In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new “nonorthogonality” type of condition. We prove global convergence to stationary points when using an Armijo line search in the given framework. We further describe three different examples of active-set gradient related directions that guarantee linear convergence rate (under suitable assumptions). Finally, we report numerical experiments showing the effectiveness of the approach.

中文翻译：

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单纯形上非凸优化问题的主动集算法框架

在本文中，我们描述了一种新的有效集算法框架，用于最小化单位单纯形上的非凸函数。在每次迭代中，该方法都使用规则来识别满足新的“非正交”类型条件的活动变量（即，在固定点处为零的变量）和特定方向（我们将其称为活动集梯度相关方向）。 。在给定框架中使用Armijo线搜索时，我们证明了全局收敛到平稳点。我们进一步描述了保证线性收敛速度（在适当的假设下）的有效集梯度相关方向的三个不同示例。最后，我们报告了数值实验，证明了该方法的有效性。