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

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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

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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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A Superadditive Lifting Result for PS

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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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Obtaining Facets of PT using Superadditive Lifting

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Example

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Obtaining Facets of PT using Superadditive Lifting

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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 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 Convex Hulls of Disjunctive Sets: An Example

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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