加速Newton-Dinkelbach方法及其在二元不等式组中的应用
An Accelerated Newton-Dinkelbach Method and its Application to Two Variables Per Inequality Systems
摘要：
我们提出了一个加速，或“前瞻性”版本的牛顿-Dinkelbach方法，一个众所周知的技术，用于解决分数和参数优化问题。这种加速在每两次迭代内将当前的最优解和最优解之间的Bregman发散减半。使用Bregman发散作为结合组合参数的势，我们在三个应用领域中获得了强多项式算法：（i）对于线性分式组合优化，我们给出了$O(m \log m)$迭代的收敛界;以前的最佳界是Wang et al.（2006）的$O(m^2 \log m)$。(ii)本文给出了求解二元线性可行不等式组（2 VPI）的一个强多项式标号校正算法。对于具有$n$变量和$m$约束的2 VPI系统，我们的算法运行在$O(mn)$迭代中。对于一般的2 VPI系统，每次迭代需要$O(mn)$时间，而对于确定性马尔可夫决策过程（DMDP）的特殊情况，每次迭代需要$O(m + n \log n)$时间。这扩展并加强了Madani（2002）的先前结果，该结果显示了用于求解DMDP的Newton-Dinkelbach方法的变体的弱多项式界。(iii)我们给出了Goemans等人（2017）的参数次模函数最小化结果的简化变体。
英文摘要：
We present an accelerated, or 'look-ahead' version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of $O(m \log m)$ iterations; the previous best bound was $O(m^2 \log m)$ by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with $n$ variables and $m$ constraints, our algorithm runs in $O(mn)$ iterations. Every iteration takes $O(mn)$ time for general 2VPI systems, and $O(m + n \log n)$ time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton-Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
