12/7/2006. Outline. Generalized Conflict Learning for Hybrid Discrete/Linear Optimization. Forward Conflict-Directed Search

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2/7/26 Massachusetts Institute of Technology Generalized Conflict Learning for Hybrid Discrete/Linear Optimization Hui Li and Brian Williams Computer Science and Artificial Intelligence Laboratory

1 2/7/26 Massachusetts Institute of Technology Generalized Conflict Learning for Hybrid Discrete/Linear Optimization Hui Li and Brian Williams Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Oct. 5 th, 25 Forward Conflict-Directed Search Backward conflict-directed search uses conflicts to select backtrack points and as a cache used to prune nodes. dependency-directed backtracking [Stallman-Sussman-77] conflict-directed backjumping [Prosser-93] dynamic backtracking [Ginsberg-93] LPSAT [Wolfman-Weld-99]. Forward conflict-directed search guides the forward step of search away from regions of the state space that are ruled out by known conflicts Conflict-directed A* [Williams-Nayak-AAAI96, Williams-Ragno-JDAM]. Assumption-based DDBT [dekleer-williams AAAI86, IJCAI89], Factor Out Failure[Freuder-IJCAI-95 Candidate Generation [dekleer-williams-aij87, Reiter-AIJ87] Introduce Generalized Forward Conflict-directed Search on Hybrid Discrete/Linear Optimization Experiments on cooperative vehicle plan execution problems demonstrates that the approach significantly outperforms branch and bound using conflicts on backtracking. 2 9 s Self-Repairing Explorers Solve COPs Using Forward, Conflict-directed Best First Search 3 Deep Space Remote Agent Experiment (May, 999) Livingstone Model-based Execution System [Williams & Nayak, AAAI96] State estimates Mode Estimation: Tracks likely States Observations Plant Commands State goals Mode Reconfiguration: Tracks least-cost state goals Optimal Satisfiability Problem OpSat uses conflicts (nogoods) in the forward direction, to substantially improve Best-first Search (CD A*) [Williams-Nayak-AAAI96,Williams-Ragno-JDAM]. Model: HMMs + CSPs 4 Increasing Cost Conflict 3 [ ] x ij = v ij Conflict Infeasible Conflict 2 Feasible s Plan-driven Agile Systems Solve Hybrid Discrete/Linear Optimization Problems via Forward Conflict-directed Search [Leaute & Williams, AAAI5] To put out the Burbank wildfires,... UAV Starts at home; {Gets fuel & water; drops on fire-} [, 5]; {Gets fuel & water; drops on fire-2} [2, 6]; Returns home. Hybrid Discrete/Linear Optimization Problems Disjunctive Linear Programs (DLPs) [Balas-ADM-79] Binary Integer Programming LCNF [Wolfman-IJCAI-99] Mixed Logical Linear Programs (MLLPs) [Hooker-JDAM-99] How do we generalize forward conflict-directed search to HDLOPs? Generalized Conflict-directed Branch and Bound (GCD-BB) s Plan-driven Agile Systems Solve Hybrid Discrete/Linear Optimization Problems via Forward Conflict-directed Search [Hoffman & Williams, ICAPS5] Hybrid Discrete/Linear Gait Poses Optimization Problems Disjunctive Linear Programs (DLPs) l r r2 l l r2 r2 l2 [Balas-ADM-79] Foot placement Binary Integer Programming l l2 Lat LCNF [Wolfman-IJCAI-99] r r2 Fwd Mixed Logical Linear Programs (MLLPs) [Hooker-JDAM-99] How do we generalize forward conflict-directed search to HDLOPs? Generalized Conflict-directed Branch and Bound (GCD-BB)

2 2/7/26 Disjunctive Linear Programs [Balas 79] Generalized Conflict-directed B&B Definition: Example: i j clause Extends traditional Branch & Bound:. Constructs conflicts from search tree nodes found Infeasible or Sub-optimal 2. Uses conflicts to guide forward search away from infeasible and sub-optimal states. 3. Performs induced unit clause relaxation Deduces (some) entailed unit clauses. 7 Demonstrates substantial performance improvement For both Best-first and depth-first Branch & Bound search, In terms of speed and memory usage. Compared with BIPs BFS and B&B without conflicts Backup on conflicts. 8 Optimal Satisfiability Problems 9 Diagnosis A B C D E [ ] x ij = v ij Or X F And Or2 Y And2 G Or3 Z f( x) = fi( xi) i f i Or Or2 Or3 And And2 fi A-G X-Z G U Conflict-directed A* Increasing Cost Select optimal state outside conflicts at each step. Conflict Infeasible Conflict 2 Conflict-directed A* Select optimal state outside conflicts at each step. Feasible subregions described by kernel assignments. Use conflicts to search for kernel assignment Increasing Cost containing the best cost candidate. Conflict Infeasible Conflict 2 Conflict 3 Feasible [Williams-Nayak-AAAI96, Williams-Ragno-JDAM?, dekleer-williams-ijca89] Conflict 3 Kernel 3 Kernel 2 Feasible Kernel 2

2/7/26 Generating candidate after two iterations Diagnosis A Or X B And F C Or2 Y D And2 G E Or3 Z Conflicts: {O=G, O2=G, A=G} is inconsistent {O=G, A=G A2=G} is inconsistent Constituent Kernels:

3 2/7/26 Generating candidate after two iterations Diagnosis A Or X B And F C Or2 Y D And2 G E Or3 Z Conflicts: {O=G, O2=G, A=G} is inconsistent {O=G, A=G A2=G} is inconsistent Constituent Kernels: {O=U, O2=U, A=U} at least one holds (Resolve one conflict - All states outside conflict) {O=U, A=U A2=U} at least one holds 3 The kernel assignments are the minimal coverings of the constituent kernels. [dekleer-williams-aij87, Reiter-AIJ87] The best kernel is found through A* search of the covering tree. [Williams-Nayak-AAAI96, Williams-Ragno-JDAM] (Kernels resolve ALL conflicts - ALL states outside ALL conflicts) st iteration {A=U} {M=U} {M2=U} 3rd iteration {A=U, M=U, M2=U} 2nd iteration A2=U Constituent Kernels M3=U {A=U, A2=U, M=U, M3=U} {A=U} {M=U} {M=U, A2=U} {M2=U, M3=U}. Branch and Bound for DLPs 2. Induced Unit Clause Relaxation 3. Generalized Conflict Learning 4. Forward Conflict-directed Search 5. Branch and Bound for DLPs. Branch on subproblems. 2. Maintain running best soln in incumbent. Binary variables 3. Bound cost using relaxed problems. Clauses 4. Prune infeasible and suboptimal branches. For DLPs For DLPs For BIPs root minimize -x-3y x y 3 y 3 V x y 2 V x A 2 A x 8 V x 3 V y x y 2 x x y 3 B 2 B x Incumbent x x y 2 y 3-6 x y x 3 x 8an upper bound C 3 x x y 6 C 2 x x x 3 C x x x 8 2. Induced Unit Clause Relaxation BIP relaxation: Binary constraint x {,} Continuous constraint x Problem: adding binary variables and constraints increases the dimensionality of the search problem. Simple DLP relaxation: Remove all non-unit clauses from the DLP Problem: ignoring all non-unit clauses forms a weak relaxation. Induced unit clause relaxation: Strengthen by adding entailed unit clauses Approach: Relax DLP to a propositional theory + apply unit propagation. (alternatively, failed literals) 7 2. Induced Unit Clause Relaxation Example: DP at a sub-problem before relaxation: s.t. x 2 y 2 x y 5 v x x > 8 v x 3 v y s.t. a a b c d e v f v g Relax to propositional clauses Relax nonunit clauses s.t. a b a c d 8 s.t. a b a c d v c e v f v g Reintroduce the linear inequalities Unit propagate Relaxed Problem: s.t. x 2 y 2 x y 5 3

2/7/26 3. Generalized Conflict Learning Extracts conflicts whenever branch pruned: Infeasibility conflict 3.

A minimal infeasibility conflict o Minimal sub-optimality conflicts: The active constraint set at the solution, assuming no degeneracy.

Set of constraints, all of whose feasible states are worse than the incumbent (- > f(x*)).

4 2/7/26 3. Generalized Conflict Learning Extracts conflicts whenever branch pruned: Infeasibility conflict 3. Minimal Conflict Extraction Extracted based on duality theory: Sub-optimality conflict Set of constraints that are not satisfiable for any value of x. A minimal infeasibility conflict o Minimal sub-optimality conflicts: The active constraint set at the solution, assuming no degeneracy. The active constraint set is identified by the non-zero terms of the optimal dual vector. [Bertsimas&Tsitsiklis97]. Set of constraints, all of whose feasible states are worse than the incumbent (- > f(x*)). o Minimal infeasibility conflicts: The extreme rays of the cone formed by the modified dual of the original LP [Gleeson&Ryan9]. A minimal sub-optimality conflict Incurs additional LP per conflict Incumbent: x*(2,) Forward Conflict-directed Search minimize -x-3y s.t. (u )x 2 () (a )y 3 V (a 2)x (b )y 2 V (b 2)x (c )x 8 V (c 2)x 3 V (c 3)y x A 2 x x C 3 x x y y For DLPs x 3 B 2 C 2 x x x 3 x 8 root y 2 B Incumbent -6 C x x x 8 y 3 A Conflicts: {x, x 3} {x, x 8} {b 2, c 2} {b 2, c } y 3 y 3 y 3 y 2 y 2 Conflicts { b 2, c 2 } { b 2, c } DLP Candidates Constituent Kernels { { b 2 }, { c 2 } } { { b 2 }, { c } } Kernels { b 2 } { c, c 2 } u a b 2 b V b 2 c V c 2 V c 3 u a b c V c 2 V c 3 unit propagation u a c c 2 b V b 2 c V c 2 V c 3 u a c 3 b V b 2 If infeasible Then prune Else unit clause relax 22 Empirical Evaluation A Mars Plane Scouting Valle Marineris Test problems: Model-based temporal plan execution for cooperative vehicles [Léauté-Williams-AAAI5]. Comparisons: GCD-BB v.s. BIP-BB v.s. algorithmic variants of GCD-BB Measures: Computation time - number of LPs & average LP size Memory use - maximum queue length Model of Vehicle & Terrain encode as disjunctive LP solve up to limited horizon extract control sequence

5 2/7/26 Generalized forward conflict-directed search on DLPs improves runtime over BIP-BB. Conflicts significantly improve best-first search runtime. 9 BIP-BB v.s. GCD-BB 3 Without conflicts v.s. with conflicts BIP-BB Series Series2 DLP+BFS+Infeas Series3 DLP+DFS+Infeas+Sub DLP+BFS+Without Conflicts DLP+BFS+With Conflicts 8/36 7/ / /48 4 8/36 7/ / / Forward conflict-directed search significantly outperforms conflicts used on backtrack. Conflicts significantly improve Best-first search memory use Backtrack with conflicts v.s. Forward conflictdirected search 45 BFS v.s. DFS DLP+BFS+Backtrack DLP+BFS+Forward Search 8/36 7/ / / /36 7/ / /48 4 BFS+Without Conflicts BFS+Infeas DFS+Infeas+Sub Memory performance similar to depth-first-search w conflicts 27 But BFS is not an anytime algorithm. Use Depth-first B&B with conflicts for infeasibility and suboptimality BFS v.s. DFS BFS+Infeas DFS+Infeas DFS+Infeas+Sub DFS+Sub Future Work Study influence of cutting planes, such as Bender s cuts, on performance. Study empirically why sub-optimality conflicts do not speed up search as much as infeasibility conflicts. Apply GCD-BB to non-clausal forms of HLLPs. Generalize to non-linear programs. 8/36 7/ / /484 Runtime performance similar to conflict-directed BFS Significantly better than B&B with infeasible conflicts alone

6 2/7/26 Generalized-Conflict-directed Branch & Bound. Learns conflicts from search tree nodes found Infeasible or Sub-optimal 2. Branches using conflicts to guide forward search away from infeasible and sub-optimal states. 3. Bounds using unit clause relaxation Deduces (some) entailed unit clauses. Demonstrates substantial performance improvement For both Best-first and depth-first Branch & Bound search, In terms of speed and memory usage. Compared with BIPs Traditional BFS and B&B Backup on conflicts. References [Gleeson&Ryan9] J. Gleeson and J. Ryan, Identifying minimally inconsistent subsystems of inequalities, ORSA J. Computing 2, 99. [Bertsimas&Tsitsiklis97] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific, 997. [Stallman77] Stallman, R. and Sussman, G.J., Forward Reasoning and Dependency-Directed Backtracking in a System for Computer-Aided Circuit Analysis. J. of Artificial Intelligence [Prosser93] Prosser, P., Hybrid Algorithms for the Constraint Satisfaction Problem. J. of Computational Intelligence, 9(3),993. [Ginsberg93] Ginsberg, M., Dynamic Backtracking. J. of Artificial Intelligence Research,, 993. [Wolfman99] Wolfman, S. and Weld, D., The LPSAT Engine & Its Application to Resource Planning. IJCAI [Williams5] Williams, B. and Ragno, R., Conflict-directed A* and its Role in Model-based Embedded Systems. JDAM, to appear 25. [Katsirelos3] Katsirelos, G. and Bacchus, F., Unrestricted Nogood Recording in CSP Search. CP, 23. [Léauté5] Léauté, T. and Williams, B., Coordinating Agile Systems Through The Model-based Execution of Temporal Plans. AAAI

Solving Hybrid Decision-Control Problems Through Conflict-Directed Branch & Bound

Solving Hybrid Decision-Control Problems Through Conflict-Directed Branch & Bound by Raj Krishnan Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the