Timo Berthold Ralf Borndörfer Gregor Hendel Heide Hoppmann Marika Karb- stein International Symposium on Mathematical Programming, July 6th, 2018, Bordeaux, France Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 1/28 Tighter LP relaxations for configuration knapsacks using extended formulations
Configuration Knapsacks
2/28 is the Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations c t x s.t. with j n g A frequent structure in many Mixed-Integer Programs (MIPs) Knapsack constraints min Ax ≤ b x ∈ { 0 , 1 } n b × Z ≥ 0 × R n c Knapsack constraint ∑ w j x j ≤ β knap( w , β ) • w j ∈ Z ≥ 0 • w j = 0 for all j > n b • β ∈ Z +
Line Planning Model [Borndörfer et al., 2013] 0 1 L x l f d e e E f F 3/28 1 F l L x F Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations f x l f l f L e l s.t. c l f x l f F f L l • • frequencies • set L of possible paths in G Example – Line Planning • Input: Graph G = ( V , E ) . • Edge demands d e ≥ 0 F = { f 1 , . . . , f d } ⊂ Z + • Operational costs c l , f > 0.
3/28 • Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations • set L of possible paths in G • frequencies s.t. Example – Line Planning • Input: Graph G = ( V , E ) . • Edge demands d e ≥ 0 F = { f 1 , . . . , f d } ⊂ Z + • Operational costs c l , f > 0. Line Planning Model [Borndörfer et al., 2013] ∑ ∑ min c l , f x l , f l ∈ L f ∈ F ∑ ∑ f · x l , f ≥ d e ∀ e ∈ E l ∈ L : e ∈ l f ∈ F ∑ x l , f ≤ 1 ∀ l ∈ L f ∈ F x ∈ { 0 , 1 } L × F
4/28 The demand inequalities can be formulated as knapsack constraint Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations Transformation into Knapsack knap( w , β ) . Let e ∈ E , use ¯ x l , f = 1 − x l , f ∈ { 0 , 1 } ∑ ∑ f · x l , f ≥ d e l ∈ L : e ∈ l f ∈ F ∑ ∑ f · ( 1 − ¯ ⇔ x l , f ) ≥ d e l ∈ L : e ∈ l f ∈ F ∑ ∑ ∑ ∑ ⇔ f · ¯ x l , f ≤ ( f ) − d e l ∈ L : e ∈ l f ∈ F l ∈ L : e ∈ l f ∈ F � �� � =: β
5/28 n weights. Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations k Central observation: Demand constraints only have ”a handful” ( d ) difgerent Configuration knapsacks Configuration knapsack Let d ∈ N . Let w ∈ Z n ≥ 0 , β ∈ Z ≥ 0 define a knapsack constraint knap( w , β ) . If there exists a partition of [ n ] into k ≤ d groups N 1 , . . . , N k [ n ] = N 1 ˙ ∪ N 2 ˙ ∪ . . . ˙ ∪ N k such that i , j ∈ N l ⇔ w i = w j (=: ω l ) , then knap( w , β ) can be written ∑ ∑ ∑ w i x i = ω l x i ≤ β i = 1 l = 1 i ∈ N l and is called a configuration knapsack.
6/28 Then, Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations P . is a valid inequality for 1 C x i C i w i C i n be minimal such that Let C Cover Inequalities [Wolsey, 1975] and Cover inequalities for knapsacks Let knap( w , β ) be a knapsack constraint. Define P := { x ∈ { 0 , 1 } n : w T x ≤ β } P LP := { x ∈ [ 0 , 1 ] n : w T x ≤ β } ⊇ conv( P )
6/28 i Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations P . is a valid inequality for 1 C x i C Then, Cover Inequalities [Wolsey, 1975] and Cover inequalities for knapsacks Let knap( w , β ) be a knapsack constraint. Define P := { x ∈ { 0 , 1 } n : w T x ≤ β } P LP := { x ∈ [ 0 , 1 ] n : w T x ≤ β } ⊇ conv( P ) Let C ⊂ [ n ] be minimal such that ∑ w i > β. i ∈ C
6/28 Cover Inequalities [Wolsey, 1975] Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations and Then, Cover inequalities for knapsacks Let knap( w , β ) be a knapsack constraint. Define P := { x ∈ { 0 , 1 } n : w T x ≤ β } P LP := { x ∈ [ 0 , 1 ] n : w T x ≤ β } ⊇ conv( P ) Let C ⊂ [ n ] be minimal such that ∑ w i > β. i ∈ C ∑ x i ≤ | C | − 1 i ∈ C is a valid inequality for conv( P ) .
• more knapsack cutting planes: • Liħted cover inequalities [Balas, Gu] • G(eneralized) U(pper) B(ound) Inequalities [Wolsey, 1990] • strengthening of cover inequalities [Carr et al, 2000] • Existence of small extended formulations for knapsack polytopes [Bienstock 2008, Bazzi et al, 2016] • a lot of very recent work presented at this conference. Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 7/28 More related work
An Extended Formulation for configuration knapsacks
• Construct higher dimensional polytope Q into the space of x -variables is tighter, [Borndörfer, Hoppmann, Karbstein, 2013] construct an extended formulation for the line planning problem. Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 8/28 A primer on extended formulations such that the projection π ideally conv { P }
Reformulation of Introduce new binary variables z y for y 9/28 i N l x i y y l z y l 1 k y z y 1 z y 0 1 y ( w x z ) Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations . 1 k x i l N l 0 y l l y l y denote all maximal points of Let w k Extended formulation for configuration knapsacks Let knap( w , β ) be a configuration knapsack of cardinality k ≤ d . ∑ ∑ ω l ≤ β l = 1 i ∈ N l � �� � =: y l
9/28 z y y y l z y l 1 k y 1 N l z y 0 1 y ( w x z ) Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations x i i k x i Extended formulation for configuration knapsacks Let knap( w , β ) be a configuration knapsack of cardinality k ≤ d . ∑ ∑ ω l ≤ β l = 1 i ∈ N l � �� � =: y l Reformulation of knap( w , β ) Let Y denote all maximal points of ∑ { y : ω l y l ≤ β, y l ∈ { 0 , . . . , | N l |} , l = 1 , . . . , k } . Introduce new binary variables z y for y ∈ Y .
9/28 x i k Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations y l z y Extended formulation for configuration knapsacks Let knap( w , β ) be a configuration knapsack of cardinality k ≤ d . ∑ ∑ ω l ≤ β l = 1 i ∈ N l � �� � =: y l Reformulation of knap( w , β ) Let Y denote all maximal points of ∑ { y : ω l y l ≤ β, y l ∈ { 0 , . . . , | N l |} , l = 1 , . . . , k } . Introduce new binary variables z y for y ∈ Y . ∑ ∑ x i ≤ ∀ l = 1 , . . . , k y ∈Y i ∈ N l ∑ ( reform( w , β, x , z ) ) z y = 1 y ∈Y z y ∈ { 0 , 1 } y ∈ Y
y z y 10/28 x 1 1 0 2 0 1 0 2 0 • Reformulation: z y 0 1 for all y , 1 x 2 • maximal points (maximal configurations) x 3 x 4 x 5 x 6 x 7 x 9 x 9 3 z y 1 2 z y 2 z y 1 2 z y 3 z y 2 Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 3 3 y l , 0 l y l 1 l 3 • weight space inequality 7 3 5 2 2 1 • three weights Example 2 x 1 + 2 x 2 + 2 x 3 + 5 x 4 + 5 x 5 + 5 x 6 + 7 x 7 + 7 x 8 + 7 x 9 ≤ 11 � �� � � �� � � �� � N 1 = { 1 , 2 , 3 } N 2 = { 4 , 5 , 6 } N 3 = { 7 , 8 , 9 }
y z y 10/28 x 4 2 0 • Reformulation: z y 0 1 for all y , 1 x 1 x 2 x 3 x 5 1 x 6 x 7 x 9 x 9 3 z y 1 2 z y 2 z y 1 2 z y 3 z y 2 Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 0 0 2 0 1 3 • maximal points (maximal configurations) Example 2 x 1 + 2 x 2 + 2 x 3 + 5 x 4 + 5 x 5 + 5 x 6 + 7 x 7 + 7 x 8 + 7 x 9 ≤ 11 � �� � � �� � � �� � N 1 = { 1 , 2 , 3 } N 2 = { 4 , 5 , 6 } N 3 = { 7 , 8 , 9 } • three weights ω 1 = 2 , ω 2 = 5 , ω 3 = 7 • weight space inequality ∑ 3 l = 1 ω l y l ≤ β , 0 ≤ y l ≤ 3
y z y 10/28 x 2 0 0 1 0 2 0 • Reformulation: z y 0 1 for all y , 1 x 1 x 3 3 x 4 x 5 x 6 x 7 x 9 x 9 3 z y 1 2 z y 2 z y 1 2 z y 3 z y 2 Hendel et al. – Tighter LP relaxations for configuration knapsacks using extended formulations 1 2 • maximal points (maximal configurations) Example 2 x 1 + 2 x 2 + 2 x 3 + 5 x 4 + 5 x 5 + 5 x 6 + 7 x 7 + 7 x 8 + 7 x 9 ≤ 11 � �� � � �� � � �� � N 1 = { 1 , 2 , 3 } N 2 = { 4 , 5 , 6 } N 3 = { 7 , 8 , 9 } • three weights ω 1 = 2 , ω 2 = 5 , ω 3 = 7 • weight space inequality ∑ 3 l = 1 ω l y l ≤ β , 0 ≤ y l ≤ 3 Y = , ,
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