In this article, we present a reliable hybrid algorithm for solving convex quadratic minimization problems. At the kth iteration, two points are computed: first, an auxiliary point ${\dot{x}}_k$ is generated by performing a gradient step using an optimal steplength, and second, the next iterate x_k + 1 is obtained by means of weighted sum of ${\dot{x}}_k$ with the penultimate iterate x_k − 1. The coefficient of the linear combination is computed by minimizing the residual norm along the line determined by the previous points. In particular, we adopt an optimal, nondelayed steplength in the first step and then use a smoothing technique to impose a delay on the scheme. Under a modest assumption, we show that our algorithm is Q-linearly convergent to the unique solution of the problem. Finally, we report numerical experiments on strictly convex quadratic problems, showing that the proposed method is competitive in terms of CPU time and iterations with the conjugate gradient method.

中文翻译：

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一种严格凸二次规划的混合梯度方法

在本文中，我们提出了一种可靠的混合算法来解决凸二次最小化问题。在第k次迭代时，计算两个点：首先，一个辅助点${\dot{X}}_{克}$是通过使用最佳步长执行梯度步骤生成的，其次，下一个迭代x _{k  + 1}是通过加权求和获得的${\dot{X}}_{克}$倒数第二次迭代x _{k  − 1}。线性组合的系数是通过沿由先前点确定的线最小化残差范数来计算的。特别是，我们在第一步中采用最佳的、非延迟的步长，然后使用平滑技术对方案施加延迟。在适度的假设下，我们表明我们的算法对问题的唯一解是 Q 线性收敛的。最后，我们报告了严格凸二次问题的数值实验，表明所提出的方法在 CPU 时间和迭代方面与共轭梯度方法相比具有竞争力。