Minimize $F(x_1, ..., x_i, y_1, ..., y_i) = \sqrt{x_1^2+y_1^2} + ... + \sqrt{x_i^2 + y_i^2}$ subject to some linear equality and inequality constraints.
I've used the active set method to solve similarly constrained quadratic objective functions before, but quadratic programming problems use the kkt conditions to find the optimum. Since I can't compute lagrange multipliers for my function due to non-continuous partial derivatives, I don't know what criteria to use to release from constraints, or how I'll know I'm at the optimum once I do.
If the active set of constraints meant the optimum was completely off the part of the function that had an undefined derivative, the Lagrange multipliers could be solved. However, it is almost guaranteed to be partially on the undefined area.
I'm guessing I could substitute a value of zero for the holes in the partial derivatives (all places where the derivative are noncontinuous are at 'folds' where the slope is symmetrical kind of) and then go on to solve the same system of equations I was solving before to find the Lagrange multipliers. This is a hack though and I don't know the math to prove it will work or not work.
How do I optimize a convex function with linear inequality constraints if it's partials are not continuous?