Exact linear programming circuits, curvature, and diameter

Natura, Bento (2022) Exact linear programming circuits, curvature, and diameter. PhD thesis, London School of Economics and Political Science.

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Abstract

We study Linear Programming (LP) and present novel algorithms. In particular, we study LP in the context of circuits, which are support-minimal vectors of linear spaces. Our results will be stated in terms of the circuit imbalance (CI), which is the worst-case ratio of nonzero entries of circuits and whose properties we study in detail. We present following results with logarithmic dependency on CI. (i) A scaling-invariant Interior-Point Method, which solves LP in time that is polynomial in the dimensions, answering an open question by Monteiro-Tsuchiya in the affirmative. This closes a long line of work by Vavasis-Ye and Monteiro-Tsuchiya; (ii)We introduce a new polynomial-time path-following interior point method where the number of iterations admits a singly exponential upper bound. This complements recent results, that path-following method must take at least exponentially many iterations; (iii)We further provide similar upper bounds on a natural notion of curvature of the central path; (iv) A black-box algorithm that requires only quadratically many calls to an approximate LP solver to solve LP exactly. This significantly strengthens the framework by Tardos, which requires exact solvers and whose runtime is logarithmic in the maximum subdeterminant of the constraint matrix. The maximum subdeterminant is exponentially bigger than CI, already for fundamental combinatorial problems such as matchings; (v) Furthermore, we obtain a circuit diameter that is quadratic in the number of variables, giving the first polynomial bound for general LP where CI is exponential. Unlike in the simplex method, one does not have to augment around the edges of the polyhedron: Augmentations can be in any circuit direction; (vi) Lastly, we present an accelerated version of the Newton–Dinkelbach method, which extends the black-box framework to certain classes of fractional and parametric optimization problems. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain improved runtimes over the non-accelerated version.

Item Type:	Thesis (PhD)
Additional Information:	© 2022 Bento Natura
Library of Congress subject classification:	Q Science > QA Mathematics
Sets:	Departments > Mathematics
Supervisor:	Végh, László A.
URI:	http://etheses.lse.ac.uk/id/eprint/4448

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