The dual of the strictly convex quadratic programming problem with unit bounds is posed as a linear ℓ1 minimization problem with quadratic terms. A smooth approximation to the linear ℓ1 function is used to obtain a parametric family of piecewise-quadratic approximation problems. The unique path generated by the minimizers of these problems yields the solution to the original problem for finite values of the approximation parameter. Thus, a finite continuation algorithm is designed. Results of extensive computational experiments are reported.