Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Decomposition for Optimal Cost Partitioning Florian Pommerening 1 oger 1 Malte Helmert 1 Gabriele R¨ Hadrien Cambazard 2 Louis-Martin Rousseau 3 Domenico Salvagnin 4 1 University of Basel, Switzerland 2 Univ. Grenoble Alpes, CNRS, Grenoble INP*, G-SCOP, 38000 Grenoble, France *Institute of Engineering Univ. Grenoble Alpes 3 Polytechnique Montreal, Canada 4 University of Padua, Italy July 14, 2019 1 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Structure In this presentation context: cost partitioning in classical planning Lagrangian decomposition simplified, specialized, ignoring assumptions see paper for details relation to cost partitioning subgradient optimization algorithm to compute optimal cost partitioning without an LP solver 2 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Decomposition 3 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Starting with a Linear Program Problem P Problem P Min c x s.t. Min c x s.t. x ≥ b 1 A 1 x i ≥ b i A i ∀ i rewrite − − − − → . . . x = x i ∀ i A k x ≥ b k x , x i ≥ 0 ∀ i x ≥ 0 4 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Starting with a Linear Program Problem P Problem P Min c x s.t. Min c x s.t. x ≥ b 1 A 1 x i ≥ b i A i ∀ i rewrite − − − − → . . . x = x i ∀ i A k x ≥ b k x , x i ≥ 0 ∀ i x ≥ 0 4 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Relaxation Problem P Problem P ( λ ) c x s.t. Min i λ i ( x i − x ) s.t. Min c x + � x i ≥ b i ∀ i A i relax − − − → x i ≥ b i ∀ i A i x = x i ∀ i x , x i ≥ 0 ∀ i x , x i ≥ 0 ∀ i for violating x = x i Penalty term λ i called Lagrangian multiplier for every choice of λ : value ( P ( λ )) ≤ value ( P ) Lagrangian dual problem: find λ that gives best lower bound best lower bound is perfect here: Max λ P ( λ ) = value ( P ) 5 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Relaxation Problem P Problem P ( λ ) c x s.t. Min i λ i ( x i − x ) s.t. Min c x + � x i ≥ b i ∀ i A i relax − − − → x i ≥ b i ∀ i A i x = x i ∀ i x , x i ≥ 0 ∀ i x , x i ≥ 0 ∀ i for violating x = x i Penalty term λ i called Lagrangian multiplier for every choice of λ : value ( P ( λ )) ≤ value ( P ) Lagrangian dual problem: find λ that gives best lower bound best lower bound is perfect here: Max λ P ( λ ) = value ( P ) 5 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Relaxation Problem P Problem P ( λ ) c x s.t. Min i λ i ( x i − x ) s.t. Min c x + � x i ≥ b i ∀ i A i relax − − − → x i ≥ b i ∀ i A i x = x i ∀ i x , x i ≥ 0 ∀ i x , x i ≥ 0 ∀ i for violating x = x i Penalty term λ i called Lagrangian multiplier for every choice of λ : value ( P ( λ )) ≤ value ( P ) Lagrangian dual problem: find λ that gives best lower bound best lower bound is perfect here: Max λ P ( λ ) = value ( P ) 5 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Lagrangian Decomposition Problem P ( λ ) i λ i ( x i − x ) s.t. Min c x + � x i ≥ b i ∀ i A i x , x i ≥ 0 ∀ i P ( λ ) decomposes into independent subproblems P ( λ ) = � i P i ( λ ) Problem P i ( λ ) Problem P 0 ( λ ) x i s.t. Min λ i � � c − � x s.t. Min i λ i x i ≥ b i A i x ≥ 0 x i ≥ 0 6 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments A closer look at P 0 ( λ ) Problem P 0 ( λ ) � � − � Min c i λ i x s.t. x ≥ 0 if all objective coefficients non-negative: value ( P 0 ( λ )) = 0 otherwise P 0 ( λ ) is unbounded Constraint encoded by P 0 ( λ ) � λ i ≤ c i 7 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Relation to Cost Partitioning 8 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Summarizing Lagrangian Decomposition Lagrangian Dual Problem Max � i P i ( λ ) s.t. � c Original Problem P λ i ≤ i Min c x s.t. Subproblem P i ( λ ) x ≥ b i ∀ i A i x i s.t. Min λ i x ≥ 0 x i ≥ b i A i x i ≥ 0 9 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Cost Partitioning of Operator-Counting Heuristics Optimal Cost Partitioning Max � i h i ( cost i ) s.t. � Heuristic h cost i ≤ cost i Min cost x s.t. Heuristic h i ( cost i ) A i x ≥ b i ∀ i x i s.t. Min cost i x ≥ 0 x i ≥ b i A i x i ≥ 0 10 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments How to Solve the Lagrangian Dual Problem Computing an optimal cost partitioning corresponds to solving the Lagrangian dual . . . but how can we solve it? P ( λ ) is concave and we want to maximize it � can use subgradient optimization 11 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 12 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (1) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) g (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (1) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) g (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (1) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) g (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (2) λ (1) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) g (2) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (2) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) g (2) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (2) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) g (2) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (2) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Subgradient Optimization 4 3 choose point λ (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (2) λ (3) 0 0 2 4 13 / 23
Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments Projected Subgradient Optimization 4 3 choose point λ (1) g (1) repeat for t = 1 , 2 . . . 2 find subgradient g ( t ) at λ ( t ) compute step length η ( t ) 1 set λ ( t +1) = λ ( t ) + η ( t ) g ( t ) λ (1) 0 0 2 4 14 / 23
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