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Sensitivity analysis for linear changes of the constraint matrix of a (mixed-integer) linear program

Guillaume Derval, Damien Ernst, Quentin Louveaux, Bardhyl Miftari

Abstract

Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly mixed-integer) linear optimisation problems in which the constraint matrix is subject to linear perturbations controlled by a scalar parameter that varies within a given interval. This is a largely unresolved question with the additional burden that the resulting value function may be largely irregular. We propose several bounding techniques that provide formal guarantees on the behaviour of the objective value across the entire parameter range. The proposed bounds rely on tools from robust optimisation, Lagrangian relaxation, and ad-hoc reformulations. Each method is assessed in terms of accuracy, precision, and computational performance. Experimental results on a large benchmark set show that the proposed bounding techniques effectively address this class of problems, delivering strong guarantees and good precision. In addition, we introduce a spatial branch-and-bound algorithm that incorporates these bounds to compute an anytime approximation of the value function within a given error tolerance, and we analyse its computational performance.

Sensitivity analysis for linear changes of the constraint matrix of a (mixed-integer) linear program

Abstract

Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly mixed-integer) linear optimisation problems in which the constraint matrix is subject to linear perturbations controlled by a scalar parameter that varies within a given interval. This is a largely unresolved question with the additional burden that the resulting value function may be largely irregular. We propose several bounding techniques that provide formal guarantees on the behaviour of the objective value across the entire parameter range. The proposed bounds rely on tools from robust optimisation, Lagrangian relaxation, and ad-hoc reformulations. Each method is assessed in terms of accuracy, precision, and computational performance. Experimental results on a large benchmark set show that the proposed bounding techniques effectively address this class of problems, delivering strong guarantees and good precision. In addition, we introduce a spatial branch-and-bound algorithm that incorporates these bounds to compute an anytime approximation of the value function within a given error tolerance, and we analyse its computational performance.

Paper Structure

This paper contains 20 sections, 14 theorems, 50 equations, 10 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Our contributions
  4. Problem formalisation
  5. Bounding methods formulations
  6. Constant robust solution
  7. Variable robust solution, affine on $\lambda$
  8. Envelope of robust linear solutions
  9. Coefficient-wise bounding methods
  10. Lower bounds using Lagrangian relaxations
  11. Spatial branch and bound
  12. Experiments
  13. Dataset
  14. Bound assessment
  15. Spatial branch and bound algorithm
  16. ...and 5 more sections

Key Result

Theorem 1

The following linear problem is equivalent to problem constant_robust_1 and provides an upper bound for $f(\lambda)$$\forall \lambda \in [\lambda_1, \lambda_2]$:

Figures (10)

  • Figure 1: Plot of $f_{\text{toy}}(\lambda)$ between $[-10, 9]$. The orange line with dots is a coarse sampling of the function, made for each $\lambda \in \{-10, 9\}$, and linearly interpolated between these points.
  • Figure 2: The feasible space of the problem given in Equation \ref{['example:toy']} for different values of $\lambda$. Each inequality partitions the space into two parts: a feasible space, coloured in white, and a non-feasible space, coloured in red. The optimum is shown as a black dot.
  • Figure 3: Bounds obtained with the different robust variable solution for the problem \ref{['eq:example-robust-1']}.
  • Figure 4: Illustration of the robust envelope algorithm on problem $\mathcal{P}_{\text{toy 1}}$ (see Annex \ref{['annex:examples']})
  • Figure 5: Illustration of the Lagrangian bounds on the Netlib problem Greenbeb where the uncertainties impact inequalities.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Example 2.1
  • Example 2.2
  • Theorem 1
  • proof
  • Example 3.1
  • Theorem 2
  • proof
  • Lemma 1
  • Corollary 1
  • Theorem 3
  • ...and 20 more