求解一类凸优化问题的区间迭代算法

Interval Iterative Algorithm for a Class of Convex Optimization

研究了一类非线性带约束的凸优化问题的求解。利用Kuhn-Tucker条件将凸优化问题等价地转化为多变元非线性方程组的求解问题。基于区间算术的包含原理及改进的Krawczyk区间迭代算法,提出一个求解凸优化问题的区间算法。对于目标函数和约束函数可微的凸优化,所提算法具有全局寻优的特性。在数值实验方面,与遗传算法、模式搜索法、模拟退火法及数学软件内置的求解器进行了比较,结果表明所提算法就此类凸优化问题能找到较多且误差较小的全局最优点。

This paper studied solving method for nonlinear constrained convex optimization and pointed out how to make use of Kuhn-Tucker condition to transfer convex programming to find roots of multivariate nonlinear systems equivalently.Our interval approach for finding optimal solutions can be combined with the principle of interval arithmetic and the modified iterative version given by R.Krawczyk.In the case of a convex programming with differentiable objective and constrained functions,the ability of global optimization can be improved.We illustrated the numerical experiments in contrast to the classical algorithms such as genetic algorithm,pattern search method,simulated annealing method and some built-in facilities of mathematical software packages.The implementation indicates that our method can provide a relative guaranteed bound on the error of the computed value and obtain more global optimal solutions.