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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

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Decomposition for Optimal Cost Partitioning

Florian Pommerening1 Gabriele R¨

  • ger1

Malte Helmert1 Hadrien Cambazard2 Louis-Martin Rousseau3 Domenico Salvagnin4

1University of Basel, Switzerland

  • 2Univ. Grenoble Alpes, CNRS, Grenoble INP*, G-SCOP, 38000 Grenoble, France

*Institute of Engineering Univ. Grenoble Alpes

3Polytechnique Montreal, Canada 4University of Padua, Italy

July 14, 2019

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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

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Decomposition

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Starting with a Linear Program

Problem P Min c x s.t. A1 x ≥ b1 . . . Ak x ≥ bk x ≥ 0

rewrite

− − − − → Problem P Min c x s.t. Ai xi ≥ bi ∀i x = xi ∀i x , xi ≥ 0 ∀i

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Starting with a Linear Program

Problem P Min c x s.t. A1 x ≥ b1 . . . Ak x ≥ bk x ≥ 0

rewrite

− − − − → Problem P Min c x s.t. Ai xi ≥ bi ∀i x = xi ∀i x , xi ≥ 0 ∀i

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Relaxation

Problem P Min c x s.t. Ai xi ≥ bi ∀i x = xi ∀i x , xi ≥ 0 ∀i

relax

− − − → Problem P(λ) Min c x +

i λi ( xi − x ) s.t.

Ai xi ≥ bi ∀i x , xi ≥ 0 ∀i Penalty term λi for violating x = xi 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: MaxλP(λ) = value(P)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Relaxation

Problem P Min c x s.t. Ai xi ≥ bi ∀i x = xi ∀i x , xi ≥ 0 ∀i

relax

− − − → Problem P(λ) Min c x +

i λi ( xi − x ) s.t.

Ai xi ≥ bi ∀i x , xi ≥ 0 ∀i Penalty term λi for violating x = xi 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: MaxλP(λ) = value(P)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Relaxation

Problem P Min c x s.t. Ai xi ≥ bi ∀i x = xi ∀i x , xi ≥ 0 ∀i

relax

− − − → Problem P(λ) Min c x +

i λi ( xi − x ) s.t.

Ai xi ≥ bi ∀i x , xi ≥ 0 ∀i Penalty term λi for violating x = xi 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: MaxλP(λ) = value(P)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Lagrangian Decomposition

Problem P(λ) Min c x +

i λi ( xi − x ) s.t.

Ai xi ≥ bi ∀i x , xi ≥ 0 ∀i P(λ) decomposes into independent subproblems P(λ) =

i Pi(λ)

Problem P0(λ) Min

  • c

−

i λi

  • x s.t.

x ≥ 0 Problem Pi(λ) Min λi xi s.t. Ai xi ≥ bi xi ≥ 0

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

A closer look at P0(λ)

Problem P0(λ) Min

  • c

−

i λi

  • x s.t.

x ≥ 0 if all objective coefficients non-negative: value(P0(λ)) = 0

  • therwise P0(λ) is unbounded

Constraint encoded by P0(λ)

  • i

λi ≤ c

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Relation to Cost Partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Summarizing Lagrangian Decomposition

Original Problem P Min c x s.t. Ai x ≥ bi ∀i x ≥ 0 Lagrangian Dual Problem Max

i Pi(λ) s.t.

  • i

λi ≤ c Subproblem Pi(λ) Min λi xi s.t. Ai xi ≥ bi xi ≥ 0

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Cost Partitioning of Operator-Counting Heuristics

Heuristic h Min cost x s.t. Ai x ≥ bi ∀i x ≥ 0 Optimal Cost Partitioning Max

i hi(costi) s.t.

  • i

costi ≤ cost Heuristic hi(costi) Min costi xi s.t. Ai xi ≥ bi xi ≥ 0

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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

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 λ(1) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(2) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(2) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 g(2) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization

2 4 1 2 3 4 λ(2) λ(3) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Projected Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Projected Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Projected Subgradient Optimization

2 4 1 2 3 4 g(1) ??? λ(1) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = λ(t) + η(t)g(t)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Projected Subgradient Optimization

2 4 1 2 3 4 g(1) λ(1) λ(2) choose point λ(1) repeat for t = 1, 2 . . .

find subgradient g(t) at λ(t) compute step length η(t) set λ(t+1) = proj((λ(t) + η(t)g(t))

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Application to Cost Partitioning

  • ver Abstractions

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization for Cost Partitioning

Analogies in cost partitioning current point λ(t)

current cost functions cost1, . . . , costk

subgradient g(t)

  • ptimal solutions of subproblems Pi(λ(t))

if subproblems are abstraction heuristics: shortest paths in abstractions

projection

project arbitrary set of cost functions to cost partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization for Cost Partitioning

Analogies in cost partitioning current point λ(t)

current cost functions cost1, . . . , costk

subgradient g(t)

  • ptimal solutions of subproblems Pi(λ(t))

if subproblems are abstraction heuristics: shortest paths in abstractions

projection

project arbitrary set of cost functions to cost partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization for Cost Partitioning

Analogies in cost partitioning current point λ(t)

current cost functions cost1, . . . , costk

subgradient g(t)

  • ptimal solutions of subproblems Pi(λ(t))

if subproblems are abstraction heuristics: shortest paths in abstractions

projection

project arbitrary set of cost functions to cost partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization for Cost Partitioning

Analogies in cost partitioning current point λ(t)

current cost functions cost1, . . . , costk

subgradient g(t)

  • ptimal solutions of subproblems Pi(λ(t))

if subproblems are abstraction heuristics: shortest paths in abstractions

projection

project arbitrary set of cost functions to cost partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Subgradient Optimization for Cost Partitioning

Anytime algorithm choose any cost partitioning cost(1) repeat for t = 1, 2 . . .

for each abstraction i

find optimal solution π∗ under cost(t)

i

set cost(t+1)

i

(o) = cost(t)

i (o) + η(t)occurrences(o, π∗)

project cost(t+1) to a cost partitioning

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Experiments

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Experiment Setup

Experiment setup IPC instances from optimal tracks (1998–2018) projections to all interesting patterns up to size 2 (and 3) non-negative cost partitioning

no good way to project to general cost partitioning

300 s time limit, 2 GB memory limit heuristic values of initial states seeded with different cost partitioning methods

uniform

  • pportunistic uniform (random/improved order)

greedy zero-one (random/improved order) saturated (random/improved order)

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Heuristic Quality

100 101 102 103 104 40% 60% 80% 100% Iterations Heuristic Quality hC/hC∗

uniform

  • pp. uniform (rand.)
  • pp. uniform (impr.)

greedy 0/1 (rand.) greedy 0/1 (impr.) saturated (rand.) saturated (impr.) 20 / 23

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Runtime

10−1 s 100 s 101 s 102 s 10−1 s 100 s 101 s 102 s unsolved uns. Computing the optimal cost partitioning 200 steps in subgradient method

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Conclusion

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Lagrangian Decomposition Relation to Cost Partitioning Subgradient Optimization Application to Cost Partitioning Experiments

Conclusion

Contributions to Cost Partitioning new interpretation as Lagrangian decomposition interesting relation to subgradient optimization anytime algorithm for suboptimal cost partitioning Future Work techniques from subgradient optimization

better stopping conditions dynamic step length functions improved updates

  • pen questions

projection for general cost partitioning consider highly different operator costs

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