结合区间和符号方法的多变元多项式的混合全局最优化方法

Abstract

提出了一种求解带边界约束的多变元多项式全局最优解的混合方法.混合是指在优化的过程中结合了区间方法、符号方法和数值方法.一方面通过区间方法在舍入误差存在的情况下得到包含最优解且满足要求的任意小区间;另一方面通过符号方法解决当Jacobi矩阵在区间内某点奇异时区间牛顿法无法验证驻点的存在性与惟一性的问题;同时,利用数值优化方法(如BFGS方法)来有效克服区间方法运算速度慢的缺点.此外,文中的算法非常有利于并行化,因此可以进一步提高算法效率.

A hybrid algorithm for global optimization of multivariate polynomials with bound constraints is proposed.Small intervals containing global optima in the presence of rounding errors are obtained by interval methods.Symbolic methods are employed when the Jacobi matrix is singular.Numerical methods are also adopted to improve the efficiency since the interval methods are usually very slow.Furthermore,the performance of our algorithm can be improved by parallelization.