Nearly-Linear Time Positive LP Solver with Faster Convergence Rate

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operation research, and optimization. Efficient algorithms for solving such LPs have received significant attention in the past 20 years [2, 3, 4, 6, 7, 9, 11, 15, 16, 18, 19, 21, 24, 25, 26, 29, 30]. Unfortunately, all known nearly-linear time algorithms for producing (1+ε)-approximate solutions to positive LPs have a running time dependence that is at least proportional to ε-2. This is also known as an O(1/√T) convergence rate and is particularly poor in many applications. In this paper, we leverage insights from optimization theory to break this longstanding barrier. Our algorithms solve the packing LP in time ~O(N ε-1) and the covering LP in time ~O(N ε-1.5). At high level, they can be described as linear couplings of several first-order descent steps. This is the first application of our linear coupling technique (see [1]) to problems that are not amenable to blackbox applications known iterative algorithms in convex optimization. Our work also introduces a sequence of new techniques, including the stochastic and the non-symmetric execution of gradient truncation operations, which may be of independent interest.