几类无线通信中的非凸矩阵优化问题及算法研究

This project mainly focuses on several kinds of nonconvex matrix optimization problems in wireless communications, with the background information as the multiple-input-multiple-output interference channel with relay aided. First, we would like to build up the sum rate maximization model with power constraints. The objective function is highly nonlinear, and thus difficult to analyze directly. We plan to approximate the objective function with different approaches and compare different results. We will solve the problem with nonlinear optimization technique like alternating iteration method, trust region idea and so on. We will also analyze the computational complexity as well as the convergence result of the algorithm. Second, we would like to consider the transmit power minimization problem with rate constraints. We plan to approximate the rate constraint with approaches that are easier to handle, and simplify the problem by alternating iteration method. We would like to propose a nonconvex sequential quadratic programming method, taking into account the special structure of the subproblems. The computational complexity and the convergence property will also be analyzed. Third, we want to solve the multiple stream transmitted problems to maximize sum rate and to minimize transmit power, aiming to improve the communication efficiency. The problems become more complicated with orthogonal constraints. We plan to shrike the feasible region, and then solve it with dual method. Further, we would like to propose the distributed versions of our algorithms for application. This project will improve the development of the interdisciplinary between optimization and wireless communications.

本项目主要研究无线通信中的几类非凸矩阵优化问题，背景是多发多收中继辅助的干扰信道。首先，我们将建立功率约束下的传输速率极大化模型。难点主要在于目标函数高度非线性，难以直接求解。为此我们将考虑多种近似函数，运用交替迭代、信赖域等方法求解和简化问题，并分析算法的计算复杂度以及收敛性质。其次，我们将考虑带有传输速率约束的发送功率极小化模型，使用近似函数近似约束中难以处理的传输速率函数，运用交替迭代的方法简化问题，并设计非凸的逐步二次规划算法求解，同时分析算法的计算复杂度和收敛性质。第三，我们希望求解多数据流传输的速率极大化和发送功率极小化问题，以提高信道的传输效率。正交约束的加入使得问题更加复杂。我们考虑用椭球近似可行域，再用对偶方法求解转化后的子问题。此外，我们还将改进算法，提供分布式实现进一步降低复杂度，更利于实际应用。本项目将促进优化与无线通信交叉学科的发展。

本项目主要针对具有无线通信应用背景和特殊数据结构的矩阵优化问题，提出高效、低复杂度的算法。对于无线通信中中继辅助多发多收干扰信道，我们主要考虑了在一定发送功率限制条件下，极大化网络传输总速率的问题。该类问题是非凸、高度非线性的矩阵优化问题。针对这一问题，我们做了一系列的工作：我们提出了全新的近似函数、全新的优化模型、低复杂度算法以及算法的分布式实现。其中，我们针对特殊结构的非凸二次约束二次规划子问题，提出了基于信赖域思想的可行压缩算法，结合逐步二次规划算法，从而得到子问题的KKT点。针对带有正交约束的二次规划子问题，我们提出了对偶算法，并证明了在一定条件下该算法收敛到子问题的最优解。我们还证明了将数据流数作为变量的优化问题的简化形式是NP难的，也针对该问题提出了相应的多初值启发式算法。我们基于上述算法，提出了两种易于实现的分布式算法，并分析了其计算和通信效率。此外，针对同一通信模型的能效极大化问题，我们也提出了相应的低复杂度算法。我们运用分式优化技巧、交替迭代方法，并引入线性最优的解码矩阵，从而将问题化简为一系列二次约束二次规划子问题的求解，并证明简化前后的问题具有相同的KKT点。本项目已按照计划进行，并完成了预期成果。