Sub-b-convex Functions and Sub-b-convex Programming Chao Mian-tao Jian Jin-bao Liang Dong-ying Abstract: The paper studied a new generalized convex function which called sub-b-convex function, and introduced a new concept of sub-b-convex set The basic properties of sub-b-convex functions were discussed in general case and differentiable case, respectively And, obtained the sufficient conditions that the sub-b convex function become quasi-convex function or pseudo-convex function Furthermore, the sufficient conditions of optimality for unconstrained and inequality constrained programming were obtained under the sub-b-convexity AMS Subject Classification: 26B25, 90C46 Key Words: convex function, sub-b-convex set, sub-b-convex function, pseudo-sub-b-convex function, optimality conditions g b à¼êúg b à5y x 7 {7 ùàl Á : ïä «g b à¼ê2âà¼ê, 0g b à8vg O3 œ/ 9Œ œ/e?øg b à¼êƒ'5ÿ, g b à¼ê [à¼ê9 à¼ê ^, 3g b à¼ê^ e ÑÃå9 تå5y `5^ Ä7 8µI[g, ÆÄ7 (10771040); 2Üg, ÆÄ7 (0832052); 2Ü Æ : 8: `zn Ø {9êÆïïÄ x 7,I,ïÄ),ù,chaomiantao@126com; 1 College of Applied Sciences, Beijing University of Technology, 100124, Beijing, China; ó ŒÆ A^ênÆ, 100124; 2 Department of Mathematics and Computer Science, Guangxi College of Education, 530023, Nanning, China; 2Ü Æ êæ OŽÅ ÆX, Hw 530023 {7,I,Æ,Ç, College of Mathematics and Information Science, Guangxi University, 530004, Nanning, China; 2ÜŒÆêÆ &E ÆÆ, Hw 530004 ÏÕŠö: ùàl,å,ïä),ù, liangdy go@126com; Guangxi Vocational and Technical College of Communications, 530023, Nanning, China; 2Ü Ï EâÆ, Hw 530023 1
aò: 90C26, 90C46 ' c:à¼ê, g b à8ü, g b à¼ê, g b à¼ê, `5^ 1 Introduction Convexity plays a vital role in many aspects of mathematical programming, for example, sufficient optimality conditions and duality theorems Over the years, many generalized convexity were presented (see [1]-[11]) In this paper, we introduce a new class of functions, which are called sub-b-convex functions and present some results about them In the following, we review several concepts of generalized convexity which have some relationships with this work Though out the paper, we assume that the set S R n is a nonempty convex set Definition 11 (see [2]) A real function f : S R n R is said to be a quasi-convex function, if f(x + (1 )y) max{f(x), f(y)}, x, y S, (0, 1) Definition 12 (see [2]) Let f : S R n R be a differentiable function on S The function f is said to be pseudo-convex function, if for each x, y S with f(x) T (y x) 0, we have f(x) f(y); or equivalently, if f(x) < f(y), then f(x) T (y x) < 0 Theorem 11 (see [2]) Let f : S R n R be differentiable on S Then f is a quasi-convex function if and only if the following equivalent statements holds: (1) If x, y S and f(x) f(y), then f(y) T (x y) 0; (2) If x, y S and f(y) T (x y) > 0, then f(x) > f(y) 2 Sub-b-convex Functions and Their Properties In this section, we present the definition of sub-b-convex function and discuss its some basic properties Definition 21 Let S be a nonempty convex set in R n The function f : S R is said to be a sub-b-convex function on S with respect to map b : S S [0, 1] R, if f(x + (1 )y) f(x) + (1 )f(y) + b(x, y, ), x, y S, [0, 1] 2
Remark 21 If one defines map b(x, y, ) = f(x + (1 )y) f(x) (1 )f(y), then, each function is sub-b-convex with this map However, our interesting is to study some b(x, y, ) with special propertiesand, obviously, each convex function f is sub-b-convex function with respect to the map b(x, y, ) = 0 Proposition 21 If f i : S R, (i = 1, 2, 3,, m) are sub-b-convex functions with respect to maps b i : S S [0, 1] R, (i = 1, 2, 3,, m) respectively Then function f = m a i f i, a i 0, i = 1, 2, 3,, m i=1 is sub-b-convex with respect to b = m i=1 a ib i Proof For all x, y S and [0, 1], f(x + (1 )y) = m i=1 a if i (x + (1 )y) m i=1 a i[f i (x) + (1 )f i (y) + b i (x, y, )] = m i=1 a if i (x) + (1 ) m i=1 a if i (y) + m i=1 a ib i (x, y, ) = f(x) + (1 )f(y) + m i=1 a ib i (x, y, ) So, from the definition of sub-b-convexity, knows f is sub-b-convex with respect to b = m i=1 a ib i Proposition 22 If functions f i : S R, (i = 1, 2, 3,, m) are sub-b-convex functions with respect to b i, (i = 1, 2, 3,, m), respectively Then f = max{f i, i = 1, 2,, m} is sub-b-convex with respect to b = max{b i, i = 1, 2,, m} Proof For forall x, y S and [0, 1], from the sub-b-convexity of f i, we have f(x + (1 )y) = max{f i (x + (1 )y), i = 1, 2,, m} = f t (x + (1 )y) f t (x) + (1 )f t (y) + b t (x, y, ) f(x) + (1 )f(y) + b(x, y, ) So, f(x) is is sub-b-convex with respect to b = max{b i, i = 1, 2,, m} Here, we introduced a new concept of sub-b-convex set Definition 22 Let X R n+1 be a nonempty set X is said to be sub-b-convex with respect to b : R n R n [0, 1] R, if (x + (1 )y, α + (1 )β + b(x, y, )) X, (x, α), (y, β) X, x, y R n, [0, 1] 3
We now give a characterization of sub-b-convex function f : S R in the term of their epigraph E(f) given by E(f) = {(x, α) x S, α R, f(x) α} Theorem 21 A function f : S R is sub-b-convex with respect to b : R n R n [0, 1] R if and only if E(f) is a sub-b-convex set with respect to b Proof Suppose that f is sub-b-convex with respect to b Let (x 1, α 1 ), (x 2, α 2 ) E(f) Then, f(x 1 ) α 1, f(x 2 ) α 2 Since f is sub-b-convex with respect to b, we have f(x 1 + (1 )x 2 ) f(x 1 ) + (1 )f(x 2 ) + b(x 1, x 2, ) α 1 + (1 )α 2 + b(x 1, x 2, ) Hence, from Definition 22 one has (x 1 + (1 )x 2, α 1 + (1 )α 2 + b(x 1, x 2, )) E(f) Thus E(f) is a sub-b-convex set with respect to b Conversely, assume that E(f) is a sub-b-convex set with respect to b Let x 1, x 2 S, then (x 1, f(x 1 )), (x 2, f(x 2 )) E(f) Thus, for [0, 1], (x 1 + (1 )x 2, f(x 1 ) + (1 )f(x 2 ) + b(x 1, x 2, )) E(f) This further follows that f(x 1 + (1 )x 2 ) f(x 1 ) + (1 )f(x 2 ) + b(x 1, x 2, ) That is, f is sub-b-convex with respect to b Proposition 23 If X i (i I) is a family of sub-b-convex set with respect to a same map b(x, y, ), then i I X i is a sub-b-convex set with respect to b(x, y, ) Proof Let (x, α), (y, β) i I X i, [0, 1] Then, for each i I, (x, α), (y, β) X i Since X i is a sub-b-convex set with respect to b, it follows that (x + (1 )y, α + (1 )β + b(x, y, )) X i, i I Thus, (x + (1 )y, α + (1 )β + b(x, y, )) i I X i Hence, i I X i is a sub-b-convex set with respect to b(x, y, ) 4
Theorem 22 If f i (i I) is a family of numerical functions, and each f i sub-b-convex with respect to the same map b(x, y, ), then the numerical function f = sup i I f i (x) is a sub-b-convex function with respect to b(x, y, ) Proof Since f i is a sub-b-convex function on S with respect to b(x, y, ), its epigraph E(f i ) = {(x, α) x S, f i (x) α} is a sub-b-convex set with respect to b Therefor, their intersection i I E(f i ) = {(x, α) x S, f i (x) α, i I} = {(x, α) x S, f(x) α} = E(f) is also a sub-b-convex set with respect to b So, from Theorem 21 and Proposition 22, one know that f = sup i I f i (x) is a sub-b-convex function with respect to b(x, y, ) In the follows, we consider continuously differentiable functions which are sub-b-convex func- b(x,y,) tion with respect to a map b Further, we assume that the limit lim 0+ x, y S is exists for fixed Theorem 23 Suppose that f : S R is differentiable and sub-b-convex with respect to map b Then f(y) T (x y) f(x) f(y) + lim 0+ b(x, y, ) Proof From Taylor expansion and the sub-b-convexity of f, we have f(y) + f(y)(x y) + o() = f(x + (1 )y) f(x) + (1 )f(y) + b(x, y, ) This implies that f(y)(x y) + o() f(x) f(y) + b(x, y, ) Dividing the inequality above by and taking 0+, we have The proof of this theorem is completed result f(y) T (x y) f(x) f(y) + lim 0+ b(x, y, ) Base on Theorem 333 of [2] and Theorem 23 of this paper, it is easy to get the following Corollary 21 Let f : S R be differentiable and sub-b-convex with respect to b If for each b(x,y,) x, y S, lim 0, then f is a convex function on S 0+ Theorem 24 Suppose that f : S R is differentiable and sub-b-convex with respect to b, then ( f(y) f(x)) T b(x, y, ) (x y) lim + lim 0+ 0+ 5 b(y, x, )
Proof From Theorem 23, we have Adding the two inequalities above, we have f(y) T (x y) f(x) f(y) + lim 0+ f(x) T (y x) f(y) f(x) + lim 0+ b(x, y, ), b(y, x, ) ( f(y) f(x)) T b(x, y, ) (x y) lim + lim 0+ 0+ b(y, x, ) The proof of this theorem is completed Theorem 25 Let f : S R be differentiable and sub-b-convex with respect to b If lim b(x,y,) f(x) f(y), x, y S, then f is quasi-convex Furthermore, if lim 0+ x, y S, then f is pseudo-convex 0+ b(x,y,) < f(x) f(y), we have If Proof For any x, y S and (0, 1) Suppose that f(x) f(y), then from Theorem 23, b(x,y,) lim 0+ f(y) T (x y) f(x) f(y) + lim 0+ b(x, y, ) b(x,y,) f(x) f(y), then f(x) f(y) + lim 0 So, f(y) T (x y) 0 0+ Therefor, we know from Theorem 11, we get that f is quasi-convex Similarly, if f(x) < f(y), we also have f(y) T (x y) < 0 So, from the Definition 12, we get that f is pseudo-convex 3 Optimality Conditions In this section, we apply the associated results above to the nonlinear programming problem First, we consider the unconstraint problem Theorem 41 Let f : S R be differentiable and sub-b-convex with respect to b Consider the optimal problem min{f(x) x S} If x S and relation f( x) T b(x, y, ) (x x) lim 0+ holds for each x S, then x is the optimal solution of f on S 0 (41) Proof For any x S, from Theorem 23, one has f( x) T b(x, x, ) (x x) lim 0+ 6 f(x) f( x)
From (41), we have f(x) f( x) 0 So, x is an optimal solution of f on S Next, we apply the associated results to the nonlinear programming problem with inequality constraints as follows: min f(x) (P g ) st g i (x) 0, i I = {1, 2,, m}, x R n Denote the feasible set of (P g ) by S g = { x R n g i (x) 0, i I} For convenience of discussion, we assume that f and g i are all differentiable and S g is a nonempty set in R n Now, we further extend the concept of sub-b-convex function, then discuss the optimality conditions of the corresponding programming Definition 44 Let S be a nonempty convex set in R n The function f : S R is said to be pseudo-sub-b-convex function on S with respect to b : S S [0, 1] R, if for each x, y S and (0, 1), from f(y) T b(x,y,) (x y) + lim 0+ 0 one can get f(x) f(y) Theorem 42 (Karush-Kuhn-Tucker Sufficient Conditions) The function f(x) is differentiable and pseudo-sub-b-convex with respect to b : S S [0, 1] R, g i (x) (i I) are differentiable and sub-b-convex with respect to b i : S S [0, 1] R (i I) Assume that x S g is a KKT point of (P g ) ie, there exist multipliers u i 0 (i I) such that f(x ) + i I u i g i (x ) = 0, u i g i (x ) = 0 (42) If b(x, x, ) lim 0+ i I u i lim 0+ b i (x, x, ), x S g (43) Then x is an optimal solution of the problem (P g ) Proof For any x S g, we have g i (x) 0 = g i (x ), i I(x ) = {i I g i (x ) = 0} Therefore, from the sub-b-convexity of g i (x) and Theorem 23, one obtain g i (x ) T (x x b(x,y,) ) lim 0 0+ for i I(x ) From (42), one has f(x ) T (x x ) = u i g i (x ) T (x x ) In view of (43), we have f(x ) T (x x ) + lim i I(x ) 0+ b(x,x,) u i ( g i (x ) T (x x b ) lim i (x,x,) ) 0+ From Theorem 0, i I(x ) So, i I(x ) i I(x ) u i g i (x ) T (x x ) + 23, one has g i (x ) T (x x b ) lim i (x,x,) 0+ f(x ) T (x x b(x, x, ) ) + lim 0+ 7 0 u i lim i I(x ) 0+ b i (x,x,) = g i (x) g i (x ) = g i (x)
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