Numerical Methods for Structured Nonsmooth Optimization

许多有趣的优化问题是非光滑的，即 要优化的函数或相关联的约束不 在其域的某个部分可微。 这个的主要目标 项目是一些重要的高效算法的设计 一类非光滑优化问题。 这只有通过 充分利用现有的结构。 因为 缺乏平滑性，标准算法不适用。 调查主要涉及三个问题。 一是 对称矩阵特征值优化问题 参数 通常，解决方案涉及多个特征值， 不是矩阵元素的光滑函数。 的应用 感兴趣的包括柱和板的优化设计， 屈曲 第二个主要问题是设计方法， 问题类：凸复合优化。 通过适当 广义梯度集的参数化，希望获得 超线性收敛的有效方法。 第三个问题 涉及非Lipschitz优化问题的特征值 非对称矩阵 例如，一个重要的应用程序 控制系统的稳定性是使谱半径最小化 一个矩阵。

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.