Numerical Methods for Structured Nonsmooth Optimization

许多有趣的优化问题都是非光滑的，即要优化的函数或相关约束在其定义域的某些部分上是不可微的。本课题的主要目标是为一些重要的非光滑优化问题设计有效的算法。只有充分利用现有的结构，才能做到这一点。由于缺乏平滑性，标准算法不适用。调查主要涉及三个问题。首先是依赖于参数的对称矩阵特征值的优化问题。通常，解涉及多个特征值，这些特征值不是矩阵元素的光滑函数。感兴趣的应用包括柱和板抗屈曲的优化设计。第二个主要问题是为一个广泛的问题类设计方法：凸组合优化。通过对广义梯度集进行适当的参数化，期望得到具有超线性收敛性的有效方法。第三个问题涉及非对称矩阵特征值的非lipschitz优化问题。例如，控制系统稳定性中的一个重要应用是最小化矩阵的谱半径。

Many interesting optimization problems are nonsmooth, i.e. either the function to be optimized or an associated constraint is not differentiable on some part of its domain. The main goal of this project is the design of efficient algorithms for some important classes of nonsmooth optimization problems. This is possible only by taking full advantage of the structure which is present. Because of the lack of smoothness, standard algorithms are not applicable. The investigation primarily addresses three issues. The first is the optimization of eigenvalues of symmetric matrices which depend on parameters. Typically, solutions involve multiple eigenvalues which are not smooth functions of the matrix elements. Applications of interest include the optimal design of columns and plates against buckling. The second major issue is the design of methods for a broad problem class: convex composite optimization. By appropriate parameterization of the generalized gradient set it is hoped to obtain efficient methods with superlinear convergence. The third issue concerns non-Lipschitz optimization problems involving eigenvalues of nonsymmetric matrices. For example, an important application arising in the stability of control systems is to minimize the spectral radius of a matrix.