We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.

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This paper studies the multiple runway aircraft landing problem. The aim is to schedule arriving aircraft to available runways at the airport. Landing times lie within predefined time windows and safety separation constraints between two successive landings must be satisfied. We propose a new approach for solving the problem. The method is based on an approximation of the separation time matrix and on time discretization. It provides lower bound or upper bound depending on the choice of the approximating matrix. These bounds are used in a cut algorithm to, exactly or heuristically, solve the problem. Computational tests, performed on publicly available problems involving up to 500 aircraft, show the efficiency of the approach.

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