Abstract
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.
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1 Introduction
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 [1] 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 [2] Youness discussed the optimality criteria of E-convex programming. Xiusu Chen [3] introduced a new concept of semi E-convex functions and discussed its properties. Yu-Ru Syan and Stanelty [4] introduced some properties of an E-convex function, while Emam and Youness in [5] 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 [6] 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.
Definition 1[1]
A set M is said to be an E-convex set with respect to an operator if and only if for each and .
Definition 2[1]
A function is said to be an E-convex function with respect to an operator on an E-convex set if and only if
for each and .
Definition 3[3]
A real-valued function is said to be semi E-convex function with respect to an operator on M if M is an E-convex set and
for each and .
Proposition 4[1]
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.
Lemma 5[1]
Letbe an- and-convex set, then M is an- and-convex set.
Lemma 6[1]
Letbe a linear map and letbe E-convex sets, thenis an E-convex set.
Definition 7[1]
Let and , we say that the set S is E-convex if for each and each , we have
2 Generalized E-convex function
Definition 8[1]
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 .
Definition 9[1]
Let be an E-convex set with respect to an operator . A function is said to be a quasi-E-convex function if and only if
for each and .
3 E-differentiable function
Definition 10 Let be a non-differentiable function at and let be an operator. A function f is said to be E-differentiable at if and only if is a differentiable function at and
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.
Let be an operator, M be an E-convex set and f be an E-differentiable function. The problem P is defined as
where f is a non-differentiable function, and the problem is defined as
where f is an E-differentiable function.
Now, we will discuss the relationship between the solutions of problems P and .
Lemma 12[11]
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 .
Proof Let be an optimal solution of the problem P. Then from Lemma 12 there is such that . Let be a not optimal solution of the problem , then there is and also , such that . Since f is strictly quasi-E-convex function, then
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 . □
Theorem 15 Let M be an E-convex set, be a one-to-one and onto operator andbe an E-differentiable function at. If there is a vectorsuch that, then there existssuch that
Proof Since f is an E-differentiable function at , then
Since and as , then there exists 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.
Proof Suppose that and let , then . By Theorem 15 there exists such that
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.
Proof Suppose that d is an arbitrary direction. Since f is a twice E-differentiable function at , then
where as .
From Corollary 16 we have , and
Since is a local minimum of , then , and
□
Example 18 Let be a non-differentiable function at , and let , then , and
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.
Proof Suppose that is not a local minimum of , and there exists a sequence is converging to such that for each k. Since , and f is twice E-differentiable at , then
where as , and
By dividing on , and letting , we get
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. □
Example 20 Let be a non-differentiable at the point , and let , then
The necessary condition for is a local minimum of is , then , and the Hessian matrix
is positive definite.
Example 21 Let be non-differentiable at the point , and let , then .
Now, let , , be an E-convex set with respect to operator E (the feasible region is shown in Figure 1) and
Then is a solution of the problem and is a solution of the problem P.
Definition 22 Let M be a nonempty E-convex set in and let . The cone of feasible direction of at denoted by D is given by
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 .
Proof Suppose that there exists a vector . Then by Theorem 15, there exists such that
By the definition of the cone of feasible direction, there exists such that
From 3.1 and 3.2 we have for each , where , which contradicts the assumption that is a local optimal solution, then . □
Lemma 24 Let M be an open E-convex set with respect to an operator, letbe E-differentiable atand letfor. Letbe a feasible solution of the problemand let. Furthermore, suppose thatforis E-differentiable atand thatforis continuous at. Ifis a local optimal solution, then, where
and E is one-to-one and onto.
Proof Let . Since and M is an open E-convex set, there exists a such that
Also, since and since is continuous at for , there exists a such that
Finally, since , for each and by Theorem 15, there exists such that
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)
Let M be an open E-convex set with respect to the one-to-one and onto operator, letbe E-differentiable atand letfor. Letbe feasible solution of the problemand let. Furthermore, suppose thatforis differentiable atand thatforis continuous at. Ifis a local optimal solution, then there exist scalarsandforsuch that
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andis a local solution of the problem P.
Proof Let be a local solution of the problem , then there is no vector d such that and . Let A be a matrix with rows and . From Gordon’s theorem [10], we have the system is inconsistent, then there exists a vector such that , where for each . And thus
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.
Proof Let be an optimal solution of the problem P. Since E is one-to-one and onto, according to Theorem 13, there exists , is an optimal solution of the problem . Hence there exist scalars satisfying the Fritz-John optimality conditions of the problem
□
Theorem 27 (Kuhn-Tucker necessary condition)
Let M be an open E-convex set with respect to the one-to-one and onto operator, letbe E-differentiable and strictly E-convex atand letfor. Letbe a feasible solution of the problemand let. Furthermore, suppose thatis continuous atforandforare linearly independent. Ifis a solution of the problem P, andis a local solution of the problem, then there exist scalarsforsuch that
Proof From the Fritz-John optimality condition theorem, there exist scalars and for each such that
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.
Proof Let be a feasible solution of the problem , then for each . Since , and is quasi-E-convex at , then
This means that does not increase by moving from along the direction . Then we must have from Theorem 15 that . Multiplying by and summing over I, we get
But since
it follows that . Since f is pseudo E-convex at , we get
Then is a global solution of the problem and from Theorem 13 is a global solution of the problem P. □
Example 29 Consider the following problem (problem P):
The feasible region of this problem is shown in Figure 2.
Let , then the problem is as follows:
We note that , where
The Kuhn-Tucker conditions are as follows:
The solution is , , and is a solution of the problem P.
4 Conclusion
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.
References
Youness EA: E -convex sets, E -convex functions and E -convex programming. J. Optim. Theory Appl. 1999, 102(3):439–450.
Youness EA: Optimality criteria in E -convex programming. Chaos Solitons Fractals 2001, 12: 1737–1745. 10.1016/S0960-0779(00)00036-9
Chen X: Some properties of semi- E -convex functions. J. Math. Anal. Appl. 2002, 275: 251–262. 10.1016/S0022-247X(02)00325-6
Syau Y-R, Lee ES: Some properties of E -convex functions. Appl. Math. Lett. 2005, 18: 1074–1080. 10.1016/j.aml.2004.09.018
Emam T, Youness EA: Semi strongly E -convex function. J. Math. Stat. 2005, 1(1):51–57.
Megahed AA, Gomma HG, Youness EA, El-Banna AH: A combined interactive approach for solving E -convex multi- objective nonlinear programming. Appl. Math. Comput. 2011, 217: 6777–6784. 10.1016/j.amc.2010.12.086
Iqbal A, Ahmad I, Ali S: Some properties of geodesic semi- E -convex functions. Nonlinear Anal., Theory Methods Appl. 2011, 74: 6805–6813. 10.1016/j.na.2011.07.005
Iqbal, A, Ali, S, Ahmad, I: On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs. J. Optim. Theory Appl. (2012), (Article Available online)
Mangasarian OL: Nonlinear Programming. Mcgraw-Hill, New York; 1969.
Bazaraa MS, Shetty CM: Nonlinear Programming Theory and Algorithms. Wiley, New York; 1979.
Youness EA: Characterization of efficient solution of multiobjective E -convex programming problems. Appl. Math. Comput. 2004, 151(3):755–761. 10.1016/S0096-3003(03)00526-5
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Megahed, A.EM.A., Gomma, H.G., Youness, E.A. et al. Optimality conditions of E-convex programming for an E-differentiable function. J Inequal Appl 2013, 246 (2013). https://doi.org/10.1186/1029-242X-2013-246
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DOI: https://doi.org/10.1186/1029-242X-2013-246