- Lagrangian-Conic Relaxations, Part II: Applications to Polynomial Optimization Problems Naohiko Arima(nao_arimame.com) Sunyoung Kim(skimewha.ac.kr) Masakazu Kojima(kojimais.titech.ac.jp) Kim-Chuan Toh(mattohkcnus.edu.sg) Abstract: We present the moment cone (MC) relaxation and a hierarchy of sparse Lagrangian-SDP relaxations of polynomial optimization problems (POPs) using the unified framework established in Part I. The MC relaxation is derived for a POP of minimizing a polynomial subject to a nonconvex cone constraint and polynomial equality constraints. It is an extension of the completely positive programming relaxation for QOPs. Under a copositivity condition, we characterize the equivalence of the optimal values between the POP and its MC relaxation. A hierarchy of sparse Lagrangian-SDP relaxations, which is parameterized by a positive integer $\omega$ called the relaxation order, is proposed for an equality constrained POP. It is obtained by combining a sparse variant of Lasserre's hierarchy of SDP relaxation of POPs and the basic idea behind the conic and Lagrangian-conic relaxations from the unified framework. We prove under a certain assumption that the optimal value of the Lagrangian-SDP relaxation with the Lagrangian multiplier $\lambda$ and the relaxation order $\omega$ in the hierarchy converges to that of the POP as $\lambda \rightarrow \infty$ and $\omega \rightarrow \infty$. The hierarchy of sparse Lagrangian-SDP relaxations is designed to be used in combination with the bisection and $1$-dimensional Newton methods, which was proposed in Part I, for solving large-scale POPs efficiently and effectively. Keywords: Polynomial optimization problem, moment cone relaxation, SOS relaxation, a hierarchy of the Lagrangian-SDP relaxations, exploiting sparsity. Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Research Report B-476, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8552, January (2014). Download: [PDF]Entry Submitted: 01/09/2014Entry Accepted: 01/09/2014Entry Last Modified: 01/09/2014Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.