Strong Convex Nonlinear Relaxations of the Pooling Problem One Relaxation to Rule Them All? Claudia D’Ambrosio LIX, ´ Ecole Polytechnique Jeff Linderoth Jim Luedtke Univ. of Wisconsin-Madison Jonas Schweiger IBM CPLEX 20th Combinatorial Optimization Workshop Aussois, France, January, 2016 D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 1 / 24
Warning! We will go an excited and dangerous quest—much like building a human pyramid D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 2 / 24
Warning! We will go an excited and dangerous quest—much like building a human pyramid Such quests are frought with peril, and it will turn out to be much more difficult than we thought D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 2 / 24
The Pooling Problem Inputs I Outputs J Pools L λ ki α kj Nodes N = I ∪ L ∪ J Arc capacities: u ij Arcs A ( i, j ) ∈ ( I × L ) ∪ ( L × J ) ∪ ( I × J ) Node capacities: C i , i ∈ N on which materials flow Attribute requirements α kj , k ∈ K, j ∈ J Material attributes: K Pooling Problem Maximize linear function of flow through network, obeying capacities, and meeting quality attribute requirements D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 3 / 24
Attribute Blending: Bilinear Inputs have associated attribute concentrations λ ki , k ∈ K, i ∈ I The concentration of an attribute in pool is the weighted average of the concentrations of its inputs—This results in bilinear constraints. D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 4 / 24
Attribute Blending: Bilinear Inputs have associated attribute concentrations λ ki , k ∈ K, i ∈ I The concentration of an attribute in pool is the weighted average of the concentrations of its inputs—This results in bilinear constraints. Variables x ij : Flow on ( i, j ) ∈ A x iℓ q iℓ : Proportion of flow to pool ℓ ∈ L from input i ∈ I . ( q iℓ = j ∈ J x ℓj ) � Note also that � i ∈ I q iℓ = 1 ∀ ℓ ∈ L w ilj = q il x lj (flow from i through pool ℓ to output j ) � � x il = w iℓj x ℓj = w iℓj j ∈ J i ∈ I D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 4 / 24
Attribute Blending: Bilinear Inputs have associated attribute concentrations λ ki , k ∈ K, i ∈ I The concentration of an attribute in pool is the weighted average of the concentrations of its inputs—This results in bilinear constraints. Variables x ij : Flow on ( i, j ) ∈ A x iℓ q iℓ : Proportion of flow to pool ℓ ∈ L from input i ∈ I . ( q iℓ = j ∈ J x ℓj ) � Note also that � i ∈ I q iℓ = 1 ∀ ℓ ∈ L w ilj = q il x lj (flow from i through pool ℓ to output j ) � � x il = w iℓj x ℓj = w iℓj j ∈ J i ∈ I Start Strong! We use the PQ-formulation (Sahinidis and Tawarmalani (2005) as our starting point D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 4 / 24
Linearizing Attribute Restriction The concentration requirement constraints can be written as � � � ( λ ki − α kj ) x ik + ( λ ki − α kj ) q ij x jk ≤ 0 ∀ k ∈ K, j ∈ J i ∈ I j ∈ J i ∈ I D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 5 / 24
Linearizing Attribute Restriction The concentration requirement constraints can be written as � � � ( λ ki − α kj ) x ik + ( λ ki − α kj ) q ij x jk ≤ 0 ∀ k ∈ K, j ∈ J i ∈ I j ∈ J i ∈ I Since w iℓj = q iℓ x ℓj , the attribute constraints have the (linear) form � � � ( λ ki − α kj ) x ij + ( λ ki − α kj ) w iℓj ≤ 0 ∀ k ∈ K, j ∈ J i ∈ I i ∈ I ℓ ∈ L D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 5 / 24
Linearizing Attribute Restriction The concentration requirement constraints can be written as � � � ( λ ki − α kj ) x ik + ( λ ki − α kj ) q ij x jk ≤ 0 ∀ k ∈ K, j ∈ J i ∈ I j ∈ J i ∈ I Since w iℓj = q iℓ x ℓj , the attribute constraints have the (linear) form � � � ( λ ki − α kj ) x ij + ( λ ki − α kj ) w iℓj ≤ 0 ∀ k ∈ K, j ∈ J i ∈ I i ∈ I ℓ ∈ L The feasible region of the pooling problem is P ∩ B where P is a polyhedron, and B = { ( x, q, w ) | w iℓj = q iℓ x ℓj ∀ i ∈ I, ℓ ∈ L, j ∈ J } The only nonlinearities/nonconvexities in the problem are in B D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 5 / 24
The “IP” approach We wish to optimize a linear function over conv ( P ∩ B ) , so we can first try to create the relaxation P ∩ conv ( B ) D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 6 / 24
The “IP” approach We wish to optimize a linear function over conv ( P ∩ B ) , so we can first try to create the relaxation P ∩ conv ( B ) But by the result of Sauron G¨ unl¨ uk et al. , the standard “McCormick relaxation” describes the convex hull of this non-convex set B w iℓj ≤ C j q il w iℓj ≤ x ℓj w iℓj ≥ C j q il + x ℓj − C D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 6 / 24
The “IP” approach We wish to optimize a linear function over conv ( P ∩ B ) , so we can first try to create the relaxation P ∩ conv ( B ) But by the result of Sauron G¨ unl¨ uk et al. , the standard “McCormick relaxation” describes the convex hull of this non-convex set B w iℓj ≤ C j q il w iℓj ≤ x ℓj w iℓj ≥ C j q il + x ℓj − C Upshot If we want to “beat” the PQ formulation (which is actually quite hard to do), we will need to look at more of the problem D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 6 / 24
D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 7 / 24
Our Quest—Seek Simple Sets Extract a simple but nontrivial set and attempt to convexify D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 7 / 24
Our Quest—Seek Simple Sets Extract a simple but nontrivial set and attempt to convexify Intuition for Pooling Will need to include more than just the nonconvexity w iℓj = q iℓ x ℓj Attribute constraints on outputs are important Idea: Focus on a single output and attribute D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 7 / 24
Make It Easier. Focus on Single Output, Single Attribute. Fix output j and attribute k . (Drop these indices) Inputs I Pools L Output J Relevant constraints � � � ( λ i − α ) x i + ( λ i − α ) w iℓ ≤ 0 � �� � � �� � i ∈ I i ∈ I ℓ ∈ L γ i γ i � x i ≤ C i ∈ I ∪ L � q iℓ = 1, ∀ ℓ ∈ L i ∈ I w iℓ = q iℓ x ℓ , ∀ i ∈ I, ℓ ∈ L D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 8 / 24
Make It Easier. Focus on Single Output, Single Attribute. Fix output j and attribute k . (Drop these indices) Inputs I Pools L Output J Relevant constraints � � � ( λ i − α ) x i + ( λ i − α ) w iℓ ≤ 0 � �� � � �� � i ∈ I i ∈ I ℓ ∈ L γ i γ i � x i ≤ C i ∈ I ∪ L � q iℓ = 1, ∀ ℓ ∈ L i ∈ I w iℓ = q iℓ x ℓ , ∀ i ∈ I, ℓ ∈ L Still Too Hard! Try to focus on a single pool ℓ ∈ L and consider rest as a single “by-pass”? D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 8 / 24
Make it Easier-er. Focus on Single Pool. Separate pool ℓ ∈ L from the rest Gray squiggles are now aggregated into variables y and z � � � � Inputs I Output j Pool ℓ γ i x i + γ i w it + γ i w iℓ ≤ 0 i ∈ I i ∈ I t ∈ L \{ ℓ } i ∈ I � �� � y � + x ℓ ≤ C, x i i ∈ I ∪ L \{ ℓ } � �� � z z ≥ 0, βz ≤ y ≤ βz β, β are the best and worst inputs that can each output β = min β := max i ∈ I γ i i ∈ I γ i D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 9 / 24
Single Output, Single Pool Set Now also drop the index ℓ for the pool, letting X be the set of ( x, y, z, q, w ) ∈ R 3 + 2 | I | that satisfy Inputs I Pool ℓ Output j � y + γ i w i ≤ 0, w i i ∈ I x z + x ≤ C, � q i = 1, i ∈ I z w i = q i x, ∀ i ∈ I βz ≤ y ≤ βz x, w, z, q ≥ 0 D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 10 / 24
Single Output, Single Pool Set Now also drop the index ℓ for the pool, letting X be the set of ( x, y, z, q, w ) ∈ R 3 + 2 | I | that satisfy Inputs I Pool ℓ Output j � y + γ i w i ≤ 0, w i i ∈ I x z + x ≤ C, � q i = 1, i ∈ I z w i = q i x, ∀ i ∈ I βz ≤ y ≤ βz x, w, z, q ≥ 0 Argh! X is still too complicated One last simplification trick D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 10 / 24
More Tricks “ BILBO BAGGINS! Do not take me for some conjuror of cheap tricks!” — Gandalf D’Ambrosio, Linderoth, Luedtke, Schweiger The Fellowship of the Pool Aussois 2016 11 / 24
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