control for safety specifications of systems with imperfect information on a partial order in this paper we consider the control problem for uncertain systems with imperfect information in which an output of interest must be kept outside an undesired region the bad set in the output space the state input output and disturbance spaces are equipped with partial orders the system dynamics are either input output order preserving with output in r2 or given by the parallel composition of input output order preserving dynamics each with scalar output we provide necessary and sufficient conditions under which an initial set of possible system states is safe that is the corresponding outputs are steerable away from the bad set with open loop controls a closed loop control strategy is explicitly constructed which guarantees that the current set of possible system states as obtained from an estimator generates outputs that never enter the bad set the complexity of algorithms that check safety of an initial set of states and implement the control map is quadratic with the dimension of the state space the algorithms are illustrated on two application examples a ship maneuver to avoid an obstacle and safe navigation of an helicopter among buildings

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

a model for stocks dynamics based on a non gaussian path integral abstract we introduce a model for the dynamics of stock prices based on a non quadratic path integral the model is a generalization of ilinski s path integral model more precisely we choose a different action which can be tuned to different time scales the result is a model with a very small number of parameters that provides very good fits of some stock prices and indices fluctuations

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

data driven chance constrained stochastic program in this paper we study data driven chance constrained stochastic programs or more specifically stochastic programs with distributionally robust chance constraints dccs in a data driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters we consider a family of density based confidence sets based on a general phi divergence measure and formulate dcc from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set we derive an equivalent reformulation for dcc and show that it is equivalent to a classical chance constraint with a perturbed risk level we also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general phi divergence measures in several special cases our results can be strengthened such that we can derive closed form expressions for the perturbed risk levels in addition we show that the conservatism of dcc vanishes as the size of historical data goes to infinity furthermore we analyze the relationship between the conservatism of dcc and the size of historical data which can help indicate the value of data finally we conduct extensive computational experiments to test the performance of the proposed dcc model and compare various phi divergence measures based on a capacitated lot sizing problem with a quality of service requirement

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

a dynamic game approach to distributionally robust safety specifications for stochastic systems abstract this paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times subject to the worst possible disturbance distribution in an ambiguity set we propose a dynamic game formulation of constructing such policies and identify conditions under which a non randomized markov policy is optimal based on this existence result we develop a practical design approach to safety oriented stochastic controllers with limited information about disturbance distributions however an associated bellman equation involves infinite dimensional minimax optimization problems since the disturbance distribution may have a continuous density to alleviate computational issues we propose a duality based reformulation method that converts the infinite dimensional minimax problem into a semi infinite program that can be solved using existing convergent algorithms we prove that there is no duality gap and that this approach thus preserves optimality the results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances while a standard stochastic safety verification tool is not

a time dependent hamilton jacobi formulation of reachable sets for continuous dynamic games we describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game the algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time dependent hamilton jacobi isaacs partial differential equation while alternative techniques for computing the reachable set have been proposed the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters because the time dependent equation s solution is continuous and defined throughout the state space methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions a numerical implementation of our formulation is described and has been released on the web its correctness is verified through a two vehicle three dimensional collision avoidance example for which an analytic solution is available

a stochastic games framework for verification and control of discrete time stochastic hybrid systems we describe a framework for analyzing probabilistic reachability and safety problems for discrete time stochastic hybrid systems within a dynamic games setting in particular we consider finite horizon zero sum stochastic games in which a control has the objective of reaching a target set while avoiding an unsafe set in the hybrid state space and a rational adversary has the opposing objective we derive an algorithm for computing the maximal probability of achieving the control objective subject to the worst case adversary behavior from this algorithm sufficient conditions of optimality are also derived for the synthesis of optimal control policies and worst case disturbance strategies these results are then specialized to the safety problem in which the control objective is to remain within a safe set we illustrate our modeling framework and computational approach using both a tutorial example with jump markov dynamics and a practical application in the domain of air traffic management

minimax control of discrete time stochastic systems this paper gives a unified self contained presentation of minimax control problems for discrete time stochastic systems on borel spaces with possibly unbounded costs the main results include conditions for the existence of minimax strategies for finite horizon problems and infinite horizon discounted and undiscounted average cost criteria the results are specialized to control systems with unknown disturbance distributions also known as games against nature two examples illustrate the theory one of them on the mold level control problem which is a key problem in the steelmaking industry

reachability analysis of discrete time systems with disturbances this paper presents new results that allow one to compute the set of states that can be robustly steered in a finite number of steps via state feedback control to a given target set the assumptions that are made in this paper are that the system is discrete time nonlinear and time invariant and subject to mixed constraints on the state and input a persistent disturbance dependent on the current state and input acts on the system existing results are not able to address state and input dependent disturbances and the results in this paper are therefore a generalization of previously published results one of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational geometry software provided the system is piecewise affine and the constraints are polygonal existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances whereas the results in this paper remove this limitation some simple examples are also given that show that even if all the relevant sets are convex and the system is linear convexity of the set of controllable states cannot be guaranteed

verification of discrete time stochastic hybrid systems a stochastic reach avoid decision problem we present a dynamic programming based solution to a probabilistic reach avoid problem for a controlled discrete time stochastic hybrid system we address two distinct interpretations of the reach avoid problem via stochastic optimal control in the first case a sum multiplicative cost function is introduced along with a corresponding dynamic recursion which quantifies the probability of hitting a target set at some point during a finite time horizon while avoiding an unsafe set during each time step preceding the target hitting time in the second case we introduce a multiplicative cost function and a dynamic recursion which quantifies the probability of hitting a target set at the terminal time while avoiding an unsafe set during the preceding time steps in each case optimal reach while avoid control policies are derived as the solution to an optimal control problem via dynamic programming computational examples motivated by two practical problems in the management of fisheries and finance are provided

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

robust solutions of optimization problems affected by uncertain probabilities in this paper we focus on robust linear optimization problems with uncertainty regions defined by divergences for example chi squared hellinger kullback leibler we show how uncertainty regions based on divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector such problems frequently occur in for example optimization problems in inventory control or finance that involve terms containing moments of random variables expected utility etc we show that the robust counterpart of a linear optimization problem with divergence uncertainty is tractable for most of the choices of typically considered in the literature we extend the results to problems that are nonlinear in the optimization variables several applications including an asset pricing example and a numerical multi item newsvendor example illustrate the relevance of the proposed approach this paper was accepted by gerard p cachon optimization

novel soft subspace clustering with multi objective evolutionary approach for high dimensional data many conventional soft subspace clustering techniques merge several criteria into a single objective to improve performance however the weighting parameters become important but difficult to set in this paper a novel soft subspace clustering with a multi objective evolutionary approach moeassc is proposed to this problem this clustering method considers two types of criteria as multiple objectives and optimizes them simultaneously by using a modified multi objective evolutionary algorithm with new encoding and operators an indicator called projection similarity validity index psvindex is designed to select the best solution and cluster number experiments on many datasets demonstrate the usefulness of moeassc and psvindex and show that our algorithm is insensitive to its parameters and is scalable to large datasets

hamilton jacobi formulation for reach avoid differential games a new framework for formulating reachability problems with competing inputs nonlinear dynamics and state constraints as optimal control problems is developed such reach avoid problems arise in among others the study of safety problems in hybrid systems earlier approaches to reach avoid computations are either restricted to linear systems or face numerical difficulties due to possible discontinuities in the hamiltonian of the optimal control problem the main advantage of the approach proposed in this paper is that it can be applied to a general class of target hitting continuous dynamic games with nonlinear dynamics and has very good properties in terms of its numerical solution since the value function and the hamiltonian of the system are both continuous the performance of the proposed method is demonstrated by applying it to a case study which involves the target hitting problem of an underactuated underwater vehicle in the presence of obstacles

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

probabilistic reachability and safety for controlled discrete time stochastic hybrid systems in this work probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs a suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations leading to dual algorithms for reachability computations in particular the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired safe region of the state space is characterized in terms of a value function and maximally safe markov policies are determined via dynamic programming these results are of interest not only for safety analysis and design but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems the temperature regulation problem presented in the paper as a case study is one such case

reachability for partially observable discrete time stochastic hybrid systems abstract when designing optimal controllers for any system it is often the case that the true state of the system is unknown to the controller imperfect state information must be taken into account in the controller s design in order to preserve its optimality the same is true when performing reachability calculations to estimate the probability that the state of a stochastic system reaches or stays within some set of interest in a given time horizon it is necessary to find a controller that drives the system to that set with maximum probability given the controller s knowledge of the true state of the system to date little work has been done on stochastic reachability calculations with partially observable states the work that has been done relies on converting the reachability optimization problem to one with an additive cost function for which theoretical results are well known our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value we develop a dynamic programming recursion for the solution of the equivalent perfect information problem proving that the recursion is valid an optimal solution exists and results in the same solution as to the original problem we also show that our results are equivalent to those for the reformulated additive cost problem and so such a reformulation is not required

robust mean covariance solutions for stochastic optimization we provide a method for deriving robust solutions to certain stochastic optimization problems based on mean covariance information about the distributions underlying the uncertain vector of returns we prove that for a general class of objective functions the robust solutions amount to solving a certain deterministic parametric quadratic program we first prove a general projection property for multivariate distributions with given means and covariances which reduces our problem to optimizing a univariate mean variance robust objective this allows us to use known univariate results in the multidimensional setting and to add new results in this direction in particular we characterize a general class of objective functions the so called one or two point support functions for which the robust objective is reduced to a deterministic optimization problem in one variable finally we adapt a result from geoffrion 1967a to reduce the main problem to a parametric quadratic program in particular our results are true for increasing concave utilities with convex or concave convex derivatives closed form solutions are obtained for special discontinuous criteria motivated by bonus and commission based incentive schemes for portfolio management we also investigate a multiproduct pricing application which motivates extensions of our results for the case of nonnegative and decision dependent returns

distributionally robust markov decision processes we consider markov decision processes where the values of the parameters are uncertain this uncertainty is described by a sequence of nested sets that is each set contains the previous one each of which corresponds to a probabilistic guarantee for a different confidence level consequently a set of admissible probability distributions of the unknown parameters is specified this formulation models the case where the decision maker is aware of and wants to exploit some yet imprecise a priori information of the distribution of parameters and it arises naturally in practice where methods for estimating the confidence region of parameters abound we propose a decision criterion based on distributional robustness the optimal strategy maximizes the expected total reward under the most adversarial admissible probability distributions we show that finding the optimal distributionally robust strategy can be reduced to the standard robust mdp where parameters are known to belong to a single uncertainty set hence it can be computed in polynomial time under mild technical conditions

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh