Are there perceptrons that maximize the average margin, rather than the minimum? A perceptron is a linear classifier.    The standard method for training a perceptron involves maximizing the minimum margin.  That is, we are trying to find:

$x^* = \text{argmax}_{x\in \text{unit vectors}} \min_{i \in 1..n} A_i x $

where $x$ is a normal vector to the separating hyperplane, $A$ is the $n\times d$ matrix of input values, $n$ is the number of input data points, $d$ is their diminsion, and $A_i x$ is the dot product between $x$ and the $i$th data point.

My question is: has there been any work on perceptrons whose hyperplane maximizes the average margin?  That is, 

$x'^* = \text{argmax}_{x\in \text{unit vectors}} \sum_{i \in 1..n} A_i x $ 

This is a much easier quantity to calculate because it does not require gradient descent.  The problem is that it is not guaranteed to find separating hyperplanes even when they do exist---the points with low margin aren't given enough weight to do this.
This is what a soft-margin SVM is already doing, except that the average has a min on the inside: $\sum_i min(h,d(x_i,c_i,w))$ where $d(x_i,c_i,w)$ is the distance of the point $x_i$ from the classification hyperplane, signed with a positive sign if it is on the correct side as determined by the target $c_i$, and $h$ is the maximum distance on the correct side you're willing to give "credit" for.  If you set $h$ to a large enough value, this is what you have above.  And, this is precisely a soft-margin linear SVM.