  This article presents a computational approach for determining the optimal
slip velocities on any given shape of an axisymmetric micro-swimmer suspended
in a viscous fluid. The objective is to minimize the power loss to maintain a
target swimming speed, or equivalently to maximize the efficiency of the
micro-swimmer. Owing to the linearity of the Stokes equations governing the
fluid motion, we show that this PDE-constrained optimization problem reduces to
a simpler quadratic optimization problem, whose solution is found using a
high-order accurate boundary integral method. We consider various families of
shapes parameterized by the reduced volume and compute their swimming
efficiency. {Among those, prolate spheroids were found to be the most efficient
micro-swimmer shapes for a given reduced volume. We propose a simple
shape-based scalar metric that can determine whether the optimal slip on a
given shape makes it a pusher, a puller, or a neutral swimmer.}
