The paper discusses the operators for the analytical solutions of problems of linear functional minimization over compact sets in a finite-dimensional space. The geometric interpretation of the results is provided through the example of a compact set in a two-dimensional real vector space defined as an intersection of a linear variety and a sphere. The piecewise-linear optimization problems are formulated and proved to possess solutions taking a form of minimization operators. Non-smooth optimization problems have been transformed into convex programming problems.