Algorithmic Classification of Constrained Extrema in Low-Dimensional Problems with Applications to Transport Location Problems

Constrained optimization plays a central role in transport and logistics location problems, such as depot siting under geometric or infrastructure-related constraints. In practice, the classification of constrained extrema by classical second-order methods, typically based on bordered Hessians and the explicit manipulation of the total differentials of the constraint functions, can be cumbersome and error-prone, especially in engineering-oriented applications. In this paper, we present algorithmic procedures for the classification of constrained extrema in low-dimensional problems (2D and 3D), with applications to transport location models. The proposed approach does not avoid the use of constraint derivatives, since first-order constraint information is necessary for any local constrained classification procedure. Rather, it avoids the explicit manipulation of the total differentials of the constraints during the application phase. The required constraint information is incorporated through first-order partial derivatives evaluated at the stationary point, leading to simple algebraic test coefficients derived from the second derivatives of the Lagrangian. The procedures apply to regular non-degenerate cases and require only the solution of Fermat-type systems together with the evaluation of low-order determinants. Their practical relevance is illustrated through a transport depot location problem with geometric constraints, showing how the proposed approach can provide a transparent and effective decision-support tool for transport and logistics engineering.