For a kind of fractional programming problem that the objective functions are the ratio of two dc (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructe...
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For a kind of fractional programming problem that the objective functions are the ratio of two dc (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal Solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper. (C) 2008 Elsevier Ltd. All rights reserved.
We present some Farkas-type results for inequality systems involving finitely many dc functions. To this end we use the so-called Fenchel-Lagrange duality approach applied to an optimization problem with dc objective ...
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We present some Farkas-type results for inequality systems involving finitely many dc functions. To this end we use the so-called Fenchel-Lagrange duality approach applied to an optimization problem with dc objective function and dc inequality constraints. Some recently obtained Farkas-type results are rediscovered as special cases of our main result.
In this paper, we deal with extended Ky Fan inequalities (EKFI) with dc functions. Firstly, a dual scheme for (EKFI) is introduced by using the method of Fenchel conjugate function. Under suitable conditions, weak and...
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In this paper, we deal with extended Ky Fan inequalities (EKFI) with dc functions. Firstly, a dual scheme for (EKFI) is introduced by using the method of Fenchel conjugate function. Under suitable conditions, weak and strong duality assertions are obtained. Then, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of (EKFI) are given. Finally, as applications, the proposed approach is applied to a convex optimization problem (COP) and a generalized variational inequality problem (GVIP).
We consider a generalized equilibrium problem involving dc functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for dc programming problems....
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We consider a generalized equilibrium problem involving dc functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for dc programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311-319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco's dual problem (Mosco in J Math Anal Appl 40:202-206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795-805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353-371, 2007, Pac J Optim 2:667-678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to dc and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for dc problems in Dinh et al. (Optimization 1-20, 2008, J Convex Anal 15:235-262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667-678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for dc and convex problems.
In this paper, by using the properties of the epigraph of the conjugate functions, we introduce some closedness conditions and investigate some characterizations of these closedness conditions. Then, by using these cl...
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In this paper, by using the properties of the epigraph of the conjugate functions, we introduce some closedness conditions and investigate some characterizations of these closedness conditions. Then, by using these closedness conditions, we obtain some Farkas-type results for a constrained fractional programming problem with dc functions. We also show that our results encompass as special cases some programming problems considered in the recent literature.
Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two ...
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Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with dc functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.
A dc. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of f...
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A dc. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in R(n) into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations-inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.
In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (dc functions) is proposed. The algorithm is based on the concept of codifferential. It is ass...
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In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (dc functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that dc decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given.
In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (dc functions) is proposed. The algorithm is based on the concept of codifferential. It is ass...
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In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (dc functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that dc decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given.
A proximal linearized algorithm for minimizing difference of two convex functions is proposed. If the sequence generated by the algorithm is bounded it is proved that every cluster point is a critical point of the fun...
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A proximal linearized algorithm for minimizing difference of two convex functions is proposed. If the sequence generated by the algorithm is bounded it is proved that every cluster point is a critical point of the function under consideration, even if the auxiliary minimizations are performed inexactly at each iteration. Linear convergence of the sequence is established under suitable additional assumptions.
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