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1. A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems

   Chapman, Margaret P. ; Lacotte, Jonathan ; Tamar, Aviv ; Lee, Donggun ; Smith, Kevin M. ; Cheng, Victoria ; Fisac, Jaime F. ; Jha, Susmit ; Pavone, Marco ; Tomlin, Claire J. ( January 2019 , ArXiv.org)

   A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set as a set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set, and provide theoretical arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivitymore »from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis).« less
2. A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems

   Chapman, Margaret P. ; Jonathan, Lacotte ; Aviv, Tamar Donggun ; Lee, Kevin M. ; Jaime, F. Fisac ; Jha, Susmit ; Marco, Pavone ; Claire, Tomlin ( July 2019 , AMERICAN CONTROL CONFERENCE)

   A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set as a set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set, and provide theoretical arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivitymore »from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis).« less
3. Reachability Analysis Using Message Passing over Tree Decompositions

   Sankaranarayanan, Sriram ( July 2020 , International Conference on Computer Aided Verification)

   In this paper, we study efficient approaches to reachability analysis for discrete-time nonlinear dynamical systems when the dependencies among the variables of the system have low treewidth. Reachability analysis over nonlinear dynamical systems asks if a given set of target states can be reached, starting from an initial set of states. This is solved by computing conservative over approximations of the reachable set using abstract domains to represent these approximations. However, most approaches must tradeoff the level of conservatism against the cost of performing analysis, especially when the number of system variables increases. This makes reachability analysis challenging for nonlinear systems with a large number of state variables. Our approach works by constructing a dependency graph among the variables of the system. The tree decomposition of this graph builds a tree wherein each node of the tree is labeled with subsets of the state variables of the system. Furthermore, the tree decomposition satisfies important structural properties. Using the tree decomposition, our approach abstracts a set of states of the high dimensional system into a tree of sets of lower dimensional projections of this state. We derive various properties of this abstract domain, including conditions under which the original high dimensionalmore »set can be fully recovered from its low dimensional projections. Next, we use ideas from message passing developed originally for belief propagation over Bayesian networks to perform reachability analysis over the full state space in an efficient manner. We illustrate our approach on some interesting nonlinear systems with low treewidth to demonstrate the advantages of our approach.« less
4. Reachability Analysis Using Message Passing over Tree Decompositions.

   https://doi.org/10.1007/978-3-030-53288-8_30  

   Sankaranarayanan, Sriram ( July 2020 , Computer Aided Verification. CAV 2020. Lecture Notes in Computer Science, vol 12224)

   In this paper, we study efficient approaches to reachability analysis for discrete-time nonlinear dynamical systems when the dependencies among the variables of the system have low treewidth. Reachability analysis over nonlinear dynamical systems asks if a given set of target states can be reached, starting from an initial set of states. This is solved by computing conservative over approximations of the reachable set using abstract domains to represent these approximations. However, most approaches must tradeoff the level of conservatism against the cost of performing analysis, especially when the number of system variables increases. This makes reachability analysis challenging for nonlinear systems with a large number of state variables. Our approach works by constructing a dependency graph among the variables of the system. The tree decomposition of this graph builds a tree wherein each node of the tree is labeled with subsets of the state variables of the system. Furthermore, the tree decomposition satisfies important structural properties. Using the tree decomposition, our approach abstracts a set of states of the high dimensional system into a tree of sets of lower dimensional projections of this state. We derive various properties of this abstract domain, including conditions under which the original high dimensionalmore »set can be fully recovered from its low dimensional projections. Next, we use ideas from message passing developed originally for belief propagation over Bayesian networks to perform reachability analysis over the full state space in an efficient manner. We illustrate our approach on some interesting nonlinear systems with low treewidth to demonstrate the advantages of our approach.« less
5. Chance-Constrained Multiple Bin Packing Problem with an Application to Operating Room Planning

   https://doi.org/10.1287/ijoc.2020.1010  

   Wang, Shanshan ; Li, Jinlin ; Mehrotra, Sanjay ( March 2021 , INFORMS Journal on Computing)

   We study the chance-constrained bin packing problem, with an application to hospital operating room planning. The bin packing problem allocates items of random sizes that follow a discrete distribution to a set of bins with limited capacity, while minimizing the total cost. The bin capacity constraints are satisfied with a given probability. We investigate a big-M and a 0-1 bilinear formulation of this problem. We analyze the bilinear structure of the formulation and use the lifting techniques to identify cover, clique, and projection inequalities to strengthen the formulation. We show that in certain cases these inequalities are facet-defining for a bilinear knapsack constraint that arises in the reformulation. An extensive computational study is conducted for the operating room planning problem that minimizes the number of open operating rooms. The computational tests are performed using problems generated based on real data from a hospital. A lower-bound improvement heuristic is combined with the cuts proposed in this paper in a branch-and-cut framework. The computations illustrate that the techniques developed in this paper can significantly improve the performance of the branch-and-cut method. Problems with up to 1,000 scenarios are solved to optimality in less than an hour. A safe approximation based on conditionalmore »value at risk (CVaR) is also solved. The computations show that the CVaR approximation typically leaves a gap of one operating room (e.g., six instead of five) to satisfy the chance constraint. Summary of Contribution: This paper investigates a branch-and-cut algorithm for a chance-constrained bin packing problem with multiple bins. The chance-constrained bin packing provides a modeling framework for applied operations research problems, such as health care, scheduling, and so on. This paper studies alternative computational approaches to solve this problem. Moreover, this paper uses real data from a hospital operating room planning setting as an application to test the algorithmic ideas. This work, therefore, is at the intersection of computing and operations research. Several interesting ideas are developed and studied. These include a strengthened big-M reformulation, analysis of a bilinear reformulation, and identifying certain facet-defining inequalities for this formulation. This paper also gives a lower-bound generation heuristic for a model that minimizes the number of bins. Computational experiments for an operating room planning model that uses data from a hospital demonstrate the computational improvement and importance of the proposed approaches. The techniques proposed in this paper and computational experiments further enhance the interface of computing and operations research.« less