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.

翻译：Invex程序是一类特殊的非凸问题，其在每个驻点处都能达到全局最小值。虽然经典的一阶梯度下降方法可以求解此类问题，但收敛速度非常缓慢。本文针对一般类别的Invex问题提出了新的一阶算法。我们识别了算法收敛的充分条件，并给出了收敛速率。此外，我们突破了无约束问题的局限，提出了一种新颖的投影梯度方法用于求解带约束的Invex程序，并保证了收敛速率。我们将我们的结果与现有的一阶算法在多种无约束和约束Invex问题上进行了比较与对比。据我们所知，我们提出的算法是首个能够求解约束Invex程序的算法。