1. Interior point methods are not worse than Simplex
- Author
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Xavier Allamigeon, Daniel Dadush, Georg Loho, Bento Natura, Laszlo A. Vegh, Mathematics of Operations Research, TROPICAL (TROPICAL), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centrum voor Wiskunde en Informatica (CWI), Centrum Wiskunde & Informatica (CWI)-Netherlands Organisation for Scientific Research, Faculty of Electrical Engineering, Mathematics and Computer Science [Twente] (EEMCS), University of Twente, Department Mathematics [London] (LSE), London School of Economics and Political Science (LSE), and Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands
- Subjects
FOS: Computer and information sciences ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,Geometry ,Time measurement ,Computer science ,Current measurement ,Partitioning algorithms ,n/a OA procedure ,Optimization and Control (math.OC) ,Linear programming ,Computer Science - Data Structures and Algorithms ,FOS: Mathematics ,Data Structures and Algorithms (cs.DS) ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Mathematics - Optimization and Control ,Upper bound - Abstract
Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds 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. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound $O(2^{n}n^{15}\log n)$ for an n-variable linear program in standard form. 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. The number of iterations of our algorithm is at most $O(n^{15}\log n)$ times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound $O(n^{15}\log n)$ that applies for arbitrarily long line segments is derived from a combinatorial progress measure. Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009).
- Published
- 2022