We introduce some approximate optimal solutions and a generalized augmented Lagrangian in nonlinear programming, establish dual function and dual problem based on the generalized augmented Lagrangian, obtain approximate KKT necessary optimality condition of the generalized augmented Lagrangian dual problem, prove that the approximate stationary points of generalized augmented Lagrangian problem...

I argue that, under the glitz, dual theories are examples of theoretically equivalent descriptions of the same underlying physical content: I distinguish them from cases of genuine underdetermination on the grounds that there is no real incompatibility involved between the descriptions. The incompatibility is at the level of unphysical structure. I argue that dual pairs are in fact very strongl...

The complete theory for Fisher and dual discriminant analysis is presented as the background of the novel algorithms. LDA is found as composition of projection onto the singular subspace for within-class normalised data with the projection onto the singular subspace for betweenclass normalised data. The dual LDA consists of those projections applied in reverse order. The experiments show that u...

Dual-hequency patch antennas may provide an alternative to large-bandwidth planar antennas, in applications in which large bandwidth is really needed for operating at two separate transmitreceive bands. When the two operating frequencies are far apart, a dual-frequency patch structure can be conceived to avoid the use of separate antennas. In this paper, a critical overview of possible solution...

We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat R. We show how all such connections lie in the orbit of the flat connection on R under the action of non-local symmetries of the self-dual Yang–Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang–Mill...

Nonlinear network flow problems with linear/nonlinear side constraints can be solved by means of Lagrangian relaxations. The dual problem is the maximization of a dual function whose value is estimated by minimizing approximately a Lagrangian function on the set defined by the network constraints. We study alternative stepsizes in the approximate subgradient methods to solve the dual problem. S...