一类最大特征值函数优化问题的UV-分解方法

UV-decomposition method for a class of maximum eigenvalue optimizations

研究一类最大特征值函数与一个仿射映射复合后的函数与一个二次连续可微的凸函数的和的无约束优化问题,许多的实际应用问题的约束优化问题可以转化为这种形式的无约束问题来求解。将处理非光滑问题的UV-分解方法应用于这一类无约束优化问题,先给出目标函数在某一点处的3种形式的UV-空间分解,证明了3种空间分解形式是等价的。其次,给出目标函数的U-Lagrange函数及它的一阶和二阶展开式。最后,基于UV-空间分解理论给出解决这样一类无约束优化问题的UV-分解算法,并证明此算法是超线性收敛的。文章结论为解决最大特征值函数的联合函数的优化问题提供了一种新的途径。

This paper is concerned with the unconstrained problem in which the object function is the sum of the composite function of the maximum eigenvalue function and an affine mapping and a finite-valued convex twice continuously differentiable function. The actual application problem of the constrained optimization problem can be solved by transforming it into this kind of unconstrained problem. We will apply UV-spaee theory which can solve nonsmooth optimization problem to the unconstrained problem. Firstly, three kinds of UV-space decomposition of the object function and the proof of the equivalency about the three kinds of UV-space decomposition were introduced. Moreover, the U- Lagrangian of the object function and its first order and second order expansions were shown later. Finally, an UV- decomposition algorithm with superlinear convergence is presented which is based on the UV-space decomposition theory. The conclusions in this paper provide a new method to solve the optimization problem of a class of maximum eigenvalue function.