly and overton currently the best search direction in terms of accuracy but each iteration requires at most half the amount of flops to leading order

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

fast semi supervised clustering with enhanced spectral embedding in recent years semi supervised clustering ssc has aroused considerable interests from the machine learning and data mining communities in this paper we propose a novel ssc approach with enhanced spectral embedding ese which not only considers the geometric structure information contained in data sets but also can make use of the given side information such as pairwise constraints specially we first construct a symmetry favored k nn graph which is highly robust to noise and outliers and can reflect the underlying manifold structures of data sets then we learn the enhanced spectral embedding towards an ideal data representation as consistent with the given pairwise constraints as possible finally by using the regularization of spectral embedding we formulate learning the new data representation as a semidefinite quadratic linear programming sqlp problem which can be efficiently solved experimental results on a variety of synthetic and real world data sets show that our ese approach outperforms the state of the art ssc algorithms in terms of speed and quality on both vector based and graph based clustering

stability of nonlinear discrete repetitive processes with markovian switching repetitive processes are a class of 2d systems that operate over a subset of the upper right quadrant of the 2d plane applications include iterative learning control where experimental verification has been reported based on a linear time invariant model approximation of the dynamics this paper considers discrete nonlinear repetitive processes with markovian switching and applies as one application the resulting stability theory to iterative learning control for a class of networked systems where time varying dynamics arise

stabilizing control law for hybrid models this note is concerned with the adaptive stabilizing control problem for jump markov systems a new control law is proposed based on a simple set of algebraic sufficient conditions it is shown that this approach can be used for a class of stochastic models for which the more restrictive solution proposed by caines and zhang 1992 does not apply

on the robust stability stabilization and stability radii of continuous time infinite markov jump linear systems this paper addresses the robust stochastic stability and stabilization of continuous time markov jump linear systems mjls with the markov jump parameters taking values in a countably infinite set it is assumed that the state and input matrices are subjected to norm bounded uncertainty with a prespecified structure which encompasses the block diagonal setting we introduce new robust analysis and synthesis characterizations such that unlike previous approaches in the mjls literature the scaling parameters are treated as decision variables in linear matrix inequalities as a by product new contributions to the theory of stability radii of mjls are provided when restricted to the finite case we further introduce new adjoint linear matrix inequality lmi characterizations for each of the robust analysis and synthesis problems as well as for stability radii besides the interest in its own right the adjoint approach allows us to verify that in the general mjls case there is a gap between the complex stability radius and what can be assessed with scaled versions of the small gain theorem this suggests a fundamental limitation of the robustness against linear perturbations that the h infty control of mjls may provide some numerical examples which include the robust control of two interconnected oscillators illustrate the main results

robust stability analysis and state feedback synthesis for discrete time systems characterized by random polytopes this paper deals with discrete time linear systems whose state transition is determined by a sequence of random matrices having uncertain distributions in representing random matrices with uncertain distributions we use random polytopes whose vertices are random matrices with given fixed distributions for such systems characterized by random polytopes we first tackle a problem of analyzing robust second moment exponential stability in particular we show a linear matrix inequality based robust stability condition that can be solved through sample based evaluation of the associated expectations the confidence level for such sample based analysis is ensured by the central limit theorem then we extend the results about analysis toward robust stabilization state feedback synthesis in such a way that the confidence level arguments can be facilitated we also provide numerical examples illustrating our analysis and synthesis framework

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

on exponential almost sure stability of random jump systems this paper is concerned with a class of random jump systems represented by transition operators which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case a series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios the stability criteria obtained are further extended to markov jump linear systems with infinite states and hence a unified approach to describing the almost sure stability of mjlss is addressed under this context all the results in the work are developed for both the continuous and discrete time systems

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

subgame perfection in positive recursive games with perfect information we consider a class of n player stochastic games with the following properties 1 in every state the transitions are controlled by one player 2 the payoffs are equal to zero in every nonabsorbing state 3 the payoffs are nonnegative in every absorbing state we propose a new iterative method to analyze these games with respect to the expected average reward we prove the existence of a subgame perfect e equilibrium in pure strategies for every e 0 moreover if all transitions are deterministic we obtain a subgame perfect 0 equilibrium in pure strategies

filtering s coupled algebraic riccati equations for discrete time markov jump systems abstract this paper deals with a set of s coupled algebraic riccati equations that arises in the study of filtering of discrete time linear jump systems with the markov chain in a general borel space s by s coupled it is meant that the algebraic riccati equations are coupled via an integral over s conditions for the existence and uniqueness of a positive semi definite solution to the filtering s coupled algebraic riccati equations are obtained in terms of the concepts of stochastic detectability and stochastic stabilizability this result is then applied to solve the infinite horizon minimum mean square linear markov jump filtering problem the obtained results generalize previous ones in the literature which considered only the case of the markov chain taking values in a finite state space

an augmented lagrangian dual optimization approach to the h weighted model updating problem abstract model updating for the quadratic eigenvalue problem aims to update the model q 2 m c k by given eigendata in this paper we consider the h weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights we propose an augmented lagrangian dual method for the h weighted model updating problem under some mild assumptions our method is shown to converge at least linearly numerical results illustrate the effectiveness of our method in addition we compare our method with the semi definite programming sdp method numerical results illustrate that when the scale of the model becomes large our method still works but the sdp method failed to converge

flexible high dimensional classification machines and their asymptotic properties classification is an important topic in statistics and machine learning with great potential in many real applications in this paper we investigate two popular large margin classification methods support vector machine svm and distance weighted discrimination dwd under two contexts the high dimensional low sample size data and the imbalanced data a unified family of classification machines the flexible assortment machine flame is proposed within which dwd and svm are special cases the flame family helps to identify the similarities and differences between svm and dwd it is well known that many classifiers overfit the data in the high dimensional setting and others are sensitive to the imbalanced data that is the class with a larger sample size overly influences the classifier and pushes the decision boundary towards the minority class svm is resistant to the imbalanced data issue but it overfits high dimensional data sets by showing the undesired data piling phenomenon the dwd method was proposed to improve svm in the high dimensional setting but its decision boundary is sensitive to the imbalanced ratio of sample sizes our flame family helps to understand an intrinsic connection between svm and dwd and provides a trade off between sensitivity to the imbalanced data and overfitting the high dimensional data several asymptotic properties of the flame classifiers are studied simulations and real data applications are investigated to illustrate theoretical findings

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

stochastic games with additive transitions abstract we deal with n player at stochastic games where at stands for additive transitions these are stochastic games in which the transition probability vector ps as for action combination a s a s 1 a s n in state s can be decomposed into player dependent components as p s a s i 1 n s i p s i a s i where s i 0 1 for all players i and i 1 n s i 1 and where p s i a s i is a probability distribution on the finite set of states s here s i reflects the influence of player i on the transitions in state s as such the class of at stochastic games covers several other well known classes such as perfect information stochastic games stochastic games with switching control and so called arat stochastic games with respect to the average reward it is not clear whether e equilibria always exist in general n player stochastic games for the class of n player at games we establish the existence of 0 equilibria although the strategies involved may be history dependent in addition we have the following results for the two player case 1 for zero sum at games stationary 0 optimal strategies always exist 2 for two player general sum at absorbing games there always exist stationary e equilibria for all e 0 several examples are provided to clarify the issues and to demonstrate the sharpness of the results

advertising models stochastic games and myopic strategies this paper analyzes dynamic advertising models we present conditions that imply that the payoffs under myopic strategies are linear in the state variable which represents the number of customers for each firm these conditions which determine a subclass of stochastic games are satisfied by several dynamic advertising models we give a criterion to check whether this model contains an equilibrium point within the class of myopic strategies that is also an equilibrium point with regard to all strategies

pure subgame perfect equilibria in free transition games we consider a class of stochastic games where each state is identified with a player at any moment during play one of the players is called active the active player can terminate the game or he can announce any player who then becomes the active player there is a non negative payoff for each player upon termination of the game which depends only on the player who decided to terminate we give a combinatorial proof of the existence of subgame perfect equilibria in pure strategies for the games in our class

a characterization of subgame perfect equilibrium plays in borel games of perfect information we provide a characterization of subgame perfect equilibrium plays in a class of perfect information games where each player s payoff function is borel measurable and has finite range the set of subgame perfect equilibrium plays is obtained through a process of iterative elimination of plays extensions to games with bounded borel measurable payoff functions are discussed as an application of our results we show that if every player s payoff function is bounded and upper semicontinuous then for every positive epsilon the game admits a subgame perfect epsilon equilibrium as we do not assume that the number of players is finite this result generalizes the corresponding result of purves and sudderth 24 purves ra sudderth wd 2011 perfect information games with upper semicontinuous payoffs math oper res 36 3 468 473

dual versus primal dual interior point methods for linear and conic programming we observe a curious property of dual versus primal dual path following interior point methods when applied to unbounded linear or conic programming problems in dual form while primal dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility unboundedness detectors

lq optimal sampled data control under stochastic delays gridding approach for stabilizability and detectability we solve a linear quadratic optimal control problem for sampled data systems with stochastic delays the delays are stochastically determined by the last few delays the proposed optimal controller

on the detectability and observability of continuous stochastic markov jump linear systems abstract this paper is concerned with the detectability and observability of continuous linear stochastic systems subject to markov jump by introducing some operators we investigate the relationship between some different concepts of detectability and observability and obtain their sufficient and necessary conditions continuous time stochastic lyapunov equations and stochastic continuous algebraic riccati equations are also discussed to serve as analysis tools

stochastic stability of jump discrete time linear systems with markov chain in a general borel space necessary and sufficient conditions for stochastic stability ss of discrete time linear systems subject to markov jumps in the parameters are considered assuming that the markov chain takes values in a general borel space s it is shown that ss is equivalent to the spectrum radius of a bounded linear operator in a banach space being less than 1 or to the existence of a solution of a lyapunov type equation these results generalize several previous results in the literature which considered only the case of the markov chain taking values in a finite or infinite countable space

simplifying optimal strategies in stochastic games we deal with zero sum limiting average stochastic games we show that the existence of arbitrary optimal strategies implies the existence of stationary varepsilon optimal strategies for all varepsilon 0 and the existence of markov optimal strategies we present such a construction for which we do not even need to know these optimal strategies furthermore an example demonstrates that the existence of stationary optimal strategies is not implied by the existence of optimal strategies so the result is sharp more generally one can evaluate a strategy pi for the maximizing player player 1 by the reward phi s pi that pi guarantees to him when starting in state s a strategy pi is called nonimproving if phi s pi geq phi s pi h for all s and for all finite histories h with final state s where pi h is the strategy pi conditional on the history h using the evaluation phi we may define the relation varepsilon better between strategies a strategy pi 1 is called varepsilon better than pi 2 if phi s pi 1 geq phi s pi 2 varepsilon for all s we show that for any nonimproving strategy pi for all varepsilon 0 there exists an varepsilon better stationary strategy and a 0 better markov strategy as well since all optimal strategies are nonimproving this result can be regarded as a generalization of the above result for optimal strategies finally we briefly discuss some other extensions among others we indicate possible simplifications of strategies that are only optimal for particular initial states by almost stationary varepsilon optimal strategies for all varepsilon 0 and by almost markov optimal strategies we also discuss the validity of the above results for other reward functions several examples clarify these issues

impedance matching for compact multiple antenna systems in random rf fields we investigate the maximization of the received power from compact multiple antenna systems in random rf fields using two novel impedance matching approaches the first approach is based on optimal impedance matching where we use convex optimization to propose a sparse optimal impedance matching technique the second approach is obtained by further relaxing the sparse optimal impedance matching method to obtain a suboptimal impedance matching som method we refer to as multiport ladder matching mlm simulation results show that the sparse optimal impedance matching approach provides the maximum received power while mlm can capture more power than conventional single port matching spm additionally they can maintain good bandwidth indicating the effectiveness of the sparse optimal and multiport ladder impedance matching approaches

exponential stability for discrete time infinite markov jump systems this technical note addresses a class of discrete time stochastic systems with infinite markov jump parameter first of all spectral criterion and lyapunov type criteria are presented for the exponential stability of considered systems further under the condition of strong detectability a barbashin krasovskii stability theorem is shown in terms of a generalized lyapunov equation with a positive semi definite term as an application of the proposed lyapunov stability criterion a sufficient condition of ell 2 input state stability is derived for the infinite markov jump systems disturbed by finite energy random disturbance

lq control of discrete time jump systems with markov chain in a general borel space in this technical note it is studied the lq optimal control problem for discrete time markov jump linear systems considering the case in which the markov chain takes values in a general borel space cal m it is shown that the solution of the lq optimal control problem is obtained in terms of the positive semi definite solution s ell ell in cal m of cal m coupled algebraic riccati equations by cal m coupled we mean that the algebraic riccati equations are coupled via an integral over a transition probability kernel cal g cdot vert cdot having a density g cdot vert cdot with respect to a sigma finite measure mu on cal m it is obtained sufficient conditions based on the concept of stochastic stabilizability and stochastic detectability for the existence and uniqueness of this positive semi definite solution these results generalize previous ones in the literature which considered only the case of the markov chain taking values in a finite or infinite countable space

improved lower bounds for the 2 page crossing numbers of k_ m n and k_n via semidefinite programming it has long been conjectured that the crossing numbers of the complete bipartite graph k m n and of the complete graph k n equal z m n bigl lfloor frac n 2 bigr rfloor bigl lfloor frac n 1 2 bigr rfloor bigl lfloor frac m 2 bigr rfloor bigl lfloor frac m 1 2 bigr rfloor and z n frac 1 4 bigl lfloor frac n 2 bigr rfloor bigl lfloor frac n 1 2 bigr rfloor bigl lfloor frac n 2 2 bigr rfloor bigl lfloor frac n 3 2 bigr rfloor respectively in a 2 page drawing of a graph the vertices are drawn on a straight line the spine and each edge is contained in one of the half planes of the spine the 2 page crossing number nu 2 g of a graph g is the minimum number of crossings in a 2 page drawing of g somewhat surprisingly there are 2 page drawings of k m n respectively k n with exactly z m n respectively z n crossings thus yielding the conjectures i nu 2 k m n stackrel z m n and ii nu 2 k n stackrel z n it

the empirical bayes envelope and regret minimization in competitive markov decision processes this paper proposes an extension of the regret minimizing framework from repeated matrix games to stochastic game models under appropriate recurrence conditions a decision maker p1 who wishes to maximize his long term average reward is facing a markovian environment which may also be affected by arbitrary actions of other agents the latter are collectively modeled as a second player p2 whose strategy is arbitrary both states and actions are fully observed by both players while p1 may obviously secure the min max value of the game he may wish to improve on that when the opponent is not playing a worst case strategy for repeated matrix games an achievable goal is presented by the bayes envelope that traces p1 s best response payoff against the observable frequencies of p2 s actions we propose a generalization to the stochastic game framework under recurrence conditions that amount to fixed state reachability the empirical bayes envelope ebe is defined as p1 s best response payoff against the stationary strategies of p2 that agree with the observed state action frequencies because the ebe may not be attainable in general we consider its lower convex hull the convex bayes envelope cbe which is proved to be achievable by p1 the analysis relies on blackwell s approachability theory the cbe is lower bounded by the value of the game and for irreducible games turns out to be strictly above the value whenever p2 s frequencies deviate from a worst case strategy in the special case of single controller games where p2 alone affects the state transitions the ebe itself is shown to be attainable