Fast Algorithms for Large Scale Convex Optimization Involving Linear Matrix Inequality

随着近年来可用计算能力的急剧提高，数值优化已成为分析和设计复杂系统的一种有吸引力的工具。在科学和工程中有大量的问题可以表述为涉及线性矩阵不等式的凸优化问题。这个问题本身就很难解决，因为不平滑通常发生在解决方案上。此外，随着系统变得越来越复杂，所产生的优化问题往往具有大量的决策变量和约束。因此，一些基本的计算组件，如矩阵的反演，特征值和特征向量的计算，或Hessian矩阵的计算，大多数方法需要的，要么被禁止，要么执行起来非常昂贵。因此，有必要使用近似值，同时保持较快的收敛速度。＊＊＊

9411664 Fan With recent dramatic increase in available computing power, numerical optimization has become an attractive tool for analysis and design of complex systems. There are a large number of problems in science and engineering which can be formulated as convex optimization problems involving linear matrix inequality. This problem is inherently difficult to solve as the nonsmoothness usually occurs at the solution. Further, as systems are becoming more complex, the resulting optimization problems tend to have a large number of decision variables as well as constraints. Therefore, several basic computation components such as inversion of a matrix, eigenvalue and eigenvector calculation, or evaluation of the Hessian matrix, that needed for most methods become either prohibited or very costly to perform. It is then necessary to use approximations and yet have the fast rate of convergence preserved. ***