Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio

Robustness, stability, recoverability, and reliability in constraint satisfaction problems

The final publication is available at Springer via http://dx.doi.org/10.1007/s10115-014-0778-3Many real-world problems in Artificial Intelligence (AI) as well as in other areas of computer science and engineering can be efficiently modeled and solved using constraint programming techniques. In many real-world scenarios the problem is partially known, imprecise and dynamic such that some effects of actions are undesired and/or several un-foreseen incidences or changes can occur. Whereas expressivity, efficiency and optimality have been the typical goals in the area, there are several issues regarding robustness that have a clear relevance in dynamic Constraint Satisfaction Problems (CSP). However, there is still no clear and common definition of robustness-related concepts in CSPs. In this paper, we propose two clearly differentiated definitions for robustness and stability in CSP solutions. We also introduce the concepts of recoverability and reliability, which arise in temporal CSPs. All these definitions are based on related well-known concepts, which are addressed in engineering and other related areas.This work has been partially supported by the research project TIN2013-46511-C2-1 (MINECO, Spain). We would also thank the reviewers for their efforts and helpful comments.Barber Sanchís, F.; Salido Gregorio, MA. (2015). Robustness, stability, recoverability, and reliability in constraint satisfaction problems. Knowledge and Information Systems. 44(3):719-734. https://doi.org/10.1007/s10115-014-0778-3S719734443Abril M, Barber F, Ingolotti L, Salido MA, Tormos P, Lova A (2008) An assessment of railway capacity. Transp Res Part E 44(5):774–806Barber F (2000) Reasoning on intervals and point-based disjunctive metric constraints in temporal contexts. J Artif Intell Res 12:35–86Bartak R, Salido MA (2011) Constraint satisfaction for planning and scheduling problems. Constraints 16(3):223–227Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53Climent L, Wallace R, Salido M, Barber F (2013) Modeling robustness in CSPS as weighted CSPS. In: Integration of AI and OR techniques in constraint programming for combinatorial optimization problems CPAIOR 2013, pp 44–60Climent L, Wallace R, Salido M, Barber F (2014) Robustness and stability in constraint programming under dynamism and uncertainty. J Artif Intell Res 49(1):49–78Dechter R (1991) Temporal constraint network. Artif Intell 49:61–295Hazewinkel M (2002) Encyclopaedia of mathematics. Springer, New YorkHebrard E (2007) Robust solutions for constraint satisfaction and optimisation under uncertainty. PhD thesis, University of New South WalesHebrard E, Hnich B, Walsh T (2004) Super solutions in constraint programming. In: Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (CPAIOR-04), pp 157–172Jen E (2003) Stable or robust? What’s the difference? Complexity 8(3):12–18Kitano H (2007) Towards a theory of biological robustness. Mol Syst Biol 3(137)Liebchen C, Lbbecke M, Mhring R, Stiller S (2009) The concept of recoverable robustness, linear programming recovery, and railway applications. In: LNCS, vol 5868Papapetrou P, Kollios G, Sclaroff S, Gunopulos D (2009) Mining frequent arrangements of temporal intervals. Knowl Inf Syst 21:133–171Rizk A, Batt G, Fages F, Solima S (2009) A general computational method for robustness analysis with applications to synthetic gene networks. Bioinformatics 25(12):168–179Rossi F, van Beek P, Walsh T (2006) Handbook of constraint programming. Elsevier, New YorkRoy B (2010) Robustness in operational research and decision aiding: a multi-faceted issue. Eur J Oper Res 200:629–638Szathmary E (2006) A robust approach. Nature 439:19–20Verfaillie G, Schiex T (1994) Solution reuse in dynamic constraint satisfaction problems. In: Proceedings of the 12th national conference on artificial intelligence (AAAI-94), pp 307–312Wallace R, Grimes D, Freuder E (2009) Solving dynamic constraint satisfaction problems by identifying stable features. In: Proceedings of international joint conferences on artificial intelligence (IJCAI-09), pp 621–627Wang D, Tse Q, Zhou Y (2011) A decentralized search engine for dynamic web communities. Knowl Inf Syst 26(1):105–125Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New YorkZhou Y, Croft W (2008) Measuring ranked list robustness for query performance prediction. Knowl Inf Syst 16:155–17

An Efficient Algorithm for Mining Frequent Sequence with Constraint Programming

The main advantage of Constraint Programming (CP) approaches for sequential pattern mining (SPM) is their modularity, which includes the ability to add new constraints (regular expressions, length restrictions, etc). The current best CP approach for SPM uses a global constraint (module) that computes the projected database and enforces the minimum frequency; it does this with a filtering algorithm similar to the PrefixSpan method. However, the resulting system is not as scalable as some of the most advanced mining systems like Zaki's cSPADE. We show how, using techniques from both data mining and CP, one can use a generic constraint solver and yet outperform existing specialized systems. This is mainly due to two improvements in the module that computes the projected frequencies: first, computing the projected database can be sped up by pre-computing the positions at which an symbol can become unsupported by a sequence, thereby avoiding to scan the full sequence each time; and second by taking inspiration from the trailing used in CP solvers to devise a backtracking-aware data structure that allows fast incremental storing and restoring of the projected database. Detailed experiments show how this approach outperforms existing CP as well as specialized systems for SPM, and that the gain in efficiency translates directly into increased efficiency for other settings such as mining with regular expressions.Comment: frequent sequence mining, constraint programmin