Nesterov Acceleration for Equality-Constrained Convex Optimization via Continuously Differentiable Penalty Functions
Nesterov Acceleration for Equality-Constrained Convex Optimization via Continuously Differentiable Penalty Functions
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DOI：
10.1109/lcsys.2020.3002687
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发表时间：
2021-03
影响因子：
3
通讯作者：
P. Srivastava;J. Cortés
中科院分区：
文献类型：
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作者：
P. Srivastava;J. Cortés
我们提出了一个使用Nesterov加速方法求解约束凸优化问题的框架。我们的方法包括首先将原始问题重新表述为使用连续可微精确惩罚函数的无约束优化问题。这种重新表述是基于用拉格朗日乘子函数代替原问题增广拉格朗日中的拉格朗日乘子。这些拉格朗日乘子函数的表达式依赖于目标函数和约束的梯度，即使原问题是凸的，也可以使无约束罚函数一般地非凸。建立了无约束罚函数为凸的目标函数和原问题约束的充分条件。这使我们能够使用Nesterov的加速梯度法进行无约束凸优化，并获得比最先进的一阶约束凸优化算法更好的保证收敛速度。仿真验证了我们的结果。
We propose a framework to use Nesterov’s accelerated method for constrained convex optimization problems. Our approach consists of first reformulating the original problem as an unconstrained optimization problem using a continuously differentiable exact penalty function. This reformulation is based on replacing the Lagrange multipliers in the augmented Lagrangian of the original problem by Lagrange multiplier functions. The expressions of these Lagrange multiplier functions, which depend upon the gradients of the objective function and the constraints, can make the unconstrained penalty function non-convex in general even if the original problem is convex. We establish sufficient conditions on the objective function and the constraints of the original problem under which the unconstrained penalty function is convex. This enables us to use Nesterov’s accelerated gradient method for unconstrained convex optimization and achieve a guaranteed rate of convergence which is better than the state-of-the-art first-order algorithms for constrained convex optimization. Simulations illustrate our results.