圆装填问题：各种凸化技术的理论比较
The circle packing problem: a theoretical comparison of various convexification techniques
AI总结：
针对单位正方形内最大等圆装填这一非凸二次约束优化难题，本文比较多种凸化技术所得 LP 与 SDP 松弛的理论强度，并证明 Anstreicher 的猜想。
摘要：
我们将在单位正方形内装填最大半径等圆的问题视为一个数学优化问题。由于存在非重叠约束，该问题是一个众所周知的困难非凸二次约束优化问题，具有许多局部最优解。我们考察了若干流行的凸化技术，由此产生了圆装填问题的线性规划松弛和半定规划松弛。我们从理论上比较了这些松弛的强度，从而证明了 Anstreicher（JOGO，2009）提出的猜想。我们的结果为非重叠约束凸化的现有机制为何无效提供了理论依据。
英文摘要：
We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle packing problem. We compare the strength of these relaxations theoretically, thereby proving the conjectures by Anstreicher (JOGO, 2009). Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.
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