12 research outputs found
Some preconditioners for systems of linear inequalities
We show that a combination of two simple preprocessing steps would generally
improve the conditioning of a homogeneous system of linear inequalities. Our
approach is based on a comparison among three different but related notions of
conditioning for linear inequalities
Rescaled coordinate descent methods for linear programming
We propose two simple polynomial-time algorithms to find a positive solution to Ax=0Ax=0 . Both algorithms iterate between coordinate descent steps similar to von Neumann’s algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax=0Ax=0 , x≥0x≥0 . This is an extended abstract. The missing proofs will be provided in the full version
Faster Margin Maximization Rates for Generic Optimization Methods
First-order optimization methods tend to inherently favor certain solutions
over others when minimizing a given training objective with multiple local
optima. This phenomenon, known as implicit bias, plays a critical role in
understanding the generalization capabilities of optimization algorithms.
Recent research has revealed that gradient-descent-based methods exhibit an
implicit bias for the -maximal margin classifier in the context of
separable binary classification. In contrast, generic optimization methods,
such as mirror descent and steepest descent, have been shown to converge to
maximal margin classifiers defined by alternative geometries. However, while
gradient-descent-based algorithms demonstrate fast implicit bias rates, the
implicit bias rates of generic optimization methods have been relatively slow.
To address this limitation, in this paper, we present a series of
state-of-the-art implicit bias rates for mirror descent and steepest descent
algorithms. Our primary technique involves transforming a generic optimization
algorithm into an online learning dynamic that solves a regularized bilinear
game, providing a unified framework for analyzing the implicit bias of various
optimization methods. The accelerated rates are derived leveraging the regret
bounds of online learning algorithms within this game framework
Rescaling algorithms for linear conic feasibility
We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ R m× n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of A T. Both algorithms iterate between simple first-order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure ρ A is negative, then the kernel problem is feasible, and the worst-case complexity of the kernel algorithm is O((m 3n + mn 2)log|ρ A| −1); if ρ A > 0, then the image problem is feasible, and the image algorithm runs in time O(m 2n 2 log ρ A −1). We also extend the image algorithm to the oracle setting. We address the degenerate case ρA = 0 by extending our algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A T. In this case, the running time bounds are expressed in the bit-size model of computation: for an input matrix A with integer entries and total encoding length L, the maximum support kernel algorithm runs in time O((m 3n + mn 2)L), whereas the maximum support image algorithm runs in time O(m 2n 2L). The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for linear programming
A New Extension of Chubanov's Method to Symmetric Cones
We propose a new variant of Chubanov's method for solving the feasibility
problem over the symmetric cone by extending Roos's method (2018) of solving
the feasibility problem over the nonnegative orthant. The proposed method
considers a feasibility problem associated with a norm induced by the maximum
eigenvalue of an element and uses a rescaling focusing on the upper bound for
the sum of eigenvalues of any feasible solution to the problem. Its
computational bound is (i) equivalent to that of Roos's original method (2018)
and superior to that of Louren\c{c}o et al.'s method (2019) when the symmetric
cone is the nonnegative orthant, (ii) superior to that of Louren\c{c}o et al.'s
method (2019) when the symmetric cone is a Cartesian product of second-order
cones, (iii) equivalent to that of Louren\c{c}o et al.'s method (2019) when the
symmetric cone is the simple positive semidefinite cone, and (iv) superior to
that of Pena and Soheili's method (2017) for any simple symmetric cones under
the feasibility assumption of the problem imposed in Pena and Soheili's method
(2017). We also conduct numerical experiments that compare the performance of
our method with existing methods by generating instances in three types:
strongly (but ill-conditioned) feasible instances, weakly feasible instances,
and infeasible instances. For any of these instances, the proposed method is
rather more efficient than the existing methods in terms of accuracy and
execution time.Comment: 44 pages; Department of Policy and Planning Sciences Discussion Paper
Series No. 1378, University of Tsukub
Post-Processing with Projection and Rescaling Algorithms for Semidefinite Programming
We propose the algorithm that solves the symmetric cone programs (SCPs) by
iteratively calling the projection and rescaling methods the algorithms for
solving exceptional cases of SCP. Although our algorithm can solve SCPs by
itself, we propose it intending to use it as a post-processing step for
interior point methods since it can solve the problems more efficiently by
using an approximate optimal (interior feasible) solution. We also conduct
numerical experiments to see the numerical performance of the proposed
algorithm when used as a post-processing step of the solvers implementing
interior point methods, using several instances where the symmetric cone is
given by a direct product of positive semidefinite cones. Numerical results
show that our algorithm can obtain approximate optimal solutions more
accurately than the solvers. When at least one of the primal and dual problems
did not have an interior feasible solution, the performance of our algorithm
was slightly reduced in terms of optimality. However, our algorithm stably
returned more accurate solutions than the solvers when the primal and dual
problems had interior feasible solutions.Comment: 78 page