By Pierre Bonami, Mustafa Kilinç, Jeff Linderoth (auth.), Jon Lee, Sven Leyffer (eds.)
Many engineering, operations, and medical functions comprise a mix of discrete and non-stop selection variables and nonlinear relationships concerning the choice variables that experience a said impression at the set of possible and optimum ideas. Mixed-integer nonlinear programming (MINLP) difficulties mix the numerical problems of dealing with nonlinear capabilities with the problem of optimizing within the context of nonconvex services and discrete variables. MINLP is among the such a lot versatile modeling paradigms on hand for optimization; yet simply because its scope is so extensive, within the such a lot basic instances it's hopelessly intractable. still, an increasing physique of researchers and practitioners — together with chemical engineers, operations researchers, commercial engineers, mechanical engineers, economists, statisticians, desktop scientists, operations managers, and mathematical programmers — have an interest in fixing large-scale MINLP instances.
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Additional info for Mixed Integer Nonlinear Programming
For each variable xj , we keep track of the number of times the variable has been branched on (τj ) and the total per-unit degradation of the objective value by branching down and up, respectively, Pj− and Pj+ . Each time variable j is branched on, Pj− and Pj+ are updated by taking into account the change of bound at that node: Pj− = i+ i− i i zL zL − zL − zL + P , and P = + Pj+ , j− j+ fji 1 − fji i i and N+ denote the nodes from where xj is the branching variable, N− i− i+ i the down and up branch, zL (resp.
Furthermore, statistical experiments have shown that pseudo-costs often provide reasonable estimates of the objective degradations caused by branching  when solving MILPs. Two diﬃculties arise with pseudo-costs. The ﬁrst one, is how to update the historical data when a node is infeasible. This matter is not settled. Typically, the pseudo-costs update is simply ignored if a node is infeasible. The second question is how the estimates should be initialized. For this, it seems that the agreed upon state-of-the art is to combine pseudocosts with strong-branching, a method that may address each of the two methods’ drawbacks— strong-branching is too slow to be performed at every node of the tree, and pseudo-costs need to be initialized.
Berthold and A. Gleixner, Undercover - a primal heuristic for MINLP based on sub-mips generated by set covering, Tech. Rep. ZIB-Report 09-40, Konrad-Zuse-Zentrum f¨ ur Informationstechnik Berlin (ZIB), 2009.  D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming, 74 (1996), pp. 121–140. ALGORITHMS AND SOFTWARE FOR CONVEX MINLP 35  R. Bixby and E. Rothberg, Progress in computational mixed integer programming. A look back from the other side of the tipping point, Annals of Operations Research, 149 (2007), pp.