The Art of Modeling

We define difficult optimization problems as problems which cannot be solved to optimality or to any guaranteed bound by any standard solver within a reasonable time limit. The problem class we have in mind are mixed integer programming (MIP) problems. Optimization and especially mixed integer optimization is often appropriate and frequently used to model real world optimization problems. While its origin started in the 1950s models became larger and more complicated. A reasonable general framework is mixed integer nonlinear programming (MINLP) problems. They are specified by the augmented vector x⊕ = x ⊕ y established by the vectors x = (x1, ..., xnc) and y T = (y1, ..., ynd) of nc continuous and nd discrete variables, an objective function f(x,y), ne equality constraints h(x,y) and ni inequality constraints g(x,y). The problem