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Additional info for Transworld Skateboarding (June 2012)
Then, E and V nodes are collapsed, and each remaining edge is transformed into an oriented arc going from the E node to the V node, obtaining a directed graph that expresses the mutual dependencies between variables and equations: an arc going from node A to node B means that equation A depends on a variable that is matched to equation B. This means that equation B should be solved first, if possible, in order to use its value when solving equation A for its matched variable. Tarjan’s algorithm is then run on this graph, to identify its strong components, which correspond to systems of implicit equations that cannot be solved sequentially by just re-ordering the equations.
J. Åkesson, C. D. Laird, G. Lavedan, K. Prölss, H. Tummescheit, S. Velut, and Y. Zhu. Nonlinear model predictive control of a CO2 post-combustion absorption unit. Chemical Engineering &Technology, 35(3):445–454, January 2012. Object-oriented modelling and simulation of physical systems               33 P. I. Barton and C. C. Pantelides. The modeling of combined continuous and discrete processes. AIChE Journal, 40:966–979, 1994. G. Berry and G. Gonthier.
In general, if the model or the constraints are non-linear, or if the control objective is not quadratic, a non-linear, possibly non-convex optimization problem will ensue, which might have many local minima besides the global optimum solution and might also be numerically ill conditioned. Ensuring the convergence of the solver to the desired solution is therefore a non-trivial problem. Some numerical issues can be addressed automatically by the tool in a fairly straightforward way, for example the correct scaling of variables and equations.