convergence of consensus models with stochastic disturbances we consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge almost surely to a consensus let a t b t rn n be possibly stochastic nonstationary matrices and x t m t 6 rn 1 be state and perturbation vectors respectively for any consensus algorithm of the form x t 1 a t x t b t m t we provide conditions under which consensus is achieved almost surely i e pr limt x t c1 1 for some c r moreover we show that this general result subsumes recently reported results for specific consensus algorithms classes including sum preserving nonsum preserving quantized and noisy gossip algorithms also provided are the e converging time for any such converging iterative algorithm i e the earliest time at which the vector x t is e close to consensus and sufficient conditions for convergence in expectation to the average of the initial node measurements finally mean square error bounds of any consensus algorithm of the form discussed above are presented

suppression and stabilisation of noise in this article we investigate the stochastic suppression and stabilisation of nonlinear systems given an unstable differential equation in which f satisfies the one sided polynomial growth condition we introduce two brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation this article shows that appropriate may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable

varying confidence levels for cvar risk measures and minimax limits conditional value at risk cvar has been widely studied as a risk measure in this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with cvar appearing in the objective and also the constraints we start by considering a problem in which cvar is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one we make use of a consistent tail condition which ensures that the cvar of a random function will converge uniformly to its supremum as the confidence level increases and establish an error bound for the cvar optimal solution under second order growth conditions the results are extended to a minimization problem with a constraint on the cvar value which in the limit as the confidence level approaches one coincides with a problem having semi infinite constraints we study the sample average approximation scheme for the cvar constraints and establish an exponential rate of convergence for the sample averaged optimal solution we propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to cvar constrained problems out of sample our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance

stochastic stability of continuous time consensus protocols a unified approach to studying convergence and stochastic stability of continuous time consensus protocols cps is presented in this work our method applies to networks with directed information flow both cooperative and noncooperative interactions networks under weak stochastic forcing and those whose topology and strength of connections may vary in time the graph theoretic interpretation of the analytical results is emphasized we show how the spectral properties such as algebraic connectivity and total effective resistance as well as the geometric properties such as the dimension and the structure of the cycle subspace of the underlying graph shape stability of the corresponding cps in addition we explore certain implications of spectral graph theory to cp design in particular we point out that expanders sparse highly connected graphs generate cps whose performance remains uniformly high when the size of the network grows unboundedly similarly we highlight the benefits of using random

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

a constraint sampling approach for multi stage robust optimization we propose a tractable approximation scheme for convex not necessarily linear multi stage robust optimization problems we approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization we obtain a priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi infinite problem arising from the decision rule approximation we demonstrate that for a suitable choice of basis functions the approximation converges as the size of the basis and the number of sampled constraints tend to infinity the approach yields an algorithm parameterized in the basis size the probability of constraint violation and the confidence that this probability will not be exceeded these three parameters serve to tune the trade off between optimality and feasibility of the decision rules and the computational cost of the algorithm we assess the convergence and scalability properties of our approach in the context of two inventory management problems

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

mitigation of complex behavior over networked systems analysis of spatially invariant structures in this paper we consider a simple distributed averaging system which incorporates various communication constraints including delays noise and link failures it has been shown in wang and elia 2012 that such networked system generates a collective levy flight behavior when part of the system loses mean square ms stability we focus on spatially invariant architectures to gain more insights into how model parameters affect emergence of this complex scale invariant behavior and to seek structures robust to communication constraints specifically we develop a computational expression for checking ms stability which is scalable with the number of unreliable links we derive the closed form formulas from this expression in the limiting case of zero and large delays and in the case of large number of nodes in the limit of large delays we derive various results that are independent of the network size and its specific interconnections we find that small inter agent coupling improves the robustness of the system networks with larger connectivity tend to be more fragile in the presence of fading connections for fixed inter agent coupling that gossiping improves the robustness and that the lattice is the most robust among the spatially invariant systems with generalized circulant interconnections

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

convergence analysis for mathematical programs with distributionally robust chance constraint convergence analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and the optimal solutions and research on this topic has been well documented in the literature of stochastic programming in this paper we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability distribution is unknown but it is possible to construct an ambiguity set of probability distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set the convergence analysis focuses on impact of the variation of the ambiguity set on the optimal value and the optimal solutions we start by deriving general convergence results under abstract conditions such as continuity of the robust probability function and uniform convergence of the robust probability functions and followed with detailed analysis of these

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

consensus conditions of multi agent systems with time varying topologies and stochastic communication noises this paper investigates the average consensus problem of first order discrete time multi agent networks in uncertain communication environments each agent can only use its own and neighbors information to design its control input to attenuate the communication noises a distributed stochastic approximation type protocol is used by using probability limit theory and algebraic graph theory consensus conditions for this kind of protocols are obtained a for the case of fixed topologies a necessary and sufficient condition for mean square average consensus is given which is also sufficient for almost sure consensus b for the case of time varying topologies sufficient conditions for mean square average consensus and almost sure consensus are given respectively especially if the network switches between jointly containing spanning tree instantaneously balanced graphs then the designed protocol can guarantee that each individual state converges both almost surely and in mean square to a common random variable whose expectation is right the average of the initial states of the whole system and whose variance describes the static maximum mean square error between each individual state and the average of the initial states of the whole system

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

fixed time synchronization of complex networks with impulsive effects via nonchattering control dealing with impulsive effects is one of the most challenging problems in the field of fixed time control in this paper we solve this challenging problem by considering fixed time synchronization of complex networks cns with impulsive effects by designing a new lyapunov function and constructing comparison systems a sufficient condition formulated by matrix inequalities is given to ensure that all the dynamical subsystems in the cns are synchronized with an isolated system in a settling time which is independent of the initial values of both the cns and the isolated system then by partitioning impulse interval and using the convex combination technique sufficient conditions in terms of linear matrix inequalities are provided our synchronization criteria unify synchronizing and desynchronizing impulses compared with the existing controllers for fixed time and finite time techniques the designed controller is continuous and does not include any sign function and hence the chattering phenomenon in most of the existing results is overcome an optimal algorithm is proposed for the estimation of the settling time numerical examples are given to show the effectiveness of our new results

multi agent consensus with relative state dependent measurement noises in this note the distributed consensus corrupted by relative state dependent measurement noises is considered each agent can mea sure or receive its neighbors state information with random noises whose intensity is a vector function of agents relative states by investigating the structure of this interaction and the tools of stochastic differential equa tions we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain the number of agents and the noise intensity function to ensure mean square m s and almost sure a s consensus and quantify the convergence rate and the steady state error especially for the case with homogeneous communication and control channels a necessary and sufficient condition to ensure m s consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology but only depends on the number of nodes and the noise coefficient constant for symmetric measurement models the almost sure convergence rate is estimated by the iterated logarithm law of brownian motions index terms distributed consensus distributed coordination fading channel measurement noises multi agent system

numerical approach to the non linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems the article is devoted to the analysis of the approximate controllability of a given type of second order infinite dimensional system defined by the hyperbolic type partial differential equation with dirichlet like type zero boundary conditions defined in the n dimensional rectangular prism following this aim spectral theory for linear unbounded operators is involved new numerical algorithms for solving some kind of non linear diofantic equations corresponding with the spectral properties of the unbounded operators are presented and proved an algorithm is optimized for eliminating the symmetrical solutions not interesting in the scope of verification of the controllability next the limit which shows that the direct application of the analytical methods of verifying the controllability in the numerical approach is impossible is proved finally so called partial approximate controllability is defined and the numerical algorithm for its verification is presented finally proven theorems are applied

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

robustifying convex risk measures for linear portfolios a nonparametric approach this paper introduces a framework for robustifying convex law invariant risk measures the robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure which represent the investors beliefs about the distribution of future asset losses it is shown that under mild conditions the infinite dimensional optimization problem of finding the worst case risk can be solved analytically and closed form expressions for the robust risk measures are obtained using these results robust versions of several risk measures including the standard deviation the conditional value at risk and the general class of distortion functionals are derived the resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems and a numerical study shows that in most cases they perform significantly better out of sample than their nonrobust variants in terms of risk expected losses and turnover

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

h consensus control with spectrum constraints for stochastic multi agent systems subject to x u v dependent noises abstract this paper is concerned with the problem of h consensus control with spectrum constraints for continuous time multi agent systems mass with x u v dependent noises by means of the spectrum technique the definition of stability is introduced and the relationship between the convergence rate of state responses and the stability is discussed for the closed loop system the aim of the addressed problem is to design a static output feedback controller such that the closed loop system is stable and the prescribed h performance requirement is satisfied in virtue of the stochastic analysis method and kronecker product sufficient conditions are obtained for the existence of h consensus controller two examples are provided to illustrate the availability of the proposed techniques

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

brief paper stability analysis for multi agent systems using the incidence matrix quantized communication and formation control the spectral properties of the incidence matrix of the communication graph are exploited to provide solutions to two multi agent control problems in particular we consider the problem of state agreement with quantized communication and the problem of distance based formation control in both cases stabilizing control laws are provided when the communication graph is a tree it is shown how the relation between tree graphs and the null space of the corresponding incidence matrix encode fundamental properties for these two multi agent control problems