engleski

On two simple decompositions of Lipschitz functions

Every continuous Lipschitz function, when considered on a compact convex set, is a plane away from the set of coordinate-wise monotone functions. If a function is smooth, and if its derivative is Lipschitz, then the function is a strictly convex quadratic away from strictly convex functions. In this article, we use these proximities to obtain possibly new results in three areas of mathematics. First, we study scalar functions and show how ordinary differential equations can be transformed to differential equations with strictly monotone solutions. In linear algebra, a basic inequality for convex functions of symmetric matrices is extended beyond convexity yielding bounds on the determinant, spectral radius and also sufficient conditions for non-singularity. Mathematical programs with generally non-smooth and non-convex functions are reduced to a partly linear coordinate-wise strictly monotone canonical form. For smooth, generally non-convex, programs the canonical form has a partly linear coordinate-wise strictly convex and increasing structure. This means, loosely speaking, that almost every continuous program of practical interest can be reduced to one of these two canonical forms.

: Lipschitz function; monotone function; convex function; ordinary differential equation; symmetric matrix; determinantal inequality; canonical form of mathematical program

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