Invex programs are a special kind of non-convex problems which attain global minima at every stationary point. While classical first-order gradient descent methods can solve them, they converge very slowly. In this paper, we propose new first-order algorithms to solve the general class of invex problems. We identify sufficient conditions for convergence of our algorithms and provide rates of convergence. Furthermore, we go beyond unconstrained problems and provide a novel projected gradient method for constrained invex programs with convergence rate guarantees. We compare and contrast our results with existing first-order algorithms for a variety of unconstrained and constrained invex problems. To the best of our knowledge, our proposed algorithm is the first algorithm to solve constrained invex programs.

我们提出了一种新的一阶算法来解决一般类别的Invex问题。我们确定了算法收敛的充分条件并给出了收敛速度。此外，我们还提供了一种针对约束Invex问题的新型投影梯度方法，并提供了收敛速度保证。我们将我们的结果与现有的一阶算法进行了比较，并对各种无约束和约束Invex问题进行了对比。据我们所知，我们的提出的算法是第一个解决约束Invex问题的算法。

INVEX程序：一阶算法及其收敛性