Research On Several Problems Related To Polynomial Optimization

Polynomial optimization is an important research area in the field of mathematical optimization.It has played an important role in a series of disciplines such as economics,management,statistics,and computer science,and therefore it has important research value and development prospects.This paper focuses on a number of problems related to polynomial optimization.See below for details:First of all,we introduce the research background,the significance of the study,and the latest research progress at home and abroad.To solve all the real solutions of polynomial equations effectively,traditional methods cannot be used.We divide the solution set of polynomial equations appropriately,and then formulate the original problem as a series of polynomial optimization subproblems.If the solution set is finite,we propose a semidefinite relaxation algorithm for solving all the real solutions,and prove the finite convergence of the algorithm.Numerical results are given.The tensor complementarity problem is an important special case of the nonlinear complementarity problem.At present,there is no effective numerical algorithm can calculate all the real solutions of the problem.In the existing literature,most of the theoretical results are also based on the assumption of some classes of structured tensors.From the viewpoint of polynomial optimization,we give a semidefinite relaxation algorithm for solving general tensor complementary problems.When the solution set is finite,the algorithm can compute all real solutions.We prove the convergence of the algorithm and verify the effectiveness of the algorithm by numerical results.The semidefinite complementarity problem with polynomial matrices is a novel polynomial optimization problem.So far,there has been very little research in this area and there is no effective numerical method which can solve it.We transform the original problem into some optimization problems with semidefinite constraints.Using the sum-of-squares decomposition of polynomial matrices,a semidefinite algorithm for such problems is proposed to generate a sequence of monotonically increasing lower bounds of the optimal value,and the sequence convergence to the optimal value.In turn,the original problem can be solved.The quadratic programming with two quadratic constraints is a classical polynomial optimization problem.We propose a modified exact Jacobian semidefinite relaxation algorithm for this problem,which has better convergence properties than Lasserre's hierarchy of semidefinite relaxation method.Compared with the exact Jacobian method proposed by Nie,we avoid introducing too many constraints in a single problem and reduce the lowest relaxation order of the relaxation problem.In addition,this method can not only be used to solve the problem with two quadratic constraints,but also be generalized to other inequality constraints problems.Based on the above-mentioned work,we propose a new subspace method.The classical Celis-Dennis-Tapia problem is a special case of polynomial optimization problem.As a subproblem of some trust region algorithms for nonlinear problem,it has important research significance.When using the semidefinite relaxation method to solve this problem,as the number of variables increases,the size of the relaxation problem increases exponentially.Therefore,when the scale of the problem is too large,many numerical methods are no longer appropriate.In order to overcome the defects,we propose a subspace approach to the Celis-Dennis-Tapia problem.We show how to find subspaces that satisfy subspace properties for this problem,by using the eigendecomposition of the Hessian matrix of the objection function.The dimensions of these subspaces are investigated.Further,we apply this method to the trust region subproblems and some special quadratically constrained quadratic programming problems.This method is different from the classical subspace methods that we are familiar with.It contains new research ideas,construction methods,and proof skills.The quadratically constrained quadratic programming problem is a generalization of the Celis-Dennis-Tapia problem.We prove the subspace properties for this problem,propose different subspace construction methods and simplify the calculation.Then we can solve the problem in a subspace instead of the original space.By analyzing the structure of these subspaces,we provide an estimate of their dimensions.When the dimension of the subspace is much smaller than the original one,the computational complexity can be greatly reduced.In addition,the subspace problem is exactly equivalent to the original problem under the same form.Therefore,this method can be combined with other algorithms.