SIAM Journal on Optimization, Volume 30, Issue 1, Page 1007-1032, January 2020.
Given a polytope $\mathcal P$, an interior point ${x}\in\mathcal P$, and a direction ${d}\in\mathbb{R}^n$, the projection of ${x}$ along ${d}$ asks to find the maximum step length $t^*$ such that ${x}+t^*{d}\in\mathcal P$; we say ${x}+t^*{d}$ is the pierce point obtained by projection. In [D. Porumbel, Math. Program., 155 (2016), pp. 147--197], we solely explored the idea of projecting the origin $0_n$ along integer directions only, focusing on dual polytopes $\mathcal P$ in Column Generation models. This work addresses a more general projection subproblem, considering arbitrary interior points ${x}\in\mathcal P$ and arbitrary noninteger directions ${d}\in\mathbb{R}^n$, in areas beyond Column Generation.The projection subproblem generalizes the separation subproblem of the well-known Cutting-Planes. We propose a new algorithm, Projective Cutting-Planes, that relies on this projection subproblem to optimize over polytopes $\mathcal P$ with prohibitively many constraints. At each iteration, this new algorithm selects a point ${x}_{new}$ on the segment joining the points ${x}$ and ${x}+t^*{d}$ determined at the previous iteration. Then, it projects ${x}_{new}$ along the direction ${d}_{new}$ pointing towards the current optimal (outer) solution (of the current outer approximation of $\mathcal P$), so as to generate a new pierce point ${x}_{new}+t^*_{new} {d}_{new}$ and a new constraint of $\mathcal P$. By reoptimizing the linear program enriched with this new constraint, the algorithm finds a new current optimal (outer) solution and moves to the next iteration by updating ${x}={x}_{new}$ and ${d}={d}_{new}$. Compared to Cutting-Planes, the main advantage of Projective Cutting-Planes is that it has a built-in functionality to generate a feasible inner solution ${new}+t^*{d}$ at each iteration. These inner solutions converge iteratively to an optimal solution ${opt}(\mathcal P)$, and so Projective Cutting-Planes is more similar to an interior point method than to the Simplex method. Numerical experiments in different optimization settings confirm the potential of the proposed ideas.

中文翻译：

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投影切面

SIAM优化杂志，第30卷，第1期，第1007-1032页，2020年1月。
给定一个多面体$ \ mathcal P $，一个内点$ {x} \ in \ mathcal P $和一个方向$ {d} \ in \ mathbb {R} ^ n $，$ {x} $的投影沿$ {d} $要求找到最大步长$ t ^ * $，使得$ {x} + t ^ * {d} \ in \ mathcal P $；我们说$ {x} + t ^ * {d} $是通过投影获得的刺穿点。在[D. Porumbel，数学。计划，第155卷（2016），第147--197页]，我们仅探讨了仅沿整数方向投影原点$ 0_n $的想法，着眼于列生成模型中的双多面体$ \ mathcal P $。这项工作解决了一个更一般的投影子问题，在列生成之外的区域中考虑了任意内部点$ {x} \ in \ mathcal P $和任意非整数方向$ {d} \ in \ mathbb {R} ^ n $。子问题概括了众所周知的“切割平面”的分离子问题。我们提出了一种新算法 投影切割平面，它依赖于此投影子问题来优化具有过多约束的多面体$ \ mathcal P $。在每次迭代中，此新算法都会在连接上一次迭代中确定的点$ {x} $和$ {x} + t ^ * {d} $的线段上选择点$ {x} _ {new} $。然后，它沿着$ {d} _ {new} $的方向投影$ {x} _ {new} $，指向当前的最佳（外部）解决方案（当前外部近似值\ P P）。生成新的刺穿点$ {x} _ {new} + t ^ * _ {new} {d} _ {new} $和$ \ mathcal P $的新约束。通过重新优化富含此新约束的线性程序，该算法找到了新的当前最佳（外部）解决方案，并通过更新$ {x} = {x} _ {new} $和$ {d} = { d} _ {new} $。与切割平面相比，投影切割平面的主要优点是它具有内置功能，可以在每次迭代时生成可行的内部解决方案$ {new} + t ^ * {d} $。这些内部解决方案迭代地收敛为最优解$ {opt}（\ mathcal P）$，因此，投影切割平面与内部点方法相比，与单纯形方法更相似。在不同优化设置下的数值实验证实了所提出思想的潜力。