In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results.

在本文中，我们考虑最小化受稀疏约束的连续可微函数的问题。我们将此问题表述为等效的析取约束优化程序。然后，我们通过使用稀疏集的条件锥和正锥来扩展一些众所周知的约束条件，并证明这些约束条件可以用于获得一阶最优条件。此外，我们通过定义一个新的广义凸性概念来给出一阶充分最优性条件。此外，我们提出了稀疏约束优化问题的二阶必要和充分最优性条件。最后，我们提供了一些例子和特殊情况来说明获得的结果。