This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work. 

　　Publication: 

　　SIAM Journal on Mathematical Analysis, Volume 54, Issue 3, pp: 2876-2905. 

　　Author: 

　　Lei Huang 

　　Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China 

　　Jiawang Nie 

　　Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA 

　　Yaxiang Yuan 

　　Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 

　　Email: yyx@lsec.cc.ac.cn