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 $\min c^t x \text{ s.t } Ax + \lambda Dx \leq b$ where $\lambda$ belongs to a set of predefined values $\Lambda$. 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 $O(m^3+pm^2)$ where $m$ is the number of constraints of the original problem and $p$ the number of values in $\Lambda$. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.
Efficient LP warmstarting for linear modifications of the constraint matrix
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