AI 中文总结
该研究提出结合熵平滑、Frank-Wolfe方法与分支定界的离散特征值优化方法，在两类问题上与SCIP-SDP对比，高维及带组合结构问题上表现更优。
AI 中文摘要
我们研究在组合与整数约束下的最小特征值最大化问题。提出一种基于分支定界的新方法，结合最小特征值函数的熵平滑技术与针对约束凹松弛的Frank-Wolfe方法，通过线性优化oracle利用组合结构。我们建立了近似与收敛性保证，包括基于部分特征分解计算的截断梯度的相关保证，并引入基于秩和特征值的剪枝及基于对偶性的变量固定策略。在E-最优实验设计与最大代数连通性问题上评估该方法，并与SCIP-SDP对比，结果表明，该方法在高维实例及具有额外组合结构的问题上表现尤为有效，而SCIP-SDP在约束更简单的中等规模实例上性能更优。
英文摘要
We study the maximization of the minimum eigenvalue under combinatorial and integrality constraints. We propose a new approach based on branch-and-bound combines entropic smoothing of the minimum eigenvalue function and Frank-Wolfe methods over concave relaxations of the constraints, thereby exploiting combinatorial structure through linear optimization oracles. We establish approximation and convergence guarantees, including for truncated gradients computed from partial eigendecompositions, and introduce rank- and eigenvalue-based pruning and duality-based variable fixing. We evaluate the method on E-optimal experimental design and maximum algebraic connectivity problems and compare it with SCIP-SDP. The results show that our approach is particularly effective for large-dimensional instances and problems with additional combinatorial structure, whereas SCIP-SDP performs better on moderately sized instances with simpler constraints.
Comments36 pages, 10 tables, 12 figures
