in this talk, we will consider the problem of designing multi-focal optical lenses. this problem has been actively studied over the past 20 years and the results obtained along the way of research have been used directly in the lens industry. we study a variational approach for the multi-focal lens design problem and find out that the euler-lagrange equation associated with the variational problem is a fourth-order nonlinear elliptic partial differential equation. we consider two linearizations of the equation along with three types of boundary conditions, and analyze the existence, uniqueness as well as regularity of the solutions under these types of boundary conditions. finally, we propose a numerical method with a special type of spline functions for solving the linearized partial differential equations, and the method is shown to be very efficient.