Numerical Algorithms For Two Polynomial Optimization Problems

Polynomial optimization problems is an important problem in nonlinear programming.In the past two decades,their theories and algorithms have been deeply developed.Based on the existing classic theories and algorithms,the paper studies numerical algorithms of the local saddle point values sorting problem of unconstrained polynomial and polynomial minimax problem.For the local saddle point values sorting problem of unconstrained polynomial,an algorithm is proposed based on optimality condition,Lasserre semidefinite relaxation method.The algorithm can determine whether local saddle points exist or not.If there exist local saddle points,the algorithm can get different local saddle points.The algorithm does not require the objective function to have concavity or convexity,Numerical experimental results show that it is effective.For the polynomial minimax problem,the paper first transforms the original problem into a minimization problem by introducing a new variable,and then solve the transformed minimization problem for obtaining the optimal solutions of the original problem by the Lasserre semidefinite relaxation method.The algorithm is convergent,Numerical experiments show that the algorithm is effective.