ABSTRACT

In this paper, we propose a projection-free accelerated method for solving convex optimization problems with unbounded feasible set. The method is an accelerated gradient scheme such that each projection subproblem is approximately solved by means of a conditional gradient scheme. Under reasonable assumptions, it is shown that an ϵ-approximate solution (concept related to the optimal value of the problem) is obtained in at most $\mathcal{O}(1/\sqrt{\epsilon })$ gradient evaluations and $\mathcal{O}(1/\epsilon )$ linear oracle calls. We also discuss a notion of approximate solution based on the first-order optimality condition of the problem and present iteration-complexity results for the proposed method to obtain an approximate solution in this sense. Finally, numerical experiments illustrating the practical behaviour of the proposed scheme are discussed.

摘要

在本文中，我们提出了一种无投影加速方法来解决具有无限可行集的凸优化问题。该方法是一种加速梯度方案，使得每个投影子问题都可以通过条件梯度方案来近似求解。在合理的假设下，最多可以得到一个ε-近似解（与问题的最优值相关的概念）$\mathcal{○}(1/\sqrt{\epsilon })$梯度评估和$\mathcal{○}(1/\epsilon )$线性预言机调用。我们还讨论了基于问题的一阶最优性条件的近似解的概念，并给出了所提出方法的迭代复杂度结果，以获得这个意义上的近似解。最后，讨论了说明所提出方案的实际行为的数值实验。