This paper formulates systems of piecewise linear equations, derived from the Karush–Kuhn–Tucker conditions for constrained convex optimization problems, as unconstrained minimization problems in which the objective function is a multivariate quadratic spline. Such formulations provide new ways of developing efficient algorithms for many optimization problems, such as the convex regression problem, the least-distance problem, the symmetric monotone linear complementarily problem, and the convex quadratic programming problem with bounded constraints. Theoretical results, a description of an algorithm and its implementation, and numerical results are presented along with a stability analysis.