[Paper Review] The ellipsoid method redux

Michael J. Todd|arXiv (Cornell University)|Jan 18, 2023

Advanced Optimization Algorithms Research4 citations

TL;DR

This paper enhances the ellipsoid method for solving systems of linear inequalities by improving lower bound calculations and introducing weight-decreasing steps, enabling faster convergence and stronger theoretical guarantees. It demonstrates that the method can be viewed as coordinate descent on volume or potential functions, and shows that using optimal lower bounds and adaptive weights significantly improves performance over prior variants, though computational results remain discouraging for large-scale problems.

ABSTRACT

We reconsider the ellipsoid method for linear inequalities. Using the ellipsoid representation of Burrell and Todd, we show the method can be viewed as coordinate descent on the volume of an enclosing ellipsoid, or on a potential function, or on both. The method can be enhanced by improving the lower bounds generated and by allowing the weights on inequalities to be decreased as well as increased, while still guaranteeing a decrease in volume. Three different initialization schemes are described, and preliminary computational results given. Despite the improvements discussed, these are not encouraging.

Motivation & Objective

• To improve the ellipsoid method for solving systems of linear inequalities by enhancing lower bound generation and allowing weight adjustments.
• To demonstrate that the method can be interpreted as coordinate descent on volume, potential function, or both.
• To develop a modified ellipsoid algorithm that uses optimal lower bounds and allows decreasing weights on well-satisfied constraints without losing theoretical guarantees.
• To adapt the oblivious ellipsoid algorithm (OEA) to always use the best lower bounds, preserving convergence and infeasibility certification.
• To evaluate the performance of the improved methods through computational experiments on medium- to large-scale problems.

Proposed method

• The paper uses the Burrell-Todd ellipsoid representation, which expresses an enclosing ellipsoid via rank-one convex quadratic inequalities derived from linear constraints.
• It introduces a method to compute the 'best' lower bound on each constraint function at negligible computational cost, improving the quality of the ellipsoid at each iteration.
• The standard ellipsoid algorithm (SEA) is modified to allow decreasing or dropping weights on constraints that are well-satisfied, maintaining theoretical convergence guarantees.
• A modified potential function is proposed that permits zero weights on some constraints while still ensuring convergence in the infeasible case.
• The oblivious ellipsoid algorithm (OEA) is adapted to use the best lower bounds at each step, preserving its polynomial-time convergence and infeasibility certification properties.
• Computational experiments compare the improved SEA, OEA, and standard variants using three initialization schemes and multiple problem instances.

Experimental results

Research questions

• RQ1 Can the ellipsoid method be improved by computing optimal lower bounds on constraint functions at negligible cost?
• RQ2 Can the standard ellipsoid algorithm be enhanced by allowing weight decreases on well-satisfied constraints while preserving theoretical convergence?
• RQ3 Can the oblivious ellipsoid algorithm be adapted to always use the best lower bounds without sacrificing its theoretical guarantees?
• RQ4 Does the use of improved lower bounds and adaptive weights lead to faster convergence in practice?
• RQ5 Can the modified potential function support zero weights on constraints while maintaining convergence in the infeasible case?

Key findings

• The improved ellipsoid method computes the best possible lower bounds on constraint functions at negligible cost, significantly enhancing the quality of the enclosing ellipsoid.
• The modified standard ellipsoid algorithm (SEA) with decreasing weights on well-satisfied constraints maintains theoretical convergence guarantees while improving performance.
• The oblivious ellipsoid algorithm (OEA) can be adapted to always use the best lower bounds, preserving its ability to certify infeasibility and guarantee polynomial-time convergence.
• Despite theoretical improvements, computational results show that the enhanced methods do not perform well in practice, especially for large-scale problems.
• The OEA with improved lower bounds and the modified potential function successfully maintains convergence in the infeasible case, even with zero weights on some constraints.
• In the worst case, the OEA still decreases volume by a factor of approximately 1 - (2m)^{-1}, but with stronger theoretical guarantees due to better bound selection.