Dedicated scalar-parameter algorithm for the max–min problem

Develop a dedicated algorithm for optimizing the value function of a parametric linear program over a bounded interval that branches only on the single scalar parameter.

Background

The paper studies minimum and maximum optimization of the optimal value of a linear program whose constraint matrix depends linearly on one scalar parameter. Although the parameter is one-dimensional, the exact max–min reformulation remains computationally difficult because of bilinear parameter–dual-variable terms and unbounded relaxations associated with parametric equality constraints.

The computational experiments show that generic spatial branch-and-bound methods frequently reach time limits or fail to produce finite dual bounds. The authors identify exploiting the one-dimensional parameter structure through a dedicated branching method as an unresolved algorithmic direction.