We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor $O(n^{1.5}\log n)$ for an $n$-variable linear program in standard form. The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in $O(2^{n} n^{1.5}\log n)$ iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.

翻译：我们开发了一种求解线性规划的新型内点法。该算法具有普适性，对于标准形式下含有n个变量的线性规划，其迭代次数与任何使用自和谐障碍函数的内点法相比，仅差一个因子O(n^{1.5} log n)。内点法的运行时间界限依赖于比特复杂度或条件测度，这些量在问题维度上可能无界，这与始终具有指数界限的单纯形法形成对比。我们的算法还承认一个组合上界，可在O(2^n n^{1.5} log n)次迭代内终止于精确解。这补充了Allamigeon、Benchimol、Gaubert和Joswig（SIAGA 2018）的先前工作，该工作展示了某一类实例，其中任何路径跟踪法必须进行指数多次迭代。