11/27/2006 Massachusetts Institute of Technology Outline - PDF document

11/27/2006 Massachusetts Institute of Technology Outline • Motivation Example Generalized Conflict Learning for • Problem Formulation Hybrid Discrete/Linear Optimization • The GCD-BB algorithm • Empirical Evaluation • Conclusion & Future Work Hui Li and Brian Williams MIT June 6, 2005 ICAPS Workshop on Plan Execution 2 Motivation Motivation Receding horizon The continuous model-based execution problem: continuous planner: • A dynamic system (a plant ) • A desired evolution of the plant state over time (a temporally flexible state plan ) • A consistent control sequence Fire fighting example [Léauté-AAAI-05] The state plan of the fire fighting example: • Two UAVs • A fire to extinguish • Avoid obstacles • Drop water • Assess the damage 3 4 Encodings Disjunctive Linear Programs In [Léauté-AAAI-05] the plant and the state plan within each limited horizon are Definition: encoded in disjunctive linear programs (DLPs) [Balas-ADM-79]. • State plan encoding example i j clause Example: • Plant model encoding example A fast algorithm is needed to solve the DLPs for each horizon. Generalized Conflict-Directed Branch and Bound 5 6 1

11/27/2006 Generalized Conflict-Directed Other Formulations Branch and Bound DLP: Binary Integer Program (BIP): • Building upon Branch and Bound (B&B) • Two key features: – Generalized Conflict Learning • infeasibility • sub-optimality • LCNF [Wolfman-IJCAI-99] “Trigger” linear inequalities with propositional variables – Forward Conflict-Directed Search • constituent kernel • Mixed Logical Linear Programs (MLLPs) • kernel [Hooker-JDAM-99] • DLP candidate - Generalization from LCNF - optimization - variables over finite domain - logic forms other than CNF 7 8 Generalized Conflict-Directed Branch and Bound Branch and Bound • Discrete + Continuous For DLPs • Building upon Branch and Bound (B&B) • “Branch” For DLPs min -x-3y s.t. x ≤ 200 root y ≤ 200 • “Bound” minimize -x-3y s.t. x ≤ 200 x ≤ 100 y ≤ 30 y ≤ 200 • Two key features: For BIPs min -x-3y y ≤ 30 V x ≤ 100 min -x-3y A 2 A 1 y ≤ 20 V x ≤ 10 s.t. x ≤ 200 s.t. x ≤ 200 y ≤ 200 x ≥ 80 V x ≥ 30 V y ≤ 0 y ≤ 20 y ≤ 200 x ≤ 10 x ≤ 10 x ≤ 100 – Generalized Conflict Learning y ≤ 30 min -x-3y min -x-3y s.t. x ≤ 200 min -x-3y B 2 B 1 s.t. x ≤ 200 y ≤ 200 • infeasibility y ≤ 200 s.t. x ≤ 200 x ≤ 100 Incumbent x ≤ 100 y ≤ 200 • sub-optimality. x ≤ 10 y ≤ 30 y ≤ 20 -160 x ≤ 10 y ≤ 0 x ≥ 80 an upper bound x ≥ 30 – Forward Conflict-Directed Search C 3 C 2 C 1 • constituent kernel min -x-3y min -x-3y min -x-3y • kernel s.t. x ≤ 200 s.t. x ≤ 200 s.t. x ≤ 200 y ≤ 200 y ≤ 200 y ≤ 200 • DLP candidate p' is a relaxed LP of an optimization x ≤ 100 x ≤ 100 x ≤ 100 x ≤ 10 x ≤ 10 problem p, if the feasible region of p’ x ≤ 10 y ≤ 0 x ≥ 30 x ≥ 80 contains the feasible region of p, and they have the same objective function. a lower bound 9 10 Generalized Conflict Learning Minimal Conflict Extraction • Infeasibility conflict • Extract minimal conflicts rather than any conflicts • Novel methods based on duality theory An relaxed LP An relaxed LP • No overhead LPs incurred LP Solver An infeasibility conflict, Reduce the constraint matrix to linearly LP Solver since the constraints are not independent rows satisfiable for any value of x. Reduce the constraint matrix to linearly independent rows Run the dual simplex method no A minimal infeasibility conflict If the dual problem is unbounded? Extract sub-optimality Output optimal Run the dual simplex method (an extreme ray is discovered?) solution conflict? no yes yes • Sub-optimality conflict Take the dual solution vector A sub-optimality conflict, since If the dual problem is unbounded? Output optimal (an extreme ray is discovered?) the best solution that satisfies it no solution no If more than n elements of the Identify the minimal conflict with has value -100 > f(x*). yes dual vector is non-zero? all the non-zero elements Identify the constraints of the minimal conflict yes with the non-zero elements of the extreme ray Identify the minimal conflict with A minimal sub-optimality conflict any n of the non-zero elements Incumbent: x*(200,0) 11 12 2

11/27/2006 Generalized Conflict-Directed Forward Conflict-Directed Search Branch and Bound • Backward conflict-directed methods use conflicts to select • Building upon Branch and Bound (B&B) backtrack points and as a cache to prune nodes without testing consistency. – dependency-directed backtracking [Stallman77] • Two key features: – conflict-directed backjumping [Prosser93] – dynamic backtracking [Ginsberg93] – Generalized Conflict Learning – LPSAT [Wolfman99]. • infeasibility • Forward conflict-directed search guides the forward step of • sub-optimality. search away from regions of the state space that are ruled out – Forward Conflict-Directed Search by known conflicts [Williams - CD-A* - JDAM05]. • constituent kernel • Our experimental results on a range of cooperative vehicle plan • kernel execution problems show that forward conflict-directed search • DLP candidate significantly outperforms backtrack search with conflicts. 13 14 Example Forward Conflict-Directed Search Conflicts Constituent Kernels minimize -x-3y min -x-3y s.t. x ≤ 200 { b 2 , c 2 } { {¬b 2 }, {¬c 2 } } s.t. (u 1 )x ≤ 200 y ≤ 200 (u 2 )y ≤ 200 For DLPs { b 2 , c 1 } { {¬b 2 }, {¬c 1 } } root (a 1 )y ≤ 30 V (a 2 )x ≤ 100 (b 1 )y ≤ 20 V (b 2 )x ≤ 10 y ≤ 30 A constituent kernel is a minimal description of x ≤ 100 min -x-3y (c 1 )x ≥ 80 V (c 2 )x ≥ 30 V (c 3 )y ≤ 0 Generate Constituent s.t. x ≤ 200 Kernels { ¬b 2 } the states that resolve a conflict. In the context min -x-3y y ≤ 200 Given the sets of constituent kernels from s.t. x ≤ 200 A 2 A 1 Kernels of DLPs, a constituent kernel of a conflict is a y ≤ 30 { ¬c 1 , ¬c 2 } y ≤ 200 multiple unresolved conflicts, kernels are x ≤ 100 y ≤ 20 linear inequality that is the negation of a linear generated, each of which resolves all the x ≤ 10 DLP Candidates constraint contained in the conflict. A DLP candidate is generated for each kernel. conflicts, by combining the constituent kernels B 2 B 1 The ones that are propositionally unsatisfiable u 1 u 1 Incumbent using minimal set covering. Conflict {x ≥ 80,x ≤ 10} → {x ≤ 80},{x ≥ 10} u 2 are pruned and the DLP is simplified, using a Generate Kernels u 2 -160 a 1 a 1 fast unit propagation test before solving any ¬c 1 y ≤ 0 x ≥ 80 ¬b 2 x ≥ 30 ¬c 2 relaxed LP. b 1 V b 2 For each unresolved conflict, a set of constituent min -x-3y min -x-3y b 1 V b 2 s.t. x ≤ 200 c 1 V c 2 V c 3 C 3 s.t. x ≤ 200 kernels are generated. C 2 c 1 V c 2 V c 3 C 1 y ≤ 200 y ≤ 200 min -x-3y min -x-3y y ≤ 30 min -x-3y y ≤ 30 unit propagation Generate DLP Candidates s.t. x ≤ 200 s.t. x ≤ 200 s.t. x ≤ 200 y ≤ 20 y ≤ 0 y ≤ 200 y ≤ 200 y ≤ 200 x ≤ 100 x ≤ 100 u 1 u 1 x ≤ 100 x ≤ 10 x ≤ 10 x ≤ 10 u 2 u 2 y ≤ 0 x ≥ 30 x ≥ 80 a 1 a 1 b 1 c 3 Conflicts: {x ≤ 10, x ≥ 30} {x ≤ 10, x ≥ 80} c 1 V c 2 V c 3 b 1 V b 2 {b 2 , c 2 } {b 2 , c 1 } 15 16 Best-First Search (BFS) v.s. Empirical Evaluation Depth-First Search (DFS) Test problems: Model-based temporal plan execution for cooperative • BFS is more efficient than DFS in time. vehicles [Léauté-AAAI-05]. Comparisons: GCD-BB v.s. BIP-BB algorithmic variants of GCD-BB • BFS can take dramatically more memory space than DFS. Measures: Computation time - number of LPs & average LP size Memory use - maximum queue length BI P-BB v.s. GCD-BB • With conflict learning and forward conflict-directed search, BFS takes similar memory space to DFS. 9000 8000 7000 6000 • The concept of sub-optimality is rooted in maintaining an BIP-BB Series1 5000 Series2 DLP+BFS+Infeas incumbent. Hence, it can be applied to DFS but not to BFS. 4000 Series3 DLP+DFS+Infeas+Sub 3000 2000 1000 0 80/36 700/144 1492/300 2456/480 1 2 3 4 Clause / Variable 17 18 3

11/27/2006 Empirical Evaluation Empirical Evaluation W ithout conflicts v.s. w ith conflicts BFS v.s. DFS 3000 800 2500 700 600 2000 DLP+BFS+Without Conflicts 500 BFS+ Infeas 1500 DFS+ Infeas 400 DFS+ Infeas+ Sub 300 DFS+ Sub 1000 200 500 DLP+BFS+With Conflicts 100 0 0 80/36 700/144 1492/300 2456/480 1 2 3 4 80/36 700/144 1492/300 2456/480 1 2 3 4 Clause / Variable Backtrack w ith conflicts v.s. Forw ard conflict- Clause / Variable BFS v.s. DFS directed search 450 1600 400 1400 350 1200 300 BFS+ Without Conflicts 1000 250 DLP+BFS+Backtrack BFS+ Infeas 800 200 DFS+ Without Conflicts DFS+ Infeas+ Sub 150 600 100 400 DLP+BFS+Forward Search 50 200 0 0 80/36 700/144 1492/300 2456/480 1 2 3 4 80/36 700/144 1492/300 2456/480 1 2 3 4 Clause / Variable Clause / Variable 19 20 Conclusion Future Work • Run GCD-BB on a range of well-known benchmark Generalized Conflict-Directed Branch and Bound (GCD-BB) problems, and compare its actual runtime against that of BIP- – branch and bound for DLPs BB. – generalized conflict learning • Study empirically the reason why sub-optimality conflicts do – forward conflict-directed search not speed up search as much as infeasibility conflicts. • Apply GCD-BB to a more general form of HDLOPs than DLPs. An order of magnitude speed-up over BIP-BB. • Extend conflict learning to non-linear programming. 21 22 4