engleski

Parametric Decompositions of Continuous Functions

Every smooth function in several variables vith a Lipschitz derivative, when considered on a compact convex set, is only a convex quadratic away from convex functions. Every continuous, possibly non-smooth, Lipschitz function is only a plane away from coordinate-wise monotone functions. In this talk we will use these proximities to study corresponding parametric decompositions of smooth and continuous non-smooth functions. In particular, several well-known results on convex functions will be extended to beyond convexity. Looking in the opposite direction, non-convex optimization problems will be formulated as partly linear convex problems. Illustrations are given in the following areas: Linear Algebra: An eigevalue inequality for convex functions of symmetric matrices is extended to nonconvex functions. The extension gives a lower bound of the celebrated Hadamard's determinantal inequality. It also yields new conditions for non-singularity of a matrix. Analysis: The classic Jensen inequality for convex functions is extended to nonconvex functions. Mpreover, both upper and lower bounds are obtained. For the product function f(x) = x1 x2 ... xn, these become bounds on the inner product of vectors. In some situations the upper bound is sharper that the Cauchy-Schwarz inequality. Optimization: We consider mathematical programs with smooth functions having Lipschitz derivatives. Such programs are reformulated as partly linear convex programs of the Liu-Floudas type. Similarly, programs with continuous non-smooth Lipschitz functions are reduced to a partly linear coordinate-wise monotone form.These two canonical forms are stable in the sense that the feasible set mappings are lower semi-continuous at optimal solutions relative to feasible perturbations.

Parametric Decompositions; Continuous Functions; Convex Functions; Smooth Functions; Non-smooth Functions

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