Abstract. This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums. Several extensions are discussed.
REF propose using the hierarchy of semi-definite relaxations to compute all the local optima which satisfy first and second order necessary conditions based on semi-definite relaxations.
The Hierarchy of Local Minimums in Polynomial Optimization
arXiv: Optimization and Control
Mathematics Computer Science