We present experimental and theoretical results on a method that applies a numerical solver iteratively to solve several non-negative quadratic programming problems in geometric optimization. The method gains efficiency by exploiting the potential sparsity of the intermediate solutions. We implemented the method to call quadprog of MATLAB iteratively. In comparison with a single call of quadprog, we obtain a 10-fold speedup on two proximity graph problems in $\mathbb{R}^d$ on some public data sets, a 10-fold speedup on the minimum enclosing ball problem on random points in a unit cube in $\mathbb{R}^d$, and a 5-fold speedup on the polytope distance problem on random points from a cube in $\mathbb{R}^d$ when the input size is significantly larger than the dimension; we also obtain a 2-fold or more speedup on deblurring some gray-scale space and thermal images via non-negative least square. We compare with two minimum enclosing ball software by G\"{a}rtner and Fischer et al.; for 1000 nearly cospherical points or random points in a unit cube, the iterative method overtakes the software by G\"{a}rtner at 20 dimensions and the software by Fischer et al. at 170 dimensions. In the image deblurring experiments, the iterative method compares favorably with other software that can solve non-negative least square, including FISTA with backtracking, SBB, FNNLS, and lsqnonneg of MATLAB. We analyze theoretically the number of iterations taken by the iterative scheme to reduce the gap between the current solution value and the optimum by a factor $e$. Under certain assumptions, we prove a bound proportional to the square root of the number of variables.

翻译：我们针对几何优化中迭代求解多个非负二次规划问题的方法进行了实验与理论分析。该方法通过利用中间解潜在的稀疏性提升计算效率。我们实现了该方法的迭代调用MATLAB中quadprog函数的程序。与单次调用quadprog相比，在$\mathbb{R}^d$空间中两个近邻图问题的公开数据集上获得10倍加速，在$\mathbb{R}^d$单位立方体内随机点的最小包围球问题上获得10倍加速，当输入规模显著大于维度时，在$\mathbb{R}^d$空间立方体内随机点的多面体距离问题上获得5倍加速；此外，通过非负最小二乘对灰度空间图像和热图像进行去模糊处理时，也实现了2倍以上加速。与Gärtner和Fischer等人开发的最小包围球软件相比，对于1000个近似共球面点或单位立方体内随机点，迭代方法在20维时超越Gärtner软件，在170维时超越Fischer软件。在图像去模糊实验中，该方法较其他可求解非负最小二乘的软件（包括带回溯的FISTA、SBB、FNNLS及MATLAB的lsqnonneg）具有更优表现。我们从理论上分析了迭代方案将当前解值与最优值差距缩小$e$倍所需的迭代次数。在特定假设下，我们证明了该次数与变量数的平方根成正比。