Abstract: A new elimination method for mixed real-integer optimization with quadratic objective function and parametric coeffcients is presented. The method combines ideas from the one found by Weispfenning about real linear and quadratic optimization and the one by Shostak on the SUP-INF method of proving Presburger formulas. It is proved that the algorithm has an almost optimal complexity, and is implemented by some test examples using a preliminary MAPLE, the symbolic computational language.