#### Pregled bibliografske jedinice broj: 347890

## Parametric Decompositions of Continuous Functions

Parametric Decompositions of Continuous Functions

*// Parametric Optimization and Related Topics PARAOPT IX Conference Program and Abstracts*

Cienfuegos, Kuba, 2007. str. 9-9 (plenarno, sažetak, znanstveni)

**Naslov**

Parametric Decompositions of Continuous Functions

**Autori**

Zlobec, Sanjo

**Vrsta, podvrsta i kategorija rada**

Sažeci sa skupova, sažetak, znanstveni

**Izvornik**

Parametric Optimization and Related Topics PARAOPT IX Conference Program and Abstracts
/ - Cienfuegos, Kuba, 2007, 9-9

**Skup**

Parametric Optimization and Related Topics PARAOPT IX

**Mjesto i datum**

Cienfuegos, Kuba, 26. - 30. 03. 2007.

**Vrsta sudjelovanja**

Plenarno

**Vrsta recenzije**

Neobjavljeni rad

**Ključne riječi**

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

**Sažetak**

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.

**Izvorni jezik**

Engleski

**Znanstvena područja**

Matematika, Ekonomija

**POVEZANOST RADA**

**Projekt / tema**

067-0000000-1076 - Modeli i metode operacijskih istraživanja u ekonomici i poslovnom odlučivanju (Zrinka Lukač, )

**Ustanove**

Ekonomski fakultet, Zagreb

**Autor s matičnim brojem:**

Sanjo Zlobec, (283526)