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Convex hull of two quadratic or a conic quadratic and a quadratic inequality

Sina Modaresi (sim23***at***pitt.edu)
Juan Pablo Vielma (jvielma***at***mit.edu)

Abstract: In this paper we consider an aggregation technique introduced by Yildiran, 2009 to study the convex hull of regions defined by two quadratic or by a conic quadratic and a quadratic inequality. Yildiran shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using Linear Matrix Inequalities (LMI). We show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic or by a strict conic quadratic and a strict quadratic inequality. We also show that in many cases under one additional assumption, these valid inequalities characterize the convex hull exactly. We also show that under certain topological conditions, the results from the open setting can be extended to characterize the closed convex hull of sets defined with non-strict conic and quadratic inequalities.

Keywords: Quadratic inequality, Conic quadratic inequality, Linear Matrix Inequality

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Global Optimization (Theory )

Category 3: Linear, Cone and Semidefinite Programming (Second-Order Cone Programming )

Citation: Technical report, Sloan School of Management, Massachusetts Institute of Technology, November 2014

Download: [PDF]

Entry Submitted: 11/17/2014
Entry Accepted: 11/17/2014
Entry Last Modified: 07/13/2016

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