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With respect to the complexity of the task to find a solution to nonlinear optimization problem minu c(u) subject to d(u) = 0, two things are important: the number of iterations until convergence and the cost per iteration. Applying a Sequential Quadratic Programming approach to this problem is equivalent to solving the system F(u, μ) := (∇u c(u) + \muTu d(u), d(u)) = 0 of nonlinear equations with a Newton-type method.

The cost per Newton iteration for the lifted problem G(u, μ, s) = 0 is higher than for the original system F(u, μ) = 0. However, the typical cubic runtime dependence on the number of variables to solve a system of linear equations can be reduced drastically by a variety of different approaches that exploit the specific structure in the function G(u, s), such as by using directional derivatives, parallelizable condensing, structure-exploiting (iterative) linear algebra, or a relaxation of the matching constraints in a dual nonsmooth Newton setting.

The lifting approach has a variety of direct advantages, such as the possibilities for parallelization of function and derivative evaluation, initialization also of lifted variables s0, e.g., in parameter estimation or tracking problems, and the avoidance of singularities of function evaluation for bad choices of u0.

But it is not clear how to choose si in an optimal way. Therefore, there is the open question whether it is possible to come up with a systematic approach to decide how many and which si should be used to minimize the (average) number of iterations.

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