a solvable stochastic control problem in hyperbolic three space abstract a stochastic optimal control problem in hyperbolic three space is described and explicitly solved the solution is obtained by finding a smooth solution to the hamilton jacobi or dynamic programming equation

the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

a relaxation column signature method for assignment problems abstract a recent computational study 7 on algorithms for assignment problems revealed that the so called modified hung rom algorithm mhra performs very well in practice the mhra is a row signature method and its effectiveness comes mainly from its capability to reduce the level by more than one unit per cycle we present an o n 3 column signature method which has the above mentioned capability the algorithm solves a sequence of relaxed transportation problems and generates strong dual feasible bases it is shown that the average number of iterations is bounded by n log n a result on balinski s strong dual feasible bases is also presented

on the length of simplex paths the assignment case this paper reports on an experimental study of the distribution of the length of simplex paths for the optimal assignment problem we study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis in this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients provided these are sufficiently spread out further the mean and standard deviation grow only moderately with the size of the problem namely asd1 8 andd1 5 respectively for ad d problem implying in particular that the pivot counts concentrate around the mean with growingd the usual simplex method on the other hand gives a growth ofd1 6 hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average at least for assignment problems

some examples of optimal stochastic controls or the stochastic maximum principle at work the stochastic maximum principle is applied to solve eight stochastic control problems four have been solved previously using much more complicated methods but the last four are new the main thrust of the work is to show that some completely observable stochastic control problems with special structure can be solved quite quickly and easily using the maximum principle

state feedback control of systems with multiplicative noise via linear matrix inequalities we consider lti systems perturbed by parametric uncertainties modeled as white noise disturbances we show how to maximize via state feedback control the smallest norm of the noise intensity vector producing instability in the mean square sense using convex optimization over linear matrix inequalities we also show how to maximize performance robustness where performance is measured by expected output energy with either bounded initial conditions and zero inputs classical lqg cost or zero initial conditions and deterministic inputs of bounded energy a generalization of the h norm

stability radii of systems with stochastic uncertainty and their optimization by output feedback we consider linear plants controlled by dynamic output feedback which are subjected to blockdiagonal stochastic parameter perturbations the stability radii of these systems are characterized and it is shown that for real data the real and the complex stability radii coincide a corresponding result does not hold in the deterministic case even for perturbations of single output feedback type in a second part of the paper we study the problem of optimizing the stability radius by dynamic linear output feedback necessary and sufficient conditions are derived for the existence of a compensator which achieves a suboptimal stability radius these conditions consist of a parametrized riccati equation a parametrized liapunov inequality a coupling inequality and a number of linear matrix inequalities one for each disturbance term the corresponding problem in the deterministic case the optimal mu synthesis problem is still unsolved

robust h infinity control in the presence of stochastic uncertainty this paper considers a robust h infinity state feedback control problem for linear uncertain systems with stochastic uncertainty the uncertainty class considered in the paper involves uncertain multiplicative white noise perturbations which satisfy a certain variance constraint a state feedback controller is presented which guarantees a prescribed level of disturbance attenuation for all admissible stochastic uncertainties

discrete time expectation maximization algorithms for markov modulated poisson processes in this paper we consider parameter estimation markov modulated poisson processes via robust filtering and smoothing techniques using the expectation maximization algorithm framework our filters and smoothers can be applied to estimate the parameters of our model in either an online configuration or an offline configuration further our estimator dynamics do not involve stochastic integrals and our new formulas in terms of time integrals are easily discretized and are written in numerically stable forms in w p malcolm r j elliott and j van der hoek ldquoon the numerical stability of time discretized state estimation via clark transformations rdquo presented at the ieee conf decision control mauii hi dec 2003

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

transportation problems and simplicial polytopes that are not weakly vertex decomposable provan and billera defined the notion of weak k decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes they showed the diameter of a weakly k decomposable simplicial complex is bounded above by a polynomial function of the number of k faces in and its dimension for weakly 0 decomposable complexes this bound is linear in the number of vertices and the dimension in this paper we exhibit the first examples of non weakly 0 decomposable simplicial polytopes our examples are in fact polar to certain transportation polytopes

a strongly polynomial algorithm for a concave production transportation problem with a fixed number of nonlinear variables we show that the production transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time the algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem monotonicity of the objective function along certain directions small proportion of nonlinear variables and combinatorial properties implied by transportation constraints

transportation problems which can be solved by the use of hirsch paths for the dual problems balinski uses his signature method for the proof of the hirsch conjecture for dual transportation polyhedra to obtain an efficient algorithm for the assignment problem we will show how to extend this method to other primal transportation problems including transportation problems with unit demands we then prove that balinski s assignment algorithm is equivalent cycle by cycle to that of hung and rom we demonstrate that under some assumptions for our probability model a modification of the latter algorithm has an average complexity of o n2logn and present some computational results confirming this we also present results that indicate that this modification compares favorably with balinski s algorithm and other codes

on the adjoint process for optimal control of diffusion processes for the markovian control problem begin gathered mathop min limits u e left int 0 t l t x u dt c x t right hfill dx f t x u dt sigma t x dw hfill end gathered it is shown that the adjoint process appearing in the maximum principle has the form p t x e tx left frac partial c partial x x t phi t t int t t frac partial l partial x s x s hat u s x s phi s t ds right where hat u is the optimal feedback control e tx denotes conditional expectation and phi is the fundamental matrix solution of dy frac partial f partial x t x t hat u t x t ydt sum k frac partial sigma k partial x t x y dw k here sigma k is the kth column of sigma and w k is thekth component of the brownian motion w it is also shown that p t x t satisfies denotes transpose begin gathered dp left frac partial f partial x t x t hat u t x t p

maximum likelihood localization of narrow band autoregressive sources via the em algorithm the authors derive an expectation maximization algorithm for the maximum likelihood estimation of the directions of arrival of multiple narrowband autoregressive ar signals embedded in noise the proposed algorithm simultaneously estimates the location parameters the ar coefficients and the signals the additional structural information that the sources are of the ar type enables one to resolve the case in which the number of sources is equal to or possibly larger than the number of sensors the authors provide examples demonstrating that the proposed algorithm can resolve two spectrally and spatially closely spaced sources by using a two sensor array

on the use of stochastic differential games against nature to ergodic control problems with unknown parameters in this paper we propose an application of the so called games against nature for giving solution to an ergodic control problem governed by a general class of markov diffusion processes whose coefficients depend on an unknown and non observable parameter to this end we assume that the values of the parameter are taken by means of actions made by some opposite player of the controller the nature then the problem reduces to finding optimality for the controller given that the nature has chosen its best strategy such a control is also known as the worst case optimal control our analysis is based on the use of the dynamic programming technique by showing among other facts the existence of classical twice differentiable solutions of the so called hamilton jacobi bellman equation we also provide an example on economic welfare to illustrate our results

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

quadratic diameter bounds for dual network flow polyhedra both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best case performance of linear programming algorithms we study the diameters of dual network flow polyhedra associated to b flows on directed graphs g v e g v e and prove quadratic upper bounds for both of them the minimum of v 1 cdot e v 1 e and frac 1 6 v 3 16 v 3 for the combinatorial diameter and frac v cdot v 1 2 v v 1 2 for the circuit diameter previously bounds on these diameters have only been known for bipartite graphs the situation is much more involved for general graphs in particular we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant

an infeasible exterior point simplex algorithm for assignment problems the so called modified hung rom algorithm based upon theoretical considerations of hirsch paths seems to be one of the most efficient algorithms for assignment problems since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest this paper presents an algorithm of this kind for assignment problems

absolute stabilization and minimax optimal control of uncertain systems with stochastic uncertainty this paper is concerned with existence and optimality properties of so called guaranteed cost controllers for an uncertain system subject to structured uncertainty the uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints it is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system for each initial state of the system this controller can be found by parametric optimization of solutions of a parameter dependent generalized matrix riccati equation arising in stochastic h infty theory

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

robustness of feedback stabilized systems in the presence of non linear and random perturbations the general robustness problem is discussed for a controlled discrete time system subject to a lur e type feedback non linearity together with random multiplicative noises effective on state input and the non linearity a sufficient condition is given utilizing the discrete stochastic version of the second method of lyapunov to guarantee that any stabilizing controller designed for the deterministic linear part of the system renders the closed loop system with perturbations asymptotically stable with probability one then specializations to systems given in controllability and controller canonical forms are made independent noise signals are assumed to be effective on individual parameters of these canonical forms whose inputs are designed to assign the closed loop poles to zero in these cases stronger results are obtained in terms of system parameters and known characteristics of perturbations

the hirsch conjecture for the fractional stable set polytope the edge formulation of the stable set problem is defined by two variable constraints one for each edge of a graph g g expressing the simple condition that two adjacent nodes cannot belong to a stable set we study the fractional stable set polytope i e the polytope defined by the linear relaxation of the edge formulation even if this polytope is a weak approximation of the stable set polytope its simple geometrical structure provides deep theoretical insight as well as interesting algorithmic opportunities exploiting a graphic characterization of the bases we first redefine pivots in terms of simple graphic operations that turn a given basis into an adjacent one these results lead us to prove that the combinatorial diameter of the fractional stable set polytope is at most the number of nodes of the given graph as a corollary the hirsch bound holds for this class of polytopes

nonconvex optimization problem the infinite horizon linear quadratic control problem with quadratic constraints abstract we present a general method for the solution of the problem of minimization of a quadratic functional in a hilbert space over the intersection of some subspace and a set defined by special quadratic constraints many optimal control problems can be formulated in this abstract form as an example a stationary infinite horizon linear quadratic optimization problem with quadratic constraints is posed and solved

robust maximum principle in minimax control this paper presents a new version of the maximum principle dealing with construction of optimal control strategies for the class of uncertain systems given by an ordinary differential equation with unknown parameters from a given finite set the problem under consideration belongs to the class of optimization problems of the minimax type the proof is based on the tent method illustrative examples conclude this study

the diameters of network flow polytopes satisfy the hirsch conjecture we solve a problem in the combinatorics of polyhedra motivated by the network simplex method we show that the hirsch conjecture holds for the diameter of the graphs of all network flow polytopes in particular the diameter of a network flow polytope for a network with n nodes and m arcs is never more than m n 1 a key step to prove this is to show the same result for classical transportation polytopes

signature classes of transportation polytopes signature algorithms solve certain classes of transportation problems in a number of steps bounded by the diameter of the dual polyhedron the class of problems to which signature algorithms may be applied the signature classes of the title are characterized and the monotonic hirsch conjecture is shown to hold for them in addition certain more precise results are given for different cases finally it is explained why a supposed proof of the hirsch conjecture for all transportation polytopes is incomplete and apparently irremedial

new finite dimensional filters for parameter estimation of discrete time linear gaussian models the authors derive a new class of finite dimensional recursive filters for linear dynamical systems the kalman filter is a special case of their general filter apart from being of mathematical interest these new finite dimensional filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the parameters of a linear dynamical system important advantages of their filter based em algorithm compared with the standard smoother based em algorithm include 1 substantially reduced memory requirements and 2 ease of parallel implementation on a multiprocessor system the algorithm has applications in multisensor signal enhancement of speech signals and also econometric modeling

hidden gauss markov models for signal classification continuous state hidden markov models cs hmms are developed as a tool for signal classification analogs of the baum 1972 viterbi 1962 and baum welch algorithms are formulated for this class of models the cs hmm algorithms are then specialized to hidden gauss markov models hgmms with linear gaussian state transition and output densities a new gaussian refactorization lemma is used to show that the baum and viterbi algorithms for hgmms are implemented by two different formulations of the fixed interval kalman smoother the measurement likelihoods obtained from the forward pass of the hgmm baum algorithm and from the kalman filter innovation sequence are shown to be equal a direct link between the baum welch training algorithm and an existing expectation maximization em algorithm for gaussian models is demonstrated a new expression for the cross covariance between time adjacent states in hgmms is derived from the off diagonal block of the conditional joint covariance matrix a parameter invariance structure is noted for the hgmm likelihood function cs hmms and hgmms are extended to incorporate mixture densities for the a priori density of the initial state application of hgmms to signal classification is demonstrated with a three class test simulation

signature methods for the assignment problem the signature of a dual feasible basis of the assignment problem is an n vector whose ith component is the number of nonbasic activities of type i j this paper uses signatures to describe a method for finding optimal assignments that terminates in at most n 1n 2 2 pivot steps and takes at most on3 work

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

fenchel type duality for matroid valuations the weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem given a pair of valuated matroidsmi v i i i 1 2 defined on a common ground setv find a common baseb 1 2 that maximizes 1 b 2 b this paper develops a fenchel type duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming a fenchel type min max theorem and a discrete separation theorem are given furthermore the subdifferentials of matroid valuations are investigated 1998 the mathematical programming society inc published by elsevier science b v

a competitive dual simplex method for the assignment problem where there is abundance of mystery and confusion in every direction the truth seldom remains hidden for long it s a matter of having plenty of angles to go at it from only the utterly simple crimes the simplex crimes you may say have the trick of remaining baffling sir john from michael innes the open house a sir john appleby mystery penguin books 1974

a risk sensitive maximum principle abstract a stochastic maximum principle is derived which is formally of general application and so not restricted to the diffusion case it moreover differs from the classic deterministic principle only by a modification of the hamiltonian and so requires no solution of stochastic differential equations or calculation of conditional expectations the two essential points in its construction are that one considers the family of criterion functions exponential in an additive cost function of which the conventional expected cost case is a degenerate member and that in the non lqg case appeal to large deviation theory must be valid

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

adaptive continuous time linear quadratic gaussian control the adaptive linear quadratic gaussian control problem where the linear transformation of the state a and the linear transformation of the control b are unknown is solved assuming only that a b is controllable and a q sub 1 sup 1 2 is observable where q sub 1 determines the quadratic form for the state in the integrand of the cost functional a weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable a diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent a lagged certainty equivalence control using this family of estimates is shown to be self optimizing for an ergodic quadratic cost functional

minimax control of switching systems under sampling we consider a general class of systems subject to two types of uncertainty a continuous deterministic uncertainty that affects the system dynamics and a discrete stochastic uncertainty that leads to jumps in the system structure at random times with the latter described by a continuous time finite state markov chain when only sampled values of the system state is available to the controller along with perfect measurements on the state of the markov chain we obtain a characterization of minimax controllers which involves the solutions of two finite sets of coupled pdes and a finite dimensional compensator for the linear quadratic case a complete characterization is given in terms of coupled generalized riccati equations which also provides the solution to a particular h optimal control problem with randomly switching system structure and sampled state measurements

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

robust output feedback control for linear stochastic systems in continuous time with time varying parameters we consider a multivariable linear system with additive inputs and noise governed by a stochastic process with independent increments we allow the parameters of the system to change in time we assume that the exact form of the coefficients is not known rather the intervals within which these coefficients fluctuate are given in addition we do not know the exact state of the system rather we observe the output process correlated with this state the set of allowable controls are feedback controls based on the observation of the output process and an ideal model dynamics with each control we associate a performance functional characterizing the quality of tracking process and the objective is to construct a robust feedback control which minimizes the maximal performance functional within the class of systems with nonstochastic coefficients fluctuating in the given intervals

modeling and control of stochastic systems with poorly known dynamics this paper is concerned with controlling poorly known systems for which only a simplified and rough model is available for control design there are many systems that cannot be reasonably probed for the sake of identification yet they are important for areas such as economy populations or medicine the ideas are developed around an alternative way to account for the bare modeling in a stochastic based setting and to heighten the control features for such a modified model the mathematical framework for the optimal control reveals important features such as the raising of a precautionary feedback policy of keep the action unchanged inaction for short on a certain state space region this feature is not seen in the robust approach but has been pointed out and permeates part of the economics literature the control problem relies on the viscosity solution for the hamilton jacobi bellman equation and the value of the problem is shown to be convex when specialized to the quadratic problem with discounted cost the exact solution inside the inaction region is given by a lyapunov type of equation and asymptotically for large state values by a riccati like equation this scenario bridges to the stochastic stability analysis for the controlled model the single control input is developed in full part analytically part numerically for the scalar case and an approximation is tested for the multidimensional case the advantage of the precautionary policy is substantial in some situations

lagrange lemma and the optimal control of diffusions ii nonlinear lagrange functionals abstract in kosmol 1991 kosmol and pavon 1993 a new elementary approach to optimal control problems relying on the lagrange lemma was described which appears to be technically and conceptually much simpler than existing methods and furthermore provides a unified variational approach in kosmol and pavon 1992 this method was further clarified and developed for linear lagrange functionals we devote this second paper to nonlinear lagrange functionals the power of this approach is here clearly demonstrated in particular it is shown that the nonlinear lagrange functional induced by the value function of the problem is just one of the many functionals which may effectively be employed to solve control problems see examples 1 and 2 in section 3 moreover in our approach we can deal in the same framework with problems with state constraints or nonsmooth problems hence the lagrange functionals approach provides new tools to solve control problems which may be readily applied even when existing approaches fail

diversity combining for ds ss systems with time varying correlated fading branches diversity combining of multiple time varying and correlated fading branches is investigated for direct sequence spread spectrum systems the correlated branches are modeled and estimated jointly as a vector autoregressive var process the joint estimation is shown to provide performance gain over separate estimation of the fading branches the parameter matrices of the var model are estimated via the method of expectation maximization em with two algorithms the first algorithm using results from kalman smoothing provides a closed form solution to the maximization problem in the iterative em procedure however the iterative em kalman algorithm operates repeatedly on a batch of training data and results in large storage requirements and long processing delays to overcome these disadvantages a new algorithm with only forward time recursions is proposed that approximates the iterative em solution and efficiently adapts to slowly changing doppler spreads as a result the new algorithm significantly reduces memory and training sequence requirements through computer simulations a near ideal bit error rate performance is found for both algorithms and the efficacy of the new adaptive algorithm for channels with changing doppler spreads is demonstrated