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A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

Sunyoung Kim(skim***at***ewha.ac.kr)
Masakazu Kojima(kojimamasakazu***at***mac.com)
Kim-Chuan Toh(mattohkc***at***nus.edu.sg)

Abstract: We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective function in a linear space $V$, and the constraint set is represented geometrically as the intersection of a nonconvex cone $K \subset V$, a face $J$ of the convex hull of $K$ and a parallel translation $L$ of a supporting hyperplane of the nonconvex cone $K$. We show that under a moderate assumption, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of $J$ and the hyperplane $L$. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems.

Keywords: Completely positive reformulation of quadratic and polynomial optimization problems, conic optimization problems, hierarchies of copositivity, faces of the completely positive cone

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Linear, Cone and Semidefinite Programming (Other )

Citation:

Download: [PDF]

Entry Submitted: 01/08/2019
Entry Accepted: 01/08/2019
Entry Last Modified: 01/08/2019

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