Dual Simplex

The dual simplex algorithm is an attractive alternative as a solution method for linear programming problems. Since the addition of new constraints to a problem typically breaks primal feasibility but not dual feasibility, the dual simplex can be deployed for rapid reoptimization, without the need of finding new primal basic feasible solutions. This is especially useful in integer programming, where the use of cutting plane techniques require the introduction of new constraints at various stages of the branch-and-bound/cut/price algorithms. In this article, we give a detailed synopsis of the dual simplex method, including its history and its relationship to the primal simplex algorithm, as well as its properties, implementation challenges, and applications. Consider the following linear programming problem P expressed in standard form: [P] min cx subject to: Ax = b x ≥ 0 where c = (c1, c2, . . . , cn) ∈ R is an n-dimensional cost vector, A = (aij) ∈ Rm×n is a matrix of constraint coefficients, b = (b1, b2, . . . , bm) T ∈ R is an m-dimensional right hand side vector, and x = (x1, x2, . . . , xn) ∈ R is an n-dimensional vector of decision variables. We refer to the total cost z = cx : R 7→ R as the objective value of the problem P. Stated in this form, P is called the primal problem, and the vector x∗ that minimizes the objective function cTx∗, while satisfying the equality condition Ax∗ = b, is called the optimal solution to the problem P. Associated with the primal problem P, there exists a dual problem D, which is formulated in the following way: [D] max bu subject to: Au ≤ c u free