# Seeking Feasibility in Linear Programs

Chapter
Part of the International Series in Operations Research and Management Science book series (ISOR, volume 118)

A general linear program has the form {min, max} cx, subject to Ax |≤,≥,=} b, lxu, where c is a 1 × n row vector, x, l, u, and b are n × 1 column vectors, and A is an m n array, all consisting of real numbers. It is simple to find an immediate feasible solution for certain linear programs. For example, the origin (all variables equal zero) is always a basic feasible solution for an LP in a variation of canonical form that consists entirely of ≤ inequalities in which every element of b is nonnegative, and all variables are nonnegative. Similarly, network LPs in which the arc flow lower bounds are all zero admit the origin as a feasible solution.

It is more difficult to find a first feasible solution when the general LP is not in this special form, e.g. includes equality or ≥ constraints, or has negative entries in b. In these cases, the origin is no longer available as a feasible solution, so more advanced methods of seeking feasibility are needed. In the simplex method, the most popular technique for reaching feasibility for general LPs is the two-phase method for reasons of numerical stability. The Big-M method, commonly presented in textbooks, is seldom used in implemented solvers.

More recently, infeasible-path interior point methods have been developed that do not necessarily reach feasibility until they also reach optimality. These techniques are beyond the scope of this book. See Wright (1997).

While reaching feasibility in LPs may seem to be a well-understood problem, there are a variety of heuristics which can speed the process considerably, such as crash starts, warm starts, and crossover from an infeasible solution.

## Keywords

Basic Variable Artificial Variable Basic Feasible Solution Pivot Element Warm Start
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.