Minimax solutions

BWh] L. P. Burton and W. M. Whyburn (1952), Minimax solutions of ordinary differential systems," Proceedings of the American Mathematical Society 3 794-803. 

In this regard, we are unaware of any exact solution to minimax problem and the so-called cyclic set of the ensuing parameters may be of help in achieving the final aims. 

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

Remark 1 The selection of weight functions u and ui is basically arbitrary. However, these functions have to satisfy certain conditions in order to ensure the existence and uniqueness of a solution. The weight functions u i and U2 utilized in this approach have been verified as meeting all these conditions (Maronna, 1976). Since u and U2 solve the minimax problem (10.36), they are also known as Huber type weights. 

A formal solution is the so-called minimax principle [354], which states that the problem of variational collapse is avoided by determining the minimum of the electronic energy with respect to the large component of the spinor, while guaranteeing a maximum of the energy with respect to the small component. How such a saddle point may look has been shown by Schwarz and Wechsel-Trakowski [217]. The minimax principle has also been discussed in great detail for the complicated two-electron problem [355] (see also Refs. [356,357]). 

In this chapter we have seen how all angular dependencies can be integrated out analytically because of the spherical symmetry of the central field potential of an atom. We are now left with the task to determine the yet unknown radial functions, for which we derived the self-consistent field equations based upon the variational minimax principle. We now address the numerical solution of these equations. The mean-field potential in the set of coupled SCF equations is the reason why we cannot solve them analytically. Hence, the radial functions need to be approximated in some way in... 

In order to apply the nonlinearity measure, the value of has to be calculated. Recalling definition (3), it becomes clear that the calculation of corresponds to the solution of a nonlinear, infinite dimensional, minimax optimization problem and belongs therewith to a class... 

---