Efficient LP warmstarting for linear modifications of the constraint matrix

We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by min ctx s.t Ax + λDx ≤ b where λ belongs to a set of predefined values Λ. Based on the information given by a precomputed basis, we present three efficient LP warm-starting algorithms. Each algorithm is either based on the eigenvalue decomposition, the Schur decomposition, or a tweaked eigenvalue decomposition to evaluate the optimal solution and optimal objective of these problems. The three algorithms have an overall complexity O(m3 + pm2) where m is
the number of constraints of the original problem and p the number of values in Λ. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.