Quadratic integer programming problems with cardinality constraint have many applications in real life. Portfolio selection is an important application in financial optimization. In this paper we develop an exact and efficient algorithm for quadratic integer programming problems with cardinality constraint. This iterative algorithm is actually a branch and bound method, which adopts a domain cut and partition scheme. Removing cardinality constraint and integrality constraints on variables, the relaxation problem is a quadratic programming problem. By solving the quadratic programming problem, we find a lower bound for the optimal objective function value of the primal problem. The domain cut technique is used to cut off some regions that do not contain any feasible solution better than the incumbent solution. Thus the optimality gap is reduced greatly. The finite number of iterations is clear from the fact that there are only a finite number of feasible integer solutions in the feasible region. Encouraging computational results are also reported in the paper.

中文翻译：

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带基数约束的二次整数规划问题的一种新算法

具有基数约束的二次整数规划问题在现实生活中有很多应用。投资组合选择是财务优化中的一个重要应用。在本文中，我们为具有基数约束的二次整数规划问题开发了一种精确有效的算法。这种迭代算法实际上是一种分支定界方法，它采用域切割和划分方案。去除变量的基数约束和完整性约束，松弛问题是二次规划问题。通过求解二次规划问题，我们找到了原始问题最优目标函数值的下界。域切割技术用于切割一些不包含任何比现有解决方案更好的可行解决方案的区域。从而极大地减小了最优性差距。从可行域中只有有限个可行整数解这一事实可以清楚地看出迭代次数是有限的。论文中还报告了令人鼓舞的计算结果。