On the existence of a short pivoting sequence for a linear program

Anders Forsgren; Fei Wang

Home/Archive/arXiv

arXiv · 1908.09735

On the existence of a short pivoting sequence for a linear program

Anders Forsgren·Fei Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, such that the pivot creates a nonsingular submatrix of the constraint matrix which increases by one row and one column at each iteration. Solving a pair of linear equations for each of these submatrices generates a sequence of optimal solutions of a primal-dual pair of linear programs of increasing dimensions, originating at the origin. The optimal solutions to the original primal-dual pair of linear programs are obtained in the final step. It is only an existence result, we have not been able to generate any rules based on properties of the problem to generate the sequence. The result is obtained by a decomposition of the final basis matrix.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Anders Forsgren, Fei Wang. 2019-08-28. On the existence of a short pivoting sequence for a linear program. https://arxiv.org/abs/1908.09735

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Riemannian Convex Bundle Method

We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallelly transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. We prove that, under mild conditions, the convex bundle method converges to a minimizer. Several numerical examples implemented using Manopt$.$jl illustrate the performance of the proposed method and compare it to the subgradient method, the cyclic proximal point algorithm, as well as the proximal bundle method.

math.OC↗

arXiv2026

Omega-Limit Sets and Input-to-State Stability in Power Grids With Switching Equilibria

This paper studies a power transmission system with both conventional generators (CGs) and distributed energy assets (DEAs) providing frequency control. We consider an operating condition with demand aggregating two dynamic components: one that switches between different values on a finite set, and one that varies smoothly over time. Such dynamic operating conditions may result from protection scheme activations, external cyber-attacks, or due to the integration of dynamic loads, such as data centers. Mathematically, the dynamics of the resulting system are captured by a system that switches between a finite number of vector fields -- or modes--, with each mode having a distinct equilibrium point induced by the demand aggregation. To analyze the stability properties of the resulting switching system, we leverage tools from hybrid dynamic inclusions and the concept of $Ω$-limit sets from sets. Specifically, we characterize a compact set that is semi-globally practically asymptotically stable under the assumption that the switching frequency and load variation rate are sufficiently slow. For arbitrarily fast variations of the load, we use a level-set argument with multiple Lyapunov functions to establish input-to-state stability of a larger set and with respect to the rate of change of the loads. The theoretical results are illustrated via numerical simulations on the IEEE 39-bus test system.

math.OC↗

arXiv2026

Cellular flow control design for mixing based on the least action principle

We consider a novel approach for the enhancement of fluid mixing via pure stirring strategies building upon the Least Action Principle (LAP) for incompressible flows. The LAP is formally analogous to the Benamou--Brenier formulation of optimal transport, but imposes an incompressibility constraint. Our objective is to find a velocity field, generated by Hamiltonian flows, that minimizes the kinetic energy while ensuring that the initial scalar distribution reaches a prescribed degree of mixedness by a finite time. This formulation leads to a ``point-to-set" type of optimization problem which relaxes the requirement on controllability of the system compared to the classic LAP framework. In particular, we assume that the velocity field is induced by a finite set of cellular flows that can be controlled in time. To establish finite time feasibility, we introduce an operator-theoretic switching argument that combines the long-time cellular flow mixing result with the von Neumann alternating-projection theorem. We then leverage the direct method to establish the existence of an optimal solution. Finally, we derive the corresponding optimality conditions for the time-dependent control problem and conduct numerical experiments demonstrating the effectiveness of the proposed control design.

math.OC↗