一类非光滑凸优化的近似乘子法
The Proximal Method of Multipliers for a Class of Nonsmooth Convex Optimization
摘要：
对一类非光滑凸优化问题给出了乘子逼近法。该方法生成一系列的最小化问题（子问题）。我们表明，序列的子问题的解决方案的近似收敛到一个鞍点的拉格朗日，即使原来的最优化问题可能拥有多个解决方案。由于福廷的增广拉格朗日出现在子问题。与标准拉格朗日量相比，增广拉格朗日量的显着性质是它总是可微的，并且通常是半光滑可微的。这一事实使我们能够采用非光滑牛顿法计算近似的子问题。近似项作为目标函数的正则化，保证了牛顿系统的可解性，而不需要假设目标函数具有强凸性。我们利用非光滑牛顿法的理论，提供了一个严格的证明所提出的算法的整体收敛。
英文摘要：
This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the subproblems converges to a saddle point of the Lagrangian even if the original optimization problem may possess multiple solutions. The augmented Lagrangian due to Fortin appears in the subproblem. The remarkable property of the augmented Lagrangian over the standard Lagrangian is that it is always differentiable, and it is often semismoothly differentiable. This fact allows us to employ a nonsmooth Newton method for computing an approximation to the subproblem. The proximal term serves as the regularization of the objective function and guarantees the solvability of the Newton system without assuming strong convexity on the objective function. We exploit the theory of the nonsmooth Newton method to provide a rigorous proof for the global convergence of the proposed algorithm.
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