Rapid Implementation of Branch-and-Cut with

Heuristics using GAMSHerman Abeledo and Hua (Edward) Ni

George Washington University

Michael Bussieck and Alex MeerausGAMS Development Corporation

Informs Annual Meeting 2003October 19-22, 2003, Atlanta, Georgia

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Branch-and-Cut

• Branch-and-cut is an established algorithm to improve the branch-and-bound search.

• Implementation facilities:– MIP solver callback functions (e.g. CPLEX,

XPRESS)– Branch-and-cut framework (e.g. ABACUS,

SYMPHONY, COIN BCP)• Heuristic algorithms can help branch-and-

bound search by strong pruning.

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Convergence – Pipeline Design

1280.7

1351.85

1423

1494.15

1565.3

1636.45

0 300 600 900 1200 1500 1800

Computation Time (seconds)

B&C&H_DualB&C&H_PrimalCPLEX_DualCPLEX_PrimalB&C_DualB&C_Primal

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Knowledge Requirements

• IT knowledge (programming in C/C++, JAVA, Solver APIs)

• Mathematical programming knowledge • Application specific knowledge

Design Goal

• Supply GAMS users with an easy access to Branch-and-cut capabilities.

• Relieve GAMS users from the burden of learning a new programming language and/or acquiring solver-specific knowledge, so that they can focus more on the design of cutting planes and heuristic algorithms.

Design Principle

GAMS SYSTEM

GAMSSolverLink

MIP Solver(e.g. CPLEX)

B&B Callback

UserCut

Generator&

Heuristics(e.g. GAMS)

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Implementation Issues

• GDX interface for information exchange• Namespace mapping facility• Option file support• DLL/SO or GAMS support• Incumbent solution validation

A Steiner Tree Problem

• Ortega, F., Wolsey, L. 2003. A Branch-and-cut Algorithm for the Single-commodity, Uncapacitated, Fixed-charge Network Flow Problem. Networks, Vol 41, 143-158

• Berlin52 – from SteinLib– 52 nodes (16 terminals)– 1326 edges

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Model Formulation

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Dicut Inequality and Separation

• Dicut inequality:

if S⊂V and b(S)>0.• Separation:

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A Simple Heuristic

• Delete the arcs with zero values in the current fractional solution.

• Solve the original model in a smaller scale.

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Cut Generator in GAMS I

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Cut Generator in GAMS II

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Convergence – Steiner Tree

626.4

835.2

1044

1252.8

0 300 600 900 1200 1500 1800

Compuation Time (seconds)

B&C&H_DualB&C&H_PrimalB&C_DualB&C_Primal

CPLEX_DualCPLEX_Primal

An Oil Pipeline Design Problem

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4

5

7

9

10

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• Brimberg, J., et. al. 2003. An Oil Pipeline Design Problem. Operations Research, Vol 51, 228-239

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South Gabon Oil field

• 33 nodes – 25 wells– 7 connections– 1 port

• 129 arcs• 5 types of pipe

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Model Formulation

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Cost Approximation

Piece-wise Linear Pipeline Cost Approximation

01020304050607080

0 20 40 60 80 100

Pipeline Capacity

Cos

t

Type 1 Type 2 Type 3 Type 4 Type 5Capacity 5 10 25 50 100

Cost 10 15 25 40 65Approx. Cost 14.5 15 20 40 80

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Two-Stage Heuristic Algorithm

• Stage One: Piecewise linear cost approximation

• Stage Two: Layout fixing

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Cutting Plane Generation

• Valid cut inequality

• Generation

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Convergence – Pipeline Design

1280.7

1351.85

1423

1494.15

1565.3

1636.45

0 300 600 900 1200 1500 1800

Computation Time (seconds)

B&C&H_DualB&C&H_PrimalCPLEX_DualCPLEX_PrimalB&C_DualB&C_Primal

Computational Results

• Overhead– Time spent within the callback functions minus

MIP computation on cuts and heuristics.– 20% ~ 25%

• Performance Improvements– Steiner: 6 hours vs. 2+ days– Pipeline Design: 20 minutes vs. 450 minutes