多项式优化的最优性条件与最优化算法及其应用

Polynomial optimization problem is a kind of optimization problems whose objective function and constraints are all described by polynomial. It is one of the foundamental problems in the field of optimization, and is NP hard. It has applications in a large range of areas, including engineering, economic and finance.Specially, it has wide applications in signal processing. Up to now, there are very few efficient algorithms to slove general polynomial optimization problems, special for large size polynomial optimization problems. Moreover, it is always very difficult to verify whether the obtained solution is a global optimal solution or not.In this project,we will do some research works on global optimality conditions for polynomial optimization problems by using the previous research results and research experience obtained by the applicant and the team members. Some global optimality sufficien conditions for some kninds of polynomial optimization problems will be obtained by using the abstract subdifferential for nonconvex functions and abstract normal cone for sets, and some global optimality necessary conditions will be obtained according to the characteristic of these polynomial optimization problems. Then some global optimization algorithms with stopping criteria will be developed based on the obtained optimality conditions. Finally, we will try to discuss the optimality conditions and optimization methods for nonlinear programming problems by using the obtained results, and solve some real problems arising in signal processing. It is anticipated that this project will lead to some important results which are internationally advanced in the area of polynomial optimization theory and algorithms, and will propel the development in the filed of global optimization.

多项式优化问题是指目标函数和约束函数都为多项式的优化问题，是最优化领域最基本的问题之一，也是NP难问题，其应用广泛见于工程、经济、金融等，在信号处理中也有着非常广泛的应用。到目前为止，解一般多项式优化问题，特别是大型多项式优化问题的比较有效的算法还很少；而且大多数算法都无法判定所得到的解是否是全局最优解。本项目将充分利用申请者及其团队在该领域所取得的前期研究成果和研究经验，采用非凸函数的抽象次梯度和集合的抽象正则锥来研究一些多项式优化问题的全局最优性充分条件，结合这些多项式优化的特点来研究他们的全局最优性必要条件，并利用所得到的全局最优性条件来设计具有良好终止准则的全局最优化算法，并利用所得到的理论与方法来研究一般非线性规划的最优性条件和最优化算法，以及解决信号处理中的一些实际问题。本项目有望在多项式优化的理论与算法方面取得一些国际领先的研究成果，以推动全局优化研究领域的进一步发展。

多项式优化问题是指目标函数和约束函数都为多项式的优化问题，是最优化领域最基本的问题之一，也是NP难问题，其应用广泛见于工程、经济、金融等，在信号处理中也有着非常广泛的应用。到目前为止，解一般多项式优化问题，特别是大型多项式优化问题的比较有效的算法还很少；而且大多数算法都无法判定所得到的解是否是全局最优解。本项目充分利用申请者及其团队在该领域所取得的前期研究成果和研究经验，研究了带箱子约束和一般约束的一般多项式优化问题的全局最优性条件和最优化算法；研究了带线性约束的三次规划问题的全局最优性条件和最优化算法；研究了一些特殊四次规划问题的最优性条件和最优化算法；研究了一种新的非线性系统的最优化方法；研究了一些非光滑凸优化问题的一些新的算法；研究了一类带有偶合线性约束的可分凸规划问题的一种快速双近似梯度算法；研究了一类具有延迟的多代理最优化问题的分布镜像下降法等等。 本项目取得一些的很好的研究成果，对全局优化研究领域的进一步发展起到了很好的作用。