Optimality conditions of E-convex programming for an E-differentiable function
© Megahed et al.; licensee Springer 2013
Received: 3 May 2012
Accepted: 30 April 2013
Published: 16 May 2013
In this paper we introduce a new definition of an E-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operator . By this definition, we can apply Kuhn-Tucker and Fritz-John conditions for obtaining the optimal solution of mathematical programming with a non-differentiable function.
KeywordsE-convex set E-convex function semi E-convex function E-differentiable function
The concepts of E-convex sets and an E-convex function have been introduced by Youness in [1, 2], and they have some important applications in various branches of mathematical sciences. Youness in  introduced a class of sets and functions which is called E-convex sets and E-convex functions by relaxing the definition of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator on the sets and the domain of the definition of functions. Also, in  Youness discussed the optimality criteria of E-convex programming. Xiusu Chen  introduced a new concept of semi E-convex functions and discussed its properties. Yu-Ru Syan and Stanelty  introduced some properties of an E-convex function, while Emam and Youness in  introduced a new class of E-convex sets and E-convex functions, which are called strongly E-convex sets and strongly E-convex functions, by taking the images of two points x and y under an operator besides the two points themselves. In  Megahed et al. introduced a combined interactive approach for solving E-convex multiobjective nonlinear programming. Also, in [7, 8] Iqbal and et al. introduced geodesic E-convex sets, geodesic E-convex and some properties of geodesic semi-E-convex functions.
In this paper we present the concept of an E-differentiable convex function which transforms a non-differentiable convex function to a differentiable function under an operator , for which we can apply the Fritz-John and Kuhn-Tucker conditions [9, 10] to find a solution of mathematical programming with a non-differentiable function.
In the following, we present the definitions of E-convex sets, E-convex functions, and semi E-convex functions.
A set M is said to be an E-convex set with respect to an operator if and only if for each and .
for each and .
for each and .
1- Let a setbe an E-convex set with respect to an operator E, , then.
2- Ifis a convex set and, then M is an E-convex set.
3- Ifandare E-convex sets with respect to E, thenis an E-convex set with respect to E.
Letbe an- and-convex set, then M is an- and-convex set.
Letbe a linear map and letbe E-convex sets, thenis an E-convex set.
2 Generalized E-convex function
Let be an E-convex set with respect to an operator . A function is said to be a pseudo E-convex function if for each with implies or for all and implies .
for each and .
3 E-differentiable function
Example 11 Let be a non-differentiable function at the point and let be an operator such that , then the function is a differentiable function at the point , and hence f is an E-differentiable function.
3.1 Problem formulation
Now, we formulate problems P and , which have a non-differentiable function and an E-differentiable function, respectively.
where f is an E-differentiable function.
Now, we will discuss the relationship between the solutions of problems P and .
Letbe a one-to-one and onto operator and let. Then, where M andare feasible regions of problems P and, respectively.
Theorem 13 Letbe a one-to-one and onto operator and let f be an E-differentiable function. If f is non-differentiable at, andis an optimal solution of the problem P, then there existssuch thatandis an optimal solution of the problem.
Proof Let be an optimal solution of the problem P. From Lemma 12 there exists such that . Let be a not optimal solution of the problem , then there is such that . Also, there exists such that . Then contradicts the optimality of for the problem P. Hence the proof is complete. □
Theorem 14 Letbe a one-to-one and onto operator, and let f be an E-differentiable function and strictly quasi-E-convex. Ifis an optimal solution of the problem P, then there existssuch thatandis an optimal solution of the problem .
Since M is an E-convex set and , then contradicts the assumption that is a solution of the problem P, then there exists , a solution of the problem , such that . □
and thus . □
Corollary 16 Let M be an E-convex set, letbe a one-to-one and onto operator, and letbe an E-differentiable and strictly E-convex function at. Ifis a local minimum of the function, then.
contradicting the assumption that is a local minimum of , and thus . □
Theorem 17 Let M be an E-convex set, be a one-to-one and onto operator, andbe twice E-differentiable and strictly E-convex function at. Ifis a local minimum of, thenand the Hessian matrixis positive semidefinite.
where as .
Then and are extremum points of , and the Hessian matrix is positive definite. And thus the point is a local minimum of the function , but the Hessian matrix is indefinite.
Theorem 19 Let M be an E-convex set, letbe a one-to-one and onto operator, and letbe a twice E-differentiable and strictly E-convex function at. Ifand the Hessian matrixis positive definite, thenis a local minimum of.
But for each k, and hence there exists an index set K such that , where . Considering this subsequence and the fact that as , then . This contradicts the assumption that is positive definite. Therefore is indeed a local minimum. □
is positive definite.
Example 21 Let be non-differentiable at the point , and let , then .
Then is a solution of the problem and is a solution of the problem P.
Lemma 23 Let M be an E-convex set with respect to an operator, and letbe E-differentiable at. Ifis a local minimum of the problem, then, where, and D is the cone of feasible direction of M at .
From 3.1 and 3.2 we have for each , where , which contradicts the assumption that is a local optimal solution, then . □
and E is one-to-one and onto.
From 3.3, 3.4 and 3.5, it is clear that points of the form are feasible to the problem for each , where . Thus , where D is the cone of feasible direction of the feasible region at . We have shown that for implies that , and hence . By Lemma 23, since is a local solution of the problem . It follows that . □
Theorem 25 (Fritz-John optimality conditions)
andis a local solution of the problem P.
holds and is a local solution of the problem P. □
Theorem 26 Letbe a one-to-one and onto operator and letbe an E-differentiable function. Ifis an optimal solution of the problem P, then there existssuch thatis an optimal solution of the problemand the Fritz-John optimality condition of the problemis satisfied.
Theorem 27 (Kuhn-Tucker necessary condition)
If , the assumption of linear independence of does not hold, then . By taking , then , holds for each . From Theorem 26, is a local solution of the problem . □
Theorem 28 Let M be an open E-convex set with respect to the one-to-one and onto operator, for, and letbe E-differentiable atand strictly E-convex at. Letbe a feasible solution of the problemand. Suppose that f is pseudo-E-convex atand thatis quasi-E-convex and differentiable atfor each. Furthermore, suppose that the Kuhn-Tucker conditions hold at. Thenis a global optimal solution of the problemand henceis a solution of the problem P.
Then is a global solution of the problem and from Theorem 13 is a global solution of the problem P. □
The solution is , , and is a solution of the problem P.
In this paper we introduced a new definition of an E-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operator , and we studied Kuhn-Tucker and Fritz-John conditions for obtaining an optimal solution of mathematical programming with a non-differentiable function. At the end, some examples have been presented to clarify the results.
The authors express their deep thanks and their respect to the referees and the Journal for these valuable comments in the evaluation of this paper.
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