@article{4dfdc923208e44a6844c0976748a4375,
title = "A geometrical analysis on convex conic reformulations of quadratic and polynomial optimization problems",
abstract = "We present a unified geometrical analysis on the completely positive programming (CPP) reformulations of quadratic optimization problems (QOPs) and their extension to polynomial optimization problems (POPs) based on a class of geometrically defined nonconvex conic programs and their convexification. The class of nonconvex conic programs minimize a linear objective function in a vector space V over the constraint set represented geometrically as the intersection of a nonconvex cone K ⊂ V, a face J of the convex hull of K, and a parallel translation L of a hyperplane. We show that under moderate assumptions, 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 L. The replacement procedure is applied for deriving the CPP reformulations of QOPs and their extension to POPs.",
keywords = "Completely positive programming reformulation, Conic optimization problems, Faces of the completely positive cone, Polynomial optimization problems, Quadratic programs",
author = "Sunyoung Kim and Masakazu Kojima and Toh, {Kim Chuan}",
note = "Funding Information: \ast Received by the editors January 8, 2019; accepted for publication (in revised form) January 23, 2020; published electronically April 28, 2020. https://doi.org/10.1137/19M1237715 Funding: The first author's research was supported by the National Research Foundation, grant 2017-R1A2B2005119. The second author's research was supported by Grant-in-Aid for Scientific Research (A) 19H00808. The third author's research was supported in part by ARF grant R-146-000-257-112 under the Ministry of Education of Singapore. \dagger Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Sudaemoon-gu, Seoul 03760, Korea (skim@ewha.ac.kr) \ddagger Department of Industrial and Systems Engineering, Chuo University, Tokyo 192-0393, Japan (kojima@is.titech.ac.jp). \S Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 (mattohkc@nus.edu.sg). Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics.",
year = "2020",
doi = "10.1137/19M1237715",
language = "English",
volume = "30",
pages = "1251--1273",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",
}