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#### Numerical Analysis and Scientific Computingmathscidoc:1911.43010

SIAM J. OPTIM., 26, (3), 1429{1464, 2016
In this paper, we consider a class of optimization problems of minimizing a (at least) twice continuously di erentiable function (probably nonconvex) f(x) : Rn ! R over a product of multiple balls/spheres constraints. Upon rescaling the balls/spheres, we cast without loss of generality such class of minimization problems in the following form: (BCOP) 8>< >: min x2Rn f(x) s:t: ci(x) := kx[i]k2 􀀀 1 = 0; i 2 E; ci(x) := kx[i]k2 􀀀 1  0; i 2 I; where E = f1; 2; : : : ;m1g, I = fm1 + 1;m1 + 2; : : : ;mg, x[i] 2 Rpi , x = (xT [1]; xT [2]; : : : ; xT [m])T , n = Pm i=1 pi, and k  k stands for the `2 vector norm. Here, we introduce the notation x[i] 2 Rpi to represent the ith subvector of x 2 Rn and formulate the product of multiple ball/sphere constraints as a set of equality and inequality constraints. To simplify subsequent presentation, we call the above programming the ball/sphere constrained optimization problem (BCOP). We emphasize that this problem does not allow overlap among the variables x[i] and therefore the constraints are separable. However, these variables x[i] may be linked together through the objective function f(x).
@inproceedings{chungen2016a,
title={A FILTER ACTIVE-SET ALGORITHM FOR BALL/SPHERE CONSTRAINED OPTIMIZATION PROBLEM},
author={Chungen Shen, LEI-HONG ZHANG, and WEI HONG YANG},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191125144004064248521},
booktitle={SIAM J. OPTIM.},
volume={26},
number={3},
pages={1429{1464},
year={2016},
}
Chungen Shen, LEI-HONG ZHANG, and WEI HONG YANG. A FILTER ACTIVE-SET ALGORITHM FOR BALL/SPHERE CONSTRAINED OPTIMIZATION PROBLEM. 2016. Vol. 26. In SIAM J. OPTIM.. pp.1429{1464. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191125144004064248521.