Local optimums in Electric Fences

Question

Hi, I'm writing a solution for the Usaco problem "Electric Fences". In the problem you have to find the optimal location for a point among a large amount of linesegments, so the sum of point-linesegment distances is smallest possible.

I had an idea, that it might be possible to do a hillclimb, and it worked for all testcases. The given analysis used a similar method, but it did not explain why this would work.

Thus I'm still unable to either prove or disprove the existence of local optimums in the given tasks. I had an idea that it could be done using induction, but I haven't been able to make it work. Can you help me?

Updated definition

Given a set of (x1,y1,x2,y2) linesegments find the (x,y) point P, that minimizes the function:

def Val(x,y):
    d = 0
    for x1,y1,x2,y2 in LineSegments:
        if triangle (x1,y1,x2,y2,x,y) is not obtuse in (x1,y1) or (x2,y2):
            d += DistPointToLine(x,y,x1,y1,x2,y2)
        else:
            d += min(DistPointToPoint(x,y,x1,y1), DistPointToPoint(x,y,x2,y2))
    return d

By some reason the problem contains only one local optima, and thus the following procedure can be used to solve it:

precision = ((-0.1,0), (0.1,0), (0,-0.1), (0,0.1))
def Solve(precision=0.1):
    x = 0; y = 0
    best = Val(x,y)
    while True:
        for dx,dy in precision:
            if Val(x+dx, y+dy) > best:
                x += dx; y += dy
                best = Val(x,y)
                break
        else:
            break
    return (x,y)

The questions is: Why does this not get stuck somewhere on the way to the global optimum? Why is there no local hilltops to bring this naive procedure to its knees?

Answer 1

It is easy to prove the algorithm's correctness if we notice that the distance function for a single line segment is a convex function. Convex in this case means that if we think of the distance function as a surface z=f(x,y), then if we filled in the volume above the surface, we'd have a convex solid. In the case of the distance from a single line segment, the solid would look like a triangular wedge with conical ends.

Since the sum of convex functions is also convex, then the sum of distances from multiple line segments will also be a convex function. Therefore, any local minimum you find must also be a global minimum by virtue of the function being convex.