TY - JOUR
T1 - Inhomogeneous polynomial optimization over a convex set
T2 - An approximation approach
AU - He, Simai
AU - Li, Zhening
AU - Zhang, Shuzhong
PY - 2015
Y1 - 2015
N2 - In this paper, we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set. The focus is on the design and analysis of polynomial-time approximation algorithms. The methods are able to deal with optimization models with polynomial objective functions in any fixed degrees. In particular, we first study the problem of maximizing an inhomogeneous polynomial over the Euclidean ball. A polynomial-time approximation algorithm is proposed for this problem with an assured (relative) worst-case performance ratio, which is dependent only on the dimensions of the model. The method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co-centered ellipsoids. Furthermore, the constraint set is extended to a general convex compact set. Specifically, we propose a polynomial-time approximation algorithm with a (relative) worst-case performance ratio for polynomial optimization over some convex compact sets, e.g., a polytope. Finally, numerical results are reported, revealing good practical performance of the proposed algorithms for solving some randomly generated instances.
AB - In this paper, we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set. The focus is on the design and analysis of polynomial-time approximation algorithms. The methods are able to deal with optimization models with polynomial objective functions in any fixed degrees. In particular, we first study the problem of maximizing an inhomogeneous polynomial over the Euclidean ball. A polynomial-time approximation algorithm is proposed for this problem with an assured (relative) worst-case performance ratio, which is dependent only on the dimensions of the model. The method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co-centered ellipsoids. Furthermore, the constraint set is extended to a general convex compact set. Specifically, we propose a polynomial-time approximation algorithm with a (relative) worst-case performance ratio for polynomial optimization over some convex compact sets, e.g., a polytope. Finally, numerical results are reported, revealing good practical performance of the proposed algorithms for solving some randomly generated instances.
KW - Approximation algorithm
KW - Inhomogeneous polynomial
KW - Polynomial optimization
KW - Tensor optimization
UR - http://www.scopus.com/inward/record.url?scp=84919343127&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84919343127&origin=recordpage
U2 - 10.1090/S0025-5718-2014-02875-5
DO - 10.1090/S0025-5718-2014-02875-5
M3 - RGC 21 - Publication in refereed journal
AN - SCOPUS:84919343127
VL - 84
SP - 715
EP - 741
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 292
ER -