## Monday, October 26, 2015

### A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization

This is wonderful news. One of the things I wonder is if this new approach will do better than some  greedy algorithms: A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization by Yin Tat Lee, Aaron Sidford, Sam Chiu-wai Wong

We improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set KRn contained in a box of radius R, we show how to either find a point in K or prove that K does not contain a ball of radius ϵ using an expected O(nlog(nR/ϵ)) oracle evaluations and additional time O(n3logO(1)(nR/ϵ)). This matches the oracle complexity and improves upon the O(nω+1log(nR/ϵ)) additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya for the current matrix multiplication constant ω<2.373 when R/ϵ=nO(1).
Using a mix of standard reductions and new techniques, our algorithm yields improved runtimes for solving classic problems in continuous and combinatorial optimization:
• Submodular Minimization: Our weakly and strongly polynomial time algorithms have runtimes of O(n2lognMEO+n3logO(1)nM) and O(n3log2nEO+n4logO(1)n), improving upon the previous best of O((n4EO+n5)logM) and O(n5EO+n6).
• Matroid Intersection: Our runtimes are O((nrlog2nTrank+n3logO(1)n)lognM) and O((n2lognTind+n3logO(1)n)lognM), achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle.
• Submodular Flow: Our runtime is O(n2lognCUEO+n3logO(1)nCU), improving upon the previous bests from 15 years ago roughly by a factor of O(n4).
• Semidefinite Programming: Our runtime is O~(n(n2+mω+S)), improving upon the previous best of O~(n(nω+mω+S)) for the regime where the number of nonzeros S is small.

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