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

Abstract

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