TY - JOUR
T1 - Reformulations for utilizing separability when solving convex MINLP problems
AU - Kronqvist, Jan
AU - Lundell, Andreas
AU - Westerlund, Tapio
PY - 2018
Y1 - 2018
N2 - Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.
AB - Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.
KW - Convex MINLP
KW - Lifted polyhedral approximation
KW - Separable MINLP
KW - Extended supporting hyperplane algorithm
KW - Outer approximation
KW - extended cutting plane algorithm
KW - Convex MINLP
KW - Lifted polyhedral approximation
KW - Separable MINLP
KW - Extended supporting hyperplane algorithm
KW - Outer approximation
KW - extended cutting plane algorithm
KW - Convex MINLP
KW - Lifted polyhedral approximation
KW - Separable MINLP
KW - Extended supporting hyperplane algorithm
KW - Outer approximation
KW - extended cutting plane algorithm
U2 - 10.1007/s10898-018-0616-3
DO - 10.1007/s10898-018-0616-3
M3 - Artikel
SN - 0925-5001
VL - 71
SP - 571
EP - 592
JO - Journal of Global Optimization
JF - Journal of Global Optimization
IS - 3
ER -