When solving large-scale cardinality-constrained Markowitz mean–variance portfolio investment problems, exact solvers may be unable to derive some efficient portfolios, even within a reasonable time limit. In such cases, information on the distance from the best feasible solution, found before the optimization process has stopped, to the true efficient solution is unavailable. In this article, I demonstrate how to provide such information to the decision maker. I aim to use the concept of lower bounds and upper bounds on objective function values of an efficient portfolio, developed in my earlier works. I illustrate the proposed approach on a large-scale data set based upon real data. I address cases where a top-class commercial mixed-integer quadratic programming solver fails to provide efficient portfolios attempted to be derived by Chebyshev scalarization of the bi-objective optimization problem within a given time limit. In this case, I propose to transform purely technical information provided by the solver into information which can be used in navigation over the efficient frontier of the cardinality-constrained Markowitz mean–variance portfolio investment problem.

当求解大规模基数受限的Markowitz均值-方差投资组合投资问题时，即使在合理的期限内，精确的求解器也可能无法获得一些有效的投资组合。在这种情况下，无法获得从最佳可行解决方案（在优化过程停止之前）到真正有效解决方案之间的距离的信息。在本文中，我演示了如何向决策者提供此类信息。我的目标是在我之前的工作中开发的有效投资组合的目标函数值上使用上下限的概念。我在基于真实数据的大规模数据集上说明了所提出的方法。我讨论的情况是，顶级商业混合整数二次规划求解器无法提供有效的投资组合，这些投资组合试图通过给定时间限制内的双目标优化问题的Chebyshev标量化来推导。在这种情况下，我建议将求解器提供的纯技术信息转换为可在基数约束的Markowitz均值-方差投资组合问题的有效边界上导航的信息。