First Order Methods for Linear Programming: Theory, Computation, and Applications

Haihao (Sean) Lu, The University of Chicago
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Linear programming (LP) is a fundamental tool in operations research with wide applications in practice. The state-of-the-art LP solvers are essentially based on either simplex method or barrier method, which are quite mature and reliable at delivering highly accurate solutions. However, it is highly challenging to further scale up these two methods. The computational bottleneck of both methods is the matrix factorization when solving linear equations, which usually requires significantly more memory usage and cannot be directly applied on the modern computing resources, i.e., distributed computing and/or GPUs. In contrast, first-order methods (FOMs) only require matrix-vector multiplications, which work very well on these modern computing infrastructures and have massively accelerated the machine learning training process during the last 15 years. In this talk, I’ll present new FOMs for LP. On the computational side, we build up a new LP solver based on the proposed FOMs and I’ll present a comprehensive numerical study on the proposed FOMs. The solver has been open-sourced through Google OR-Tools. On the theory side, I’ll present new techniques that improve the existing complexity of FOMs for LP and show that the proposed algorithms achieve the optimal convergence rate in the class of FOMs. I’ll conclude the talk with open questions and new directions on this line of research. Part of this research was done at Google.

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