Generalized Conflict Learning for Hybrid Discrete/Linear Optimization

Conflict-directed search algorithms have formed the core of practical, model-based reasoning systems for the last three decades. At the core of many of these applications is a series of discrete constraint optimization problems and a conflict-directed search algorithm, which uses conflicts in the forward search step to focus search away from known infeasibilities and towards the optimal feasible solution. In the arena of model-based autonomy, deep space probes have given way to more agile vehicles, such as coordinated vehicle control, which must robustly control their continuous dynamics. Controlling these systems requires optimizing over continuous, as well as discrete variables, using linear as well as logical constraints. This paper explores the development of algorithms for solving hybrid discrete/linear optimization problems that use conflicts in the forward search direction, carried from the conflict-directed search algorithm in model-based reasoning. We introduce a novel algorithm called Generalized Conflict-Directed Branch and Bound (GCD-BB). GCD-BB extends traditional Branch and Bound (B&B), by first constructing conflicts from nodes of the search tree that are found to be infeasible or sub-optimal, and then by using these conflicts to guide the forward search away from known infeasible and sub-optimal states. Evaluated empirically on a range of test problems of coordinated air vehicle control, GCD-BB demonstrates a substantial improvement in performance compared to a traditional B&B algorithm applied to either disjunctive linear programs or an equivalent binary integer programming encoding. This research is funded by The Boeing Company grant MIT-BA-GTA-1 and by NASA grant NNA04CK91A.

[1]  Jonathan Cagan,et al.  Activity Analysis: The Qualitative Analysis of Stationary Points for Optimal Reasoning , 1994, AAAI.

[2]  Brian Charles Williams,et al.  Robust Execution of Temporally Flexible Plans for Bipedal Walking Devices , 2006, ICAPS.

[3]  Robert J. Ragno,et al.  Solving Optimal Satisfiability Problems Through Clause-Directed A* , 2002 .

[4]  María Auxilio Osorio Lama,et al.  Mixed Logical-linear Programming , 1999, Discret. Appl. Math..

[5]  Brian C. Williams,et al.  Diagnosis with Behavioral Modes , 1989, IJCAI.

[6]  Dana S. Nau,et al.  On the Use of Integer Programming Models in AI Planning , 1999, IJCAI.

[7]  Jon Doyle,et al.  A Truth Maintenance System , 1979, Artif. Intell..

[8]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[9]  Edward M. Reingold,et al.  Backtrack programming techniques , 1975, CACM.

[10]  A. Neumaier.  Complete search in continuous global optimization and constraint satisfaction , 2004, Acta Numerica.

[11]  Brian C. Williams,et al.  Coordinating Agile Systems through the Model-based Execution of Temporal Plans , 2005, AAAI.

[12]  Brian C. Williams,et al.  Conflict-directed A* and its role in model-based embedded systems , 2007, Discret. Appl. Math..

[13]  Francesca Rossi,et al.  Principles and Practice of Constraint Programming – CP 2003 , 2003, Lecture Notes in Computer Science.

[14]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Real-World SAT Instances , 1997, AAAI/IAAI.

[15]  Patrick Prosser,et al.  HYBRID ALGORITHMS FOR THE CONSTRAINT SATISFACTION PROBLEM , 1993, Comput. Intell..

[16]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[17]  Raj Krishnan.  Solving hybrid decision-control problems through conflict-directed branch & bound , 2004 .

[18]  Pascal Van Hentenryck,et al.  Safe and tight linear estimators for global optimization , 2005, Math. Program..

[19]  Andreas G. Hofmann,et al.  To appear in ICAPS 2005 Workshop on Plan Execution Safe Execution of Temporally Flexible Plans for Bipedal Walking Devices , 2005 .

[20]  Pascal Van Hentenryck,et al.  Numerica: A Modeling Language for Global Optimization , 1997, IJCAI.

[21]  Rina Dechter,et al.  Enhancement Schemes for Constraint Processing: Backjumping, Learning, and Cutset Decomposition , 1990, Artif. Intell..

[22]  David Detlefs,et al.  Simplify: a theorem prover for program checking , 2005, JACM.

[23]  Brian C. Williams,et al.  Diagnosing Multiple Faults , 1987, Artif. Intell..

[24]  P. Pandurang Nayak,et al.  A Model-Based Approach to Reactive Self-Configuring Systems , 1996, AAAI/IAAI, Vol. 2.

[25]  B. Moor,et al.  Mixed integer programming for multi-vehicle path planning , 2001, 2001 European Control Conference (ECC).

[26]  Brian C. Williams,et al.  Model-based programming of intelligent embedded systems and robotic space explorers , 2003, Proc. IEEE.

[27]  Rina Dechter,et al.  Generalized best-first search strategies and the optimality of A* , 1985, JACM.

[28]  Henry A. Kautz,et al.  State-space Planning by Integer Optimization , 1999, AAAI/IAAI.

[29]  Daniel S. Weld,et al.  The LPSAT Engine & Its Application to Resource Planning , 1999, IJCAI.

[30]  Matthew L. Ginsberg,et al.  Dynamic Backtracking , 1993, J. Artif. Intell. Res..

[31]  Panos Y. Papalambros.  Monotonicity in Goal and Geometric Programming , 1980 .

[32]  Peter J. Stuckey,et al.  Solving linear arithmetic constraints for user interface applications , 1997, UIST '97.

[33]  John N. Hooker,et al.  Logic, Optimization, and Constraint Programming , 2002, INFORMS J. Comput..

[34]  Fahiem Bacchus,et al.  Unrestricted Nogood Recording in CSP Search , 2003, CP.

[35]  Panos Y. Papalambros,et al.  Global Non-Iterative Design Optimization Using Monotonicity Analysis , 1978 .

[36]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Exceptionally Hard SAT Instances , 1996, CP.

[37]  Gerald J. Sussman,et al.  Forward Reasoning and Dependency-Directed Backtracking in a System for Computer-Aided Circuit Analysis , 1976, Artif. Intell..