Methods of Nonsmooth Analysis as Applied to the Problem of Minimizing the Sum of Moduli of Affine Functions

An application of constructive nonsmooth analysis methods to the problem of minimizinga convex piecewise affine function defined as the sum of absolute values of affine functions isdemonstrated. Hypodifferential calculus was used in the general (multidimensional) case, whilesubdifferential calculus was employed in the scalar case. Analyzing the optimality criterion, onecan reveal that the point delivering the global minimum can be found by solving thecorresponding linear programming problem. In the scalar case, the solution can also be found inclosed form as the weighted median of the nodes of a broken line.