MIP Lifting Techniques for Mixed Integer Nonlinear Programs

MIP Lifting Techniques for Mixed Integer Nonlinear Programs

Jean-Philippe P. Richard*School of Industrial Engineering, Purdue University

Mohit TawarmalaniKrannert School of Management, Purdue University

*Supported by NSF DMI0348611

MIP 2006, Thursday June 8th 2006

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Structure of the Talk


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A Motivation in Integer Programming


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A Motivation in Integer Programming


6

What is MIP Lifting?


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Some Literature on MIP lifting


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What is Hard about MINLP Lifting?


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Overview & Goal of Our Work


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Some Nice Features of MINLP Lifting


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Goal of Part II


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Mixed Integer Nonlinear Knapsack


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Mixed Integer Nonlinear Knapsack


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Generating Valid Inequalities for PS


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A General Lifting Result for PS


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Advantages and Limitations of the Lifting Scheme


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A Superadditive Lifting Result for PS


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Application: Bilinear Mixed Integer Knapsack Problem (BMIKP)


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BMIKP: Comments


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The Convex Hull of PT’ is a Polyhedron


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Obtaining Facets of PT using Superadditive Lifting


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Example


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Another Family of Strong Inequalities for PT


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Another Family of Strong Inequalities for PT


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Example


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An Equivalent Integer Programming Formulation for PT


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An Equivalent Integer Programming Formulation for PT


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Strong Rank-1 Inequalities


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High Rank Certificate


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Lifted Cover Cuts for BMIKP are not Strong Rank-1


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Towards the Next Step…


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Goal of Part III


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A General Procedure


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Deriving Nonlinear Cuts for Mixed Integer Programs: Applications


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Obtaining the Convex Hull of a Simple Bilinear Knapsack Set


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Obtaining the Convex hull of a Simple Bilinear Knapsack Set


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Obtaining Convex Hulls of Disjunctive Sets: An Example


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Obtaining the Convex Hull of Disjunctive Sets: A Result


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Obtaining the Convex Hull of Disjunctive Sets: An Example


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Obtaining the Convex Hull of Disjunctive Sets: An Example


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Some Comments


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Goal of Part IV


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Generalizing the Theory…


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Generalizing the Superadditive Lifting Theory…


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Application: Sequence Independent lifting for single-constraint problems


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Application: Sequence Independent lifting for single-constraint problems


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Conclusion & Future Work

MIP Lifting Techniques for Mixed Integer Nonlinear Programs

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