# Can gradient descent be applied to non-convex functions?

I'm just learning about optimization, and having trouble understanding the difference between convex and non-convex optimization. From my understanding, a convex function is one where "the line segment between any two points on the graph of the function lies above or on the graph". In this case, a gradient descent algorithm could be used, because there is a single minimum and the gradients will always take you to that minimum.

However, what about the function in this figure: Here, the blue line segment crosses below the red function. However, the function still has a single minimum, and so gradient descent would still take you to this minimum.

So my questions are:

1) Is the function in this figure convex, or non-convex?

2) If it is non-convex, then can convex optimization methods (gradient descent) still be applied?

Thanks!

• Gradient descent leads to a minimum, which is local or global depending on the nature of the function, the position of the starting point, and the direction. – Khue Sep 17 '15 at 12:57