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- W2171077402 abstract "We introduce a family of new transforms based on the imitation of the Breg- man proximal mapping to minimize the difference of two convex functions. It can be seen as a descent scheme which takes into consideration the convex proper- ties of the two convex functions separately. A direct application of the proposed scheme to variational inclusion is given. 1 Preliminaries In non-convex programming, the fundamental property of convex problems that lo- cal solutions are global ones is no longer true. Therefore, methods using only local information are insufficient to locate global minima. Thus, optimality conditions for non-convex optimization problems have to take into account the form and the structure of the model. Here in this work, we are interested in a certain class of models called d.c. problems. These problems deal with a minimization or maximization of the difference of two or more convex functions. It is well known that with two convex functions g and h the sum g +h is again a convex function, as is the maximum max{g,h} and the multiple λg for any positive λ. The difference g−h, however, is not a convex function. This is why d.c. problems are difficult as they are non-convex problems (see (11, 8, 4) and references therein). In this work, we are presenting a modified Bregman proximal scheme to find out the minimum of the difference of two convex functions, using the convexity of both functions involved in the d.c model. The presented scheme is a type of descent method. Let E be a finite-dimensional vector space and let k ·,·ldenote the inner product in E.L etΘ(E) denote the set of convex proper and lower semi-continuous functions on E.L etf be a d.c function on E, that means that there exist two functions g and h both in Θ(E )s uch thatf(x )= g(x) − h(x), ∀ x ∈ E. Moreover, suppose that h(x) 9 ∞ for all x ∈ E and that ridom(g) ∩ ridom(h) 9 ∅. In this paper we are concerned with minimizing a d.c. function f on E: min∈E f(x)." @default.
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- W2171077402 title "A modified Bregman proximal scheme to minimize the difference of two convex functions." @default.
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