Piecewise Parametric Structure in the Pooling Problem - from Sparse - PowerPoint PPT Presentation

Piecewise Parametric Structure in the Pooling Problem - from Sparse Strongly-Polynomial Solutions to NP-Hardness - Radu Baltean-Lugojan* radu.baltean-lugojan09@imperial.ac.uk Ruth Misener r.misener@imperial.ac.uk Computational Optimisation Group Centre for Process Systems Engineering Department of Computing 29 Mar 2017 http://www.optimization-online.org/DB_HTML/2016/05/5457.html *EPSRC EP/P008739/1 funding for RBL Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 1 / 27

Pooling Problem - Where does it appear? Inputs Outputs Pools in 1 p 1 o 1 x 11 in 2 y 11 x 22 p 2 o 2 Directs y 22 di 1 z 12 Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 2 / 27

Globally Optimizing Pooling Problems in 1 p 1 o 1 x 11 Previous work in 2 y 11 in 3 p 2 o 2 Explore P / NP boundary Alfaki and . . . Haugland [2013] • Haugland [2016] • in I p 3 o 3 Haugland and Hendrix [2016] • x IL Boland et al. [2016] . . . . z di 1 1 . . MIP Approximation Dey and Gupte 1 di 2 p L o O [2015] • Gupte et al. [2016] di 3 y LO . Cutting Planes D’Ambrosio et al. . . [2011] • Gupte et al. [2016] di H z HO Finding Patterns Ceccon et al. [2016] This work: Parameterize with respect to pool concentration Develop strongly polynomial algorithms for several pooling subproblems; Use patterns of dominating topologies to find piecewise active network structure (nodes with flows strictly > 0) and motivate problem sparsity. Exploit sparsity to decompose large scale instances in smaller sub-problems Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 3 / 27

How to solve problems multiplying quality times flowrate? Haverly [1978] Solve pooling problem via sequential linear programming, i.e., approximate nonconvex bilinear terms as linear. Beale et al. [1965] In Iron-Making Track manganese concentration between two blast furnaces and a sinter plant; To Solve Reformulate as a separable program and apply a piecewise linear approximation; Key Assumption Only a few variables are active at once and the associated functions can be approximated linearly. Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 4 / 27

Previous Piecewise Schemes - Bilinear Envelopes McCormick [1976], Al-Khayyal and Falk [1983]  z ≥ x · y L + x L · y − x L · y L   z ≥ x · y U + x U · y − x U · y U    Envelope of z = x · y z ≤ x · y L + x U · y − x U · y L    z ≤ x · y U + x L · y − x L · y U   Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 5 / 27

Previous Piecewise Schemes - Piecewise Bilinear Envelopes Balas [1979], Floudas [1995], Karuppiah and Grossmann [2006], Meyer and Floudas [2006], Wicaksono and Karimi [2008], Misener and Floudas [2012]   z ≥ x · y L + a ( n ) · y − a ( n ) · y L       z ≥ x · y U + a ( n + 1) · y − a ( n + 1) · y U         z ≤ x · y L + a ( n + 1) · y − a ( n + 1) · y L     Piecewise   ∨ n z ≤ x · y U + a ( n ) · y − a ( n ) · y U Envelope of z = x · y         a ( n ) ≤ x ≤ a ( n + 1)       y L ≤ y ≤ y U       Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 6 / 27

Standard Pooling Network p-Formulation � � � � Objective max d j · y lj + d j · z ij − γ i · x il − γ i · z ij x il , y lj , z ij , p lk ( l , j ) ∈ T Y ( i , j ) ∈ T Z ( i , l ) ∈ T X ( i , j ) ∈ T Z  Feed � �  A L z ij ≤ A U i ≤ x il + ∀ i Inputs [1,. . . ,I] Pools [1,. . . ,L] Outputs [1,. . . ,O] i Avail in 1 p 1 x 11 o 1 l :( i , l ) ∈ T X j :( i , j ) ∈ T Z in 2 y 11  Pool  S L � x il ≤ S U in 3 p 2 o 2 l ≤ ∀ l l Capacity . . . i :( i , l ) ∈ T X in I p 3 o 3  Product x IL  D L � � z ij ≤ D U j ≤ y lj + ∀ j j Demand Directs [1,. . . ,H] l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z . . . . z di 1 11 . .  Material � � di 2 x il − y lj = 0 ∀ l p L o O  Balance di 3 y LO i :( i , l ) ∈ T X j :( l , j ) ∈ T Y . . .  Quality � � di H C ik x il = p lk y lj ∀ l , k z  H O Balance i :( i , l ) ∈ T X j :( l , j ) ∈ T Y  �  � � p lk y lj ≥ P L � y lj + � z ij   jk  Product  l :( l , j ) ∈ T Y  l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z ∀ j , k  � � Quality  � + C ik z ij ≤ P U � y lj + � z ij    jk   l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z i :( i , j ) ∈ T Z Bounds [ x il , y lj , z ij ≥ 0 ∀ i , l , j Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 7 / 27

Standard Pooling Network p-Formulation � � � � Objective max d j · y lj + d j · z ij − γ i · x il − γ i · z ij x il , y lj , z ij , p lk ( l , j ) ∈ T Y ( i , j ) ∈ T Z ( i , l ) ∈ T X ( i , j ) ∈ T Z  Feed � �  A L z ij ≤ A U i ≤ x il + ∀ i Inputs [1,. . . ,I] Pools [1,. . . ,L] Outputs [1,. . . ,O] i Avail in 1 p 1 x 11 o 1 l :( i , l ) ∈ T X j :( i , j ) ∈ T Z in 2 y 11  Pool  S L � x il ≤ S U in 3 p 2 o 2 l ≤ ∀ l l Capacity . . . i :( i , l ) ∈ T X in I p 3 o 3  Product x IL  D L � � z ij ≤ D U j ≤ y lj + ∀ j j Demand Directs [1,. . . ,H] l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z . . . . z di 1 11 . .  Material � � di 2 x il − y lj = 0 ∀ l p L o O  Balance di 3 y LO i :( i , l ) ∈ T X j :( l , j ) ∈ T Y . . .  Quality � � di H C ik x il = p lk y lj ∀ l , k z  H O Balance i :( i , l ) ∈ T X j :( l , j ) ∈ T Y  �  � � p lk y lj ≥ P L � y lj + � z ij   jk  Product  l :( l , j ) ∈ T Y  l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z ∀ j , k  � � Quality  � + C ik z ij ≤ P U � y lj + � z ij    jk   l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z i :( i , j ) ∈ T Z Bounds [ x il , y lj , z ij ≥ 0 ∀ i , l , j Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 7 / 27

Standard Pooling Network p-Formulation � � � � Objective max d j · y lj + d j · z ij − γ i · x il − γ i · z ij x il , y lj , z ij , p lk ( l , j ) ∈ T Y ( i , j ) ∈ T Z ( i , l ) ∈ T X ( i , j ) ∈ T Z  Feed  A L � � z ij ≤ A U i ≤ x il + ∀ i Inputs [1,. . . ,I] Pools [1,. . . ,L] Outputs [1,. . . ,O] i Avail in 1 p 1 x 11 x 11 o 1 l :( i , l ) ∈ T X j :( i , j ) ∈ T Z in 2 y 11  Pool  S L � x il ≤ S U in 3 p 2 o 2 l ≤ ∀ l l Capacity . . . i :( i , l ) ∈ T X in I p 3 o 3  Product x IL  D L � � z ij ≤ D U j ≤ y lj + ∀ j j Demand Directs [1,. . . ,H] l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z . . . . z di 1 11 . .  Material � � di 2 x il − y lj = 0 ∀ l p L o O  Balance di 3 y LO i :( i , l ) ∈ T X j :( l , j ) ∈ T Y . . .  Quality � � di H C ik x il = p lk y lj ∀ l , k z  H O Balance i :( i , l ) ∈ T X j :( l , j ) ∈ T Y  �  � � p lk y lj ≥ P L � y lj + � z ij   jk  Product  l :( l , j ) ∈ T Y  l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z ∀ j , k  � � Quality  � + C ik z ij ≤ P U � y lj + � z ij    jk   l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z i :( i , j ) ∈ T Z Bounds [ x il , y lj , z ij ≥ 0 ∀ i , l , j Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 7 / 27

Standard Pooling Network p-Formulation � � � � Objective max d j · y lj + d j · z ij − γ i · x il − γ i · z ij x il , y lj , z ij , p lk ( l , j ) ∈ T Y ( i , j ) ∈ T Z ( i , l ) ∈ T X ( i , j ) ∈ T Z  Feed � �  A L z ij ≤ A U i ≤ x il + ∀ i Inputs [1,. . . ,I] Pools [1,. . . ,L] Outputs [1,. . . ,O] i Avail in 1 p 1 x 11 x 11 o 1 l :( i , l ) ∈ T X j :( i , j ) ∈ T Z in 2 y 11  Pool  S L � x il ≤ S U in 3 p 2 o 2 l ≤ ∀ l l Capacity . . . i :( i , l ) ∈ T X in I p 3 o 3  Product x IL  D L � � z ij ≤ D U j ≤ y lj + ∀ j j Demand Directs [1,. . . ,H] l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z . . . . z di 1 11 . .  Material � � di 2 x il − y lj = 0 ∀ l p L o O  Balance di 3 y LO i :( i , l ) ∈ T X j :( l , j ) ∈ T Y . . .  Quality � � di H C ik x il = p lk y lj ∀ l , k z  H O Balance i :( i , l ) ∈ T X j :( l , j ) ∈ T Y  �  � � p lk y lj ≥ P L � y lj + � z ij   jk  Product  l :( l , j ) ∈ T Y  l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z ∀ j , k  � � Quality  � + C ik z ij ≤ P U � y lj + � z ij    jk   l :( l , j ) ∈ T Y i :( i , j ) ∈ T Z i :( i , j ) ∈ T Z Bounds [ x il , y lj , z ij ≥ 0 ∀ i , l , j Baltean-Lugojan, Misener Piecewise Parametric Pooling Problem 29 Mar 2017 7 / 27