Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection Aggregation Technique of Mixed Integer Rounding Cut and Cutting Plane Selection *Koichi Fujii NTT DATA Mathematical Systems Inc. 2019/03/29 1 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection Background : Joint work with SCIP team • Visited Zuse Institute Berlin from April 2018 to March 2019, joined SCIP team • SCIP : Non-commercial Constraint Integer Programs Solver • Idea: push forward optimization together to tackle the practical problems 2 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection Summary : Aggregation Technique of Mixed Integer Rounding Cut Motivation Cutting plane is an essential feature of branch-and-bound Goal Improve Mixed integer rounding cut , one of the most important cutting plane Our Results Improve heuristics of generating Mixed integer rounding cut 3 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection Summary : Cutting Plane Selection Motivation How to select cutting planes is an important issue in the implementation Goal Improve the strategy of cutting planes selection. Our Results Assess the strategy cloud points selection / one step selection . 4 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection Agenda 1 Introduction 2 The Cutting Planes in Branch and Bound 3 Aggregation Technique of Mixed Integer Rounding Cut Modified Bound Substitution Capacitated Facility Location Cut 4 Cutting Plane Selection 5 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection The Cutting Plane MILP { c T x | Ax = b , x ∈ Z n ≥ 0 × R m } min x ≥ 0 • LP relaxation is solved at each node • Node can be pruned if (local) lower bound exceeds upper bound (integer feasible solution) 6 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection The Cutting Plane MILP { c T x | Ax = b , x ∈ Z n ≥ 0 × R m } min x ≥ 0 7 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection The Cutting Plane 8 / 37
Introduction The Cutting Planes in Branch and Bound Aggregation Technique of Mixed Integer Rounding Cut Cutting Plane Selection The Cutting Plane Cutting plane improves lower bound (not always). 9 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts Heuristic generating procedure mixed integer rounding cut [Marchand, Wolsey, 1998]: • Start with one constraint of the problem • [ Bound Substitution ] Complement variables if LP solution is closer to upper bound • [ Rounding ] Apply MIR formula • [ Aggregation ] If no violated cut is found, add another problem constraint to the current aggregated inequality, go to Bound Substitution 10 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Rounding Proposition 3.1 (Wolsey Marchand ’98) y Let S := { x , y ∈ Z × R + | x – y ≤ b } . Then, x − 1 − f 0 ≤ ⌊ b ⌋ is valid for S with f 0 = b − ⌊ b ⌋ 11 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Bound Substitution Basic strategy : choose the closest bound distance base on LP solution • Substitute with simple lower bound ¯ y j = y j − l j • Substitute with simple upper bound ¯ y j = u j − y j • Substitute with variable lower bound ¯ y j = y j − l j x j when y j ≥ l j x j • Substitute with variable upper bound ¯ y j = u j x j − y j when y j ≤ u j x j What is the best choice for degenerate case? l j ≤ y j ≤ u j (1) l j x j ≤ y j ≤ u j x j (2) y ∗ j = x ∗ j = 0 (3) 12 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Experiment of Bound Substitution ∑ ∑ minimize c i x i + d i y i (4) i i subject to x i ≤ u i y i , ∀ i ∈ I (5) x i ≥ l i y i , ∀ i ∈ I (6) ∑ x i ≥ b (7) i x i ∈ R , y i ∈ { 0 , 1 } (8) 13 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Experiment of Bound Substitution Regard knapsack constraint as a base inequality. There are five choices for each variables to aggregate. • (case1) Substitute with variable lower bound • (case2) Substitute with variable lower bound + substitute integer varibale • (case3) Substitute with variable upper bound • (case4) Substitute with variable upper bound + substitute integer variable • (case5) No substitution 14 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Experiment of Bound Substitution Regard knapsack constraint as a base inequality. There are five choices for each variables to aggregate. • Test on 1000 random generated instances • Generate cutting plane on each case • Implemented by PySCIPOpt (SCIP Python interface) 15 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Experiment of Bound Substitution • The experiment shows that case3 is the best choice 100 case1 case2 80 case3 case4 case5 60 gap% 40 20 0 0 500 1 , 000 1 , 500 2 , 000 2 , 500 patterns 16 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Bound Substitution l i x i ≤ y i ≤ u i x i (9) l i ≤ y i ≤ u i (10) Modifed rules for mixed integer rounding cuts (work with Robert Gottwald): • Substitute with variable upper bound if u i − u i x ∗ i ≥ l i − l i x ∗ i • Substitute with variable lower bound if l i − l i x ∗ i ≥ u i − u i x ∗ i i f ( (( ∗ b e s t l b t y p e ) > = 0 | | ( ∗ bestubtype ) > = 0) && ! SCIPisEQ ( scip , ∗ b e s t l b − s i m p l e l b , simpleub − ∗ bestub ) ) { i f ( ∗ b e s t l b − s i m p l e l b > simpleub − ∗ bestub ) ∗ selectedbound = SCIP BOUNDTYPE LOWER; e l s e ∗ selectedbound = SCIP BOUNDTYPE UPPER ; } 17 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Computational Results on Unit Commitment Problems • p h t : power production of the thermal units • u h t , v h t , w h t : status of the thermal units ∑ ∑ ∑ α h p h β h u h γ h v h minimize t + t + (11) t h h h p h u h t ≤ p h t ≤ p h u h (12) subject to t ( p h − ∆ h t ≤ p h + p h + ) u h t − 1 + p h (13) t − 1 ( p h − ∆ h t − 1 ≤ p h + p h − ) u h t + p h (14) t u h t − u h t − 1 = v h t − w h (15) t ∑ v h k ≤ u h ∑ w h k ≤ 1 − u h (16) t , t t ≥ k ≥ t − τ min t ≥ k ≥ t − τ max ∑ p h t ≥ D t (17) h 18 / 37
Introduction The Cutting Planes in Branch and Bound Modified Bound Substitution Aggregation Technique of Mixed Integer Rounding Cut Capacitated Facility Location Cut Cutting Plane Selection Mixed Integer Rounding Cuts: Computational Results on Unit Commitment Problems modified bound substitution bracket model faster slower time node ≥ 0 360 276 26 0.44 0.17 ≥ 10 147 138 5 0.33 0.04 ≥ 100 5 5 0 0.11 0.01 Table: computational results on UC instances • Implemented in SCIP 5.0.1 • Achieve overall 2.27x speedup on 360 randomly generated UC instances. • 3.03x speedup on relatively hard problem ( both case take ≥ 10 sec) • 2 more intances can be solved ( 3600 sec time limit). 19 / 37
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