necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

a maximum principle for nonsmooth optimal control problems with state constraints abstract optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints the conditions which apply in the absence of differentiability assumptions on the data are stated in terms of clarke s generalized jacobians although not the most general available the conditions are derived by a novel method this involves removal of the state constraints by introduction of a penalty term and application of ekeland s variational principle

necessary optimality conditions for differential difference inclusions with state constraints abstract we consider the mayer optimal control problem with dynamics given by a nonconvex differential difference inclusion whose trajectories are constrained to a closed set necessary optimality conditions in the form of the maximum principle are obtained

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

a combined approach for evaluating papers authors and scientific journals an integrated model for ranking scientific publications together with authors and journals recently presented in bini del corso romani etna 2008 is closely analyzed the model which relies on certain adjacency matrices h k and f obtained from the relations of citation authorship and publication provides the ranking by means of the perron vector of a stochastic matrix obtained by combining h k and f some perturbation theorems concerning the perron vector previously introduced by the authors are extended to more general cases and a counterexample to a property previously addressed by the authors is presented the theoretical results confirm the consistency and effectiveness of our model some paradigmatic examples are reported together with some results obtained on a real set of data

the adjoint arc in nonsmooth optimization we extend the theory of necessary conditions for nonsmooth problems of bolza in three ways first we incorporate state constraints of the intrinsic type x t e x t for all t second we make no assumption of calmness or normality and third we show that a single adjoint function of bounded variation simultaneously satisfies the hamiltonian inclusion the euler lagrange inclusion and the weierstrass pontryagin maximum condition along with the usual transversality relations

variational approach to second order sufficient optimality conditions in optimal control we consider a bolza optimal control problem with convex pure state constraints and general control constraints and propose new second order sufficient optimality conditions for a weak local minimum that can be applied in a general framework in particular the reference control may be merely measurable and the dynamics may be discontinuous in the time variable our approach is based on a new decomposition result for control constraints involving second order tangents

third order phase transition in random tilings we consider the domino tilings of an aztec diamond with a cut off corner of macroscopic square shape and given size and address the bulk properties of tilings as the size is varied we observe that the free energy exhibits a third order phase transition when the cut off square increasing in size reaches the arctic ellipse the phase separation curve of the original unmodified aztec diamond we obtain this result by studying the thermodynamic limit of a certain nonlocal correlation function of the underlying six vertex model with domain wall boundary conditions the so called emptiness formation probability efp we consider efp in two different representations as a function for toda chains and as a random matrix model integral the latter has a discrete measure and a linear potential with hard walls the observed phase transition shares properties with both gross witten wadia and douglas kazakov phase transitions

maximizing pagerank via outlinks we analyze linkage strategies for a set j of webpages for which the webmaster wants to maximize the sum of google s pagerank scores the webmaster can only choose the hyperlinks starting from the webpages of j and has no control on the hyperlinks from other webpages we provide an optimal linkage strategy under some reasonable assumptions c 2008 elsevier inc all rights reserved

generalized emptiness formation probability in the six vertex model in the six vertex model with domain wall boundary conditions the emptiness formation probability is the probability that a rectangular region in the top left corner of the lattice is frozen we generalize this notion to the case where the frozen region has the shape of a generic young diagram we derive here a multiple integral representation for this correlation function

new enumeration formulas for alternating sign matrices and square ice partition functions abstract the refined enumeration of alternating sign matrices asms of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study starting with the refined alternating sign matrix conjecture of mills robbins rumsey 1983 25 its proof by zeilberger 1996 31 and more recent work on doubly refined and triply refined enumerations by several authors in this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the top left bottom triply refined and top left bottom right quadruply refined enumerations the latter case solves the problem of computing the full boundary correlation function for asms the enumeration formulas are proved by deriving new representations which are of independent interest for the partition function of the square ice model with domain wall boundary conditions at the combinatorial point 2 3

necessary optimality conditions for infinite dimensional state constrained control problems abstract this paper is concerned with first order necessary optimality conditions for state constrained control problems in separable banach spaces assuming inward pointing conditions on the constraint we give a simple proof of pontryagin maximum principle relying on infinite dimensional neighboring feasible trajectories theorems proved in 20 further we provide sufficient conditions guaranteeing normality of the maximum principle we work in the abstract semigroup setting but nevertheless we apply our results to several concrete models involving controlled pdes pointwise state constraints as positivity of the solutions are allowed

degenerate optimal control problems with state constraints standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest e g when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set in these cases a nonzero but nevertheless trivial set of multipliers exists we give conditions for the existence of nontrivial multipliers a feature of these conditions is that they allow nonconvex velocity sets and measurably time dependent data the proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint originating in the dynamic programming literature

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

emptiness formation probability of the six vertex model and the sixth painleve equation we show that the emptiness formation probability of the six vertex model with domain wall boundary conditions at its free fermion point is a tau function of the sixth painleve equation using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit

a connection between the maximum principle and dynamic programming for constrained control problems we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function this additional relation is applied in turn to show that the maximum principle is nondegenerate we also provide a sufficient condition for the normality of the maximum break principle to derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories then duality theory of convex analysis is applied to derive necessary conditions for optimality in this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case

google pageranking problem the model and the analysis the spectral and jordan structures of the web hyperlink matrix g c cg 1 c ev t have been analyzed when g is the basic stochastic google matrix c is a real parameter such that 0 1y c nv for all v this limit may fail to exist for some v if l is not semisimple as a special case of our results we obtain a complex analog of pagerank for the web hyperlink matrix g c with a complex parameter c we study regularity limits expansions and conditioning of y c and we propose algorithms e g complex extrapolation power method on a modified matrix etc that may provide an efficient way to compute pagerank also with c close or equal to 1 an interpretation of the limit vector nv and a related critical discussion on the model on its adherence to reality and possible ways for its improvement represent the contribution of the paper on modeling issues

boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions abstract boundary correlation functions of the six and nineteen vertex models on an n n lattice with domain wall boundary conditions are studied the general expression of the boundary correlation functions is obtained for the six vertex model by the use of the quantum inverse scattering method for the nineteen vertex model the boundary correlation functions are shown to be expressed in terms of those for the six vertex model

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

filippov s and filippov wazewski s theorems on closed domains abstract the celebrated filippov s theorem implies that given a trajectory x1 0 r n of a differential inclusion x f t x with the set valued map f measurable in t and k lipschitz in x for any initial condition x2 0 r n there exists a trajectory x2 starting from x2 0 such that x1 t x2 t ekt x1 0 x2 0 filippov wazewski s theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x co f t x by a trajectory of the original inclusion x f t x starting from the same initial condition in the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset r n the latter is allowed to be non smooth we impose a generalized soner type condition on f and yielding extensions of the above classical results to infinite horizon constrained problems applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well

contingent cones to reachable sets of control systems high order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory it is shown that the adjoint vector of pontryagin s maximum principle is normal to the set of variations of reachable sets results are applied to study optimal control problems for dynamical systems described by 1 closed loop control systems 2 nonlinear implicit systems 3 differential inclusions 4 control systems with jumps

boundary spontaneous polarization in the six vertex model with a reflecting boundary we consider the six vertex model on a 2n n lattice with domain wall boundary conditions for the lower upper and right boundaries and a reflecting end for the left boundary the boundary one point correlation functions which describe the boundary spontaneous polarization are calculated and expressed as some determinants of n n matrices

some characterizations of optimal trajectories in control theory several characterizations of optimal trajectories for the classical mayer problem in optimal control are provided for this purpose the regularity of directional derivatives of the value function is studied for instance it is shown that for smooth control systems the value function v is continuously differentiable along an optimal trajectory x t 0 1 to bf r n provided v is differentiable at the initial point t 0 x t 0 then the upper semicontinuity of the optimal feedback map is deduced the problem of optimal design is addressed obtaining sufficient conditions for optimality finally it is shown that the optimal control problem may be reduced to a viability one

boundary polarization in the six vertex model vertical arrow fluctuations near the boundaries in the six vertex model on the two dimensional n n square lattice with the domain wall boundary conditions are considered the one point correlation function boundary polarization is expressed via the partition function of the model on a sublattice the partition function is represented in terms of standard objects in the theory of orthogonal polynomials this representation is used to study the large n limit the presence of the boundary affects the macroscopic quantities of the model even in this limit the logarithmic terms obtained are compared with predictions from conformal field theory

a note on the value function for constrained control problems abstract we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a given set and we obtain a relation between the costate function that appears in the maximum principle and the value function this relation extends the known conditions existing in the literature for unconstrained problems to those for problems under state constraints

combining pontryagin s principle and dynamic programming for linear and nonlinear systems to study optimal control and disturbance attenuation problems two prominent and somewhat alternative strategies have emerged in the last century dynamic programming dp and pontryagin s minimum principle pmp the former characterizes the solution by shaping the dynamics in a closed loop a priori unknown via the selection of a feedback input at the price however of the solution to typically daunting partial differential equations the latter instead provides extended dynamics that must be satisfied by the optimal process for which boundary conditions a priori unknown should be determined the results discussed in this article combine the two approaches by matching the corresponding trajectories i e combining the underlying sources of information knowledge of the complete initial condition from dp and of the optimal dynamics from pmp the proposed approach provides insights for linear as well as nonlinear systems in the case of linear systems the derived conditions lead to matrix algebraic equations similar to the classic algebraic riccati equations ares although with coefficients defined as polynomial functions of the input gain matrix with the property that the coefficient of the quadratic term of such equation is sign definite even if the corresponding coefficient of the original are is sign indefinite as it is typically the case in the mathcal h infty control problem this feature is particularly appealing from the computational point of view since it permits the use of standard minimization techniques for convex functions such as the gradient algorithm in the presence of nonlinear dynamics the strategy leads to algebraic equations that allow us to locally construct the optimal feedback by considering the behavior of the closed loop dynamics at a single point in the state space

investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints in this paper we study the degeneracy phenomenon in optimal control problems with state constraints it is shown that this phenomenon occurs because of the incompleteness of the standard variants of pontryagin s maximum principle for problems with state constraints a new maximum principle containing additional information about the behavior of the hamiltonian at the endtimes is developed we also obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle the results obtained pertain to optimal control problems with systems described by differential inclusions and ordinary differential equations

an integral type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints abstract for optimal control problems involving ordinary differential equations and functional inequality state constraints the maximum principle may degenerate producing no useful information about minimizers this is known as the degeneracy phenomenon several non degenerate forms of the maximum principle valid under different constraint qualifications have been proposed in the literature in this paper we propose a new constraint qualification under which a nondegenerate maximum principle is validated in contrast with existing results our constraint qualification is of an integral type an advantage of the proposed constraint qualification is that it is verified on a larger class of problems with nonsmooth data and convex velocity sets

normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints abstract there has been a longstanding interest in deriving conditions under which dynamic optimization problems are normal that is the necessary conditions of optimality nco can be written with a nonzero multiplier associated with the objective function this paper builds upon previous results on nondegenerate nco for trajectory constrained optimal control problems to provide even stronger normal forms of the conditions the nco developed may address problems with nonsmooth less regular data the particular case of calculus of variations problems is here explored to show a favorable comparison with existent results

necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints the main purpose of this work is to establish some first and second order necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints the control may affect both the drift and the diffusion terms of the systems and the control regions are allowed to be nonconvex a stochastic inward pointing condition is proposed to ensure the normality of the corresponding necessary conditions

the relationship between the maximum principle and dynamic programming let v t x be the infimum cost of an optimal control problem viewed as a function of the initial time and state t x dynamic programming is concerned with the properties of v cdot cdot and in particular with its characterization as a solution to the hamilton jacobi bellman equation heuristic arguments have long been advanced relating the maximum principle to dynamic programming according to p t v x left t x 0 t right here x 0 cdot is the minimizing state function under consideration and p cdot is the costate function of the maximum principle in this paper we examine the validity of such claims and find that this relationship interpreted as a differential inclusion involving the generalized gradient is indeed true almost everywhere and at the endpoints for a very large class of nonsmooth optimal control problems

sensitivity interpretations of the costate variable for optimal control problems with state constraints in optimal control theory it is well known that the costate arc and the associated maximized hamiltonian function can be interpreted in terms of gradients of the value function evaluated along the optimal state trajectory such relations have been referred to as sensitivity relations in the literature we provide in this paper new sensitivity relations for state constrained optimal control problems for the class of optimal control problems considered there is no guarantee that the costate arc is unique a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid the proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem the introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer the extra information thereby obtained yields the desired set of sensitivity relations

comparison of perturbation bounds for the stationary distribution of a markov chain abstract the purpose of this paper is to review and compare the existing perturbation bounds for the stationary distribution of a finite irreducible homogeneous markov chain

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints abstract in this paper the value function of a bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a hamilton jacobi equation the state constraints are given by an arbitrary closed set with possibly empty interior in particular soner s inward pointing condition is extended here to the case of degenerate state constraints

conditioning of the entries in the stationary vector of a google type matrix abstract using combinatorial and analytic techniques we give conditioning bounds for the stationary vector t of a stochastic matrix of the form ca 1 c b where c 0 1 is a scalar and a and b are stochastic matrices the latter being rank one such matrices and their stationary vectors arise as a key component in google s pagerank algorithm the conditioning bounds considered include normwise absolute componentwise and relative componentwise and the bounds depend on c and on quantities such as the number of dangling nodes which correspond to rows of a having all entries equal or the lengths of certain cycles in the directed graph associated with a it is shown that if vertex j is on only long cycles in that directed graph then the corresponding entry in t exhibits better conditioning properties and that for dangling nodes the sensitivity of the corresponding entries in t decreases as the number of dangling nodes increases conditions are given that are sufficient to ensure that an iterate of the power method accurately reflects the relative ordering of two entries in t

dynamic programming principle of control systems on manifolds and its relations to maximum principle abstract we study the dynamic programming principle dpp for short on manifolds obtain the hamilton jacobi bellman hjb for short equation and prove that the value function is the only viscosity solution to the hjb equation then we investigate the relation between dpp and pontryagin s maximum principle pmp for short from which we obtain pmp on manifolds

nondegenerate necessary conditions for nonconvex optimal control problems with state constraints standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set and so give no useful information about optimal controls recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions in these papers it is typically shown that if a constraint qualification is imposed requiring existence of inward pointing velocities then sets of multipliers exist in addition to the trivial ones a simple new approach for deriving nondegenerate necessary conditions is presented which permits relaxation of hypotheses previously imposed concerning data regularity and convexity of the velocity set

the raise and peel model of a fluctuating interface we propose a one dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter the adsorption rate at a special value of this parameter the model has some interesting features for example the spectrum is given by conformal field theory and the stationary non equilibrium probability distribution is given by the two dimensional equilibrium distribution of the ice model with domain wall type boundary conditions this connection is used to find exact analytic expressions for several quantities of the stochastic model vice versa some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model at the special point we study several properties of the model such as the height fluctuations as well as cluster and avalanche distributions the latter has a long tail which shows that the model exhibits self organized criticality we also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent 2 3 furthermore we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken