Unconstraint minimization of continuous convex functions. Application to LP

TL;DR

This paper introduces a novel ellipsoid-based algorithm for unconstrained minimization of continuous convex functions, with applications to linear programming, providing bounds, feasibility checks, and solutions without prior radius knowledge.

Contribution

It presents a new algorithm that searches for convex function minima without prior search radius knowledge and applies it to linear programming, including feasibility testing.

Findings

Provides a polynomial upper bound on flop count for finding the optimum.

Develops a method to estimate the distance to feasible points in LP.

Offers an algorithm that can prove the non-existence of an optimal point if stopped early.

Abstract

Our contribution in this paper is two folded. We consider first the case of linear programming with real coefficients and give a method which allows the computation of a new upper bound on the distance from the origin to a feasible point. Next we present an application of the ellipsoid method to form a novel algorithm which allows one to search for the continuous convex functions minimum without the prior knowledge on the search radius. If stopped early, the proposed algorithm can give proofs that the optimal value has not been reached yet, hence the user can opt for more iterations. However, if prior guarantees exist on the existence of an optimum point in a given ball, the algorithm is guaranteed to find it. For such a case we prove a polynomial upper bound on the number of flops required to obtain the solution. The presented algorithm is then applied to linear programming. We further develop our algorithm and provide a method to answer the feasibility question of linear programs with real coefficients in a number of flops bounded above by a polynomial in the number of variables and the number of constraints. However, in case of feasibility, our method does not generate an actual feasible point. For obtaining such a point, we have to make some assumptions on the subgradients of a certain function.