Scaling nonlinear programs

Bilevel optimization formulation for parameter estimation in liquid–liquid phase equilibrium problems

“…Proper scaling of variables is important in optimization, even when scale-free methods are used (Bazaraa et al, 1993). Methods for automated scaling exist, but are not perfect (Lasdon and Beck, 1981) and therefore scaling should be performed in the problem formulation. Finally, after the translation there is smaller correlation between the parameters, compare the discussion on linear regression given by Beck and Arnold (1977, Chapter 5).…”

Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems

“…Like the logarithmic scaling, the inverse-power scaling is only applied to the constraint function G. Hence the Jacobian of an inverse-power scaled NLP is of the form (8), where ∇G is given by (15).…”

“…The evolution of these automatic scaling techniques leads to the development of the projected Jacobian rows normalization (PJRN) which we propose here, and by extension, to the nonlinear scaling methodologies. For completeness, it is important to stress that convex optimization [4] or pre-optimization [8] could also be used to automatically scale the NLP, although this is out of the scope of this work, since we are interested in simple techniques which do not require additional off-the-shelf software. This paper is organized as follows.…”

On the importance of scaling in equation-based modelling

“…In fact, the authors could only find a couple of references on the related topic of non-linear optimization solvers, for which the numerical solution is even more sensitive to bad scaling than the solution of algebraic equations. In [4], the authors attempted to devise a method for automatic scaling, which however did not provide consistently good performance, while in [2] the authors discussed the impact of correct scaling on the convergence of some nonlinear programming problems. To the authors' knowledge, no references exist on this problem in the context of EOOLTs.…”