Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is still significantly harder to solve than a similar-size Linear Program (LP). It is well-known that a semidefinite program can be written as an LP with infinitely-many cuts that could be solved by repeated separation in a Cutting-Planes scheme; this approach is likely to end up in failure. We proposed in [Projective Cutting-Planes, Daniel Porumbel, Siam Journal on Optimization, 2020] the Projective Cutting-Planes method that upgrades t he well-known separation sub-problem to the projection sub-problem: given a feasible $y$ inside a polytope $P$ and a direction $d$, find the maximum $t^*$ so that $y+t^*d\in P$. Using this new sub-problem, one can generate a sequence of both inner and outer solutions that converge to the optimum over $P$. This paper shows that the projection sub-problem can be solved very efficiently in a semidefinite programming context, enabling the resulting method to compete very well with state-of-the-art semidefinite optimization software (refined over decades). Results suggest it may the fastest method for matrix sizes larger than $2000\times 2000$.

翻译：为寻求组合优化问题更紧的松弛，半定规划作为线性规划的推广，能够提供更优的界限且仍可在多项式时间内求解。然而在实际中，半定规划求解难度仍显著高于同等规模的线性规划（LP）。众所周知，半定规划可被表述为具有无穷多个切割的线性规划，这些切割可通过切割平面方案中的重复分离过程求解；但该方法往往以失败告终。我们在文献[Projective Cutting-Planes, Daniel Porumbel, Siam Journal on Optimization, 2020]中提出了投影切割平面法，将经典的分离子问题升级为投影子问题：给定可行点$y$位于多面体$P$内部及方向$d$，求最大$t^*$使得$y+t^*d\in P$。通过利用这一新子问题，可生成收敛于$P$上最优解的内点与外点交替序列。本文证明，在半定规划背景下该投影子问题可被高效求解，使得所提方法能够与经过数十年优化的最先进半定优化软件竞争。结果表明，对于矩阵规模超过$2000\times 2000$的情形，该方法可能为目前最快的求解方案。