This paper focuses on the cardinality constrained mean-variance portfolio optimization, in which only a small number of assets are invested. We first treat the covariance matrix of asset returns as a diagonal matrix with a special matrix processing technique. Using the dual theory, we formulate the lower bound problem of the original problem as a max-min optimization. For the inner minimization problem with the cardinality constraint, we obtain its analytical solution for the portfolio weights. Then, the lower bound problem turns out to be a simple concave optimization with respect to the Lagrangian multipliers. Thus, the interval split method and the supergradient method are developed to solve it. Based on the precise lower bound, the depth-first branch and bound method are designed to find the global optimal investment selection strategy. Compared with other lower bounds and the current popular mixed integer programming solvers, such as CPLEX and SCIP, the numerical experiments show that our method has a high searching efficiency. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12101317, 12271071, and 11991024], the Natural Science Foundation of Jiangsu Province [Grant BK20200819], the Team Project of Innovation Leading Talent in Chongqing [Grant CQYC20210309536], the Contract System Project of Chongqing Talent Plan [Grant cstc2022ycjh-bgzxm0147], and the Philosophy and Social Science Fund of Education Department of Jiangsu Province [Grant 2020SJA0168]. K.F.C. Yiu is supported in part by the Research Grants Council of Hong Hong [Grant PolyU 15223419], the Hong Kong Polytechnic University [Grants 4-ZZPT and 1-WZ0E], and the Research Centre for Quantitative Finance [Grant 1-CE03]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0344 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0344 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

One of the most crucial distinctions between Chinese and Western languages is that the former is based on ideograms, whereas the latter is based on phonograms. Due to this distinction, Western learners of Chinese often experience more difficulties in grasping correct character stroke sequence and/or stroke direction relative to native Chinese speakers. In this paper, we designed a HanZi writing environment with automatic feedback to address the above issue. Before the collection of HanZi characters on a massive scale, we conducted a pilot study to collect handwritten Chinese samples from 160 college students in the U.S. The findings from this study enabled us to further refine the learning environment and design optimal learning and teaching strategies for learners and teachers.