Research On The Number Of The Minimum And The Approximation Bound Of Polynomial Optimization Problem

Polynomial optimization problem is a kind of special optimization problem which is expressed by polynomials in objective function and constraint condition,and it is widely used in practical problems such as signal processing and medical imaging,but the polynomial optimization problem is generally non-convex and NP hard,and the global optimal problem is not easy to solve,so study of polynomial optimization problem has important theoretical significance and application value.The number of extremes of the polynomial optimization problem is related to the 16th problem in 23 questions raised by Hilbert.Since 1993,Durfee?Kronenfeld et al studied the minimum number of two variables and no further research results,until 2003,Qi Liqun and Koklay give the guess of the minimum number,but the proof is not given at that time,and in the article they put forward whether give the proof of their guess.So the paper first give the proof of the conjecture of Qi Liqun and Koklay that a 2n-degree or 2n+1-degree polynomial of r variables has at mostn~r isolated local minima when n?2.Secondly,the original tight constraints are transformed into sum of squares by Lasserre,and its founded conditions are given.We prove that the founded conditions make the sum of squares feasible.Then we find the lower bound of the objective function which is in a set of constraints.Finally,on the basis of the original approximation bound theorem,it is further transformed and a new approximation bound theorem is obtained.The new approximation theorem reduces the parameters by the original theorem and makes it easy to calculate.