Int. J. Anal. Appl. (2023), 21:63 Received: Oct. 31, 2020. 2020 Mathematics Subject Classification. 90C26, 90C29, 90C48. Key words and phrases. generalized subconvexlike maps; presubconvexlike maps; set-valued optimization; optimality conditions. https://doi.org/10.28924/2291-8639-21-2023-63 © 2023 the author(s) ISSN: 2291-8639 1 Optimality Conditions for Set-Valued Optimization Problems Renying Zeng* Mathematics Department, Saskatchewan Polytechnic, Canada *Correspondence: renying.zeng@saskpolytech.ca ABSTRACT. In this paper, we first prove that the generalized subconvexlikeness introduced by Yang, Yang and Chen [1] and the presubconvelikeness introduced by Zeng [2] are equivalent. We discuss set-valued nonconvex optimization problems and obtain some optimality conditions. 1. Introduction Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objectives and/or the constraints are set-valued maps. Corley [3] pointed out that the dual problem of a multiobjective optimization involves the optimization of a set- valued map, while Klein and Thompson [4] gave some examples in Economics where it is necessary to use set-valued maps instead of single-valued maps. There are many recent developments about set-values optimization problems, e.g., [5-9]. Convex and generalized convex optimization is a rich branch of mathematics. Many interesting and useful definitions of generalized convexities were introduced. Borwein [10] proposed the definition of cone convexity, Fan [11] introduced the definition of convexlikeness. Yang, Yang and Chen [1] defined the generalized subconvexlike functions, while Zeng [2] introduced a presubconvexlikeness. In this paper, we first prove that the generalized subconvexlikeness introduced by Yang, Yang, and Chen [1] and the presubconvexlikeness introduced by Zeng [2] are equivalent, in locally convex https://doi.org/10.28924/2291-8639-21-2023-63 2 Int. J. Anal. Appl. (2023), 21:63 topological spaces. And then, we deal with set-valued optimization problems and obtain some optimality conditions. A subset Y + of a real linear topological space Y is a cone if y Y +  for all y Y +  and 0  . We denote by 0 Y the zero element in the linear topological space Y and simply by 0 if there is no confusion. A convex cone is one for which 1 1 2 2 y y Y  + +  for all 1 2 ,y y Y +  and 1 2 , 0   . A pointed cone is one for which ( ) {0}Y Y + + − = . Let Y be a real linear topological space with pointed convex cone Y + . We denote the partial order induced by Y + as follows: 1 2 y y iff 1 2 y y Y + −  , 1 2 y y iff 1 2 inty y Y + −  , where intY + denotes the topological interior of a set Y + . Let X, Zi, Wj be real linear topological spaces and Y be an ordered linear topological space with the partial order induced by a pointed convex cone Y + . We recall some notions of generalized convexity of set-valued maps. First we recall the notion of cone-convexity of a set-valued map introduced by Borwein [10]. Definition 1.1 (Convexity) Let X, Y be real linear topological spaces, D X a nonempty convex set and Y + a convex cone in Y. A set-valued map f : X Y→ is said to be Y + -convex on D if and only if 1 2 ,x x D  , [0,1]  , there holds 1 2 1 2 ( ) (1 ) ( ) ( (1 ) ) .f x f x f x x Y    + + −  + − + The following notion of generalized convexity is a set-valued map version of Ky Fan convexity [11] (Ky Fan’s definition was for vector-valued optimization problems). Definition 1.2 (Convexlike) Let X, Y be real linear topological spaces, D X a nonempty set and Y + be a convex cone in Y. A set-valued map f : X Y→ is said to be Y + -convexlike on D if and only if 1 2 ,x x D  , [0,1]  , 3 x D  such that 1 2 3 ( ) (1 ) ( ) ( ) .f x f x f x Y  + + −  + The following concept of generalized subconvexlikeness was introduced by Yang, Yang and Chen [1] ([1] introduced subconvexlikeness for vector-valued optimization). 3 Int. J. Anal. Appl. (2023), 21:63 Definition 1.3 (Generalized subconvexlike) Let Y be a linear topological space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is said to be generalized Y + -subconvexlike on D if intu Y +   such that 1 2 ,x x D  , 0  , [0,1]  , 3 x D  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y    + + + −  + . The following Lemma 1.1 is from Chen and Rong [12, Proposition 3.1]. Lemma 1.1 A function f : YD → is generalized Y + -subconvexlike on D if  + Yu int , 1 2 ,x x D  ,  ]1,0[ ,  Dx 3 ,  0 such that 1 2 3 ( ) (1 ) ( ) ( ) .u f x f x f x Y   + + + −  + A bounded function in a real linear topological space can be fined as following Definition 1.4 (e.g,, see Yosida [13]). Definition 1.4 (Bounded set-valued map) A subset M of a real linear topological space Y is said to be a bounded subset if for any given neighbourhood U of 0,  positive scalar  such that 1 M U −  , where 1 1 { ; ; }M y Y y v v M  − − =  =  . A set-valued map f : D Y→ is said to bounded map if f (Y) is a bounded subset of Y. The following Definition 1.5 was introduced by Zeng [2] for single-valued functions. Definition 1.5 (Presubconvexlike) Let Y be a linear topological space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is said to be Y + -presubconvexlike on D if 1 2 ,x x D  , [0,1]  , 0  , 3 x D  , 0  ,  bounded set-valued map u: D Y→ such that 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y    + + + −  + . It is obvious that Y + -convex  Y+ -convexlike  generalized Y+ -subconvexlike  Y+ - presubconvexlike. It is important to note that the concept of convexlike or any weaker concepts are only nontrivial if Y is not the one-dimensional Euclidean space since any real-valued function is R+-convexlike. 4 Int. J. Anal. Appl. (2023), 21:63 2. The Equivalence of Generalized Subconvexlikeness and Presubconvexlikeness In this section, we are going to prove that Definition 1.4 (Generalized subconvexlikeness) and Definition 1.5 (Presubconvexlikeness) are equivalent. Definition 2.1 (1) A subset M of Y is said to be convex, if 1 2 ,y y M and 0 1  implies 1 2 (1 )y y M + −  ; (2) M is said to be balanced if y M and | | 1  implies y M  ; (3) M is said to be absorbing if for any given neighbourhood U of 0, there exists a positive scalar  such that 1 M U −  , where 1 1 { ; ; }M y Y y v v M  − − =  =  . Definition 2.2 A real linear topological space Y is called a locally convex, linear topological space (we call it a locally convex topological space, in the sequel) if any neighborhood of 0 Y contains a convex, balanced, and absorbing open set. From [13, pp.26 Theorem, pp.33 Definition 1] one has Lemma 2.1. Lemma 2.1 Banach spaces are locally convex topological spaces, so are finite dimensional Euclidean spaces. Proposition 2.1 Let Y be a locally convex topological space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is generalized Y + -subconvexlike on D if and only if 0 ( ( ) int ) t tf D Y  + + is convex. Proof. The necessity. See [1, Theorem 2.1]. The sufficiency. Assume that 0 ( ( ) int ) t tf D Y  + + is convex, aim to show that f : D Y→ is generalized Y + -subconvexlike on D. From Lemma 1.1, we are going to show that,  + Yu int , 1 2 ,x x D  ,  ]1,0[ ,  Dx 3 ,  0 such that 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y   + + + −  + . int ,y Y +   0,t  since int Y + is a cone, one has int .ty Y +  1 1 2 2 ( ), ( ),y f x y f x   ,R  one has 1 2 0 ( ) , ( ) ( ( ) int ). t f x ty f x ty tf D Y + + +  + From the convexity of 0 ( ( ) int ) t tf D Y  + + , 3 3 , intx D y Y +     , 0  such that 5 Int. J. Anal. Appl. (2023), 21:63 1 2 1 2 3 3 0 ( ( ) ) (1 )( ( ) ) ( ) (1 ) ( ) ( ) ( ( ) int ). t f x ty f x ty f x f x ty f x y tf D Y      + + + − +  + − +  +  + For the given intu Y +  , From Definition 2.2, neighbourhood U of 0 such that U is convex, balanced, and absorbing, and intu U Y + +  , where u U+ is a neighbourhood of u. Therefore, we may take 0t  small enough, such that ty U−  . Then, intty u u U Y + − +  +  . This and the convexity of int Y + imply that 3 int .y ty u Y + − +  And so 1 2 3 3 3 3 ( ) (1 ) ( ) ( ) ( ) int ( ) .u f x f x f x y ty f x Y f x Y     + + + + −  + −  +  + Proposition 2.2 Let Y be a locally convex topological space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is Y + -presubconvexlike on D if and only if 0 ( ( ) int ) t tf D Y  + + is convex. Proof. The necessity. Suppose that f is Y + -presubconvexlike on, aim to show that 0 ( ( ) int ) t tf D Y  + + is convex. 1 2 1 1 1 2 2 2 0 , ( ( ) int ) t v t y y v t y y tf D Y + +  +  = + = +  + , 1 2 ,x x D  such that 1 1 ( ),y f x 2 2 ( )y f x . Let 0 1 2 (1 ) ,y y y  + + + = + − then 0 int .y Y + +  Therefore, neighbourhood U of 0 such that 0 y U + + is a neighbourhood of 0 y + and 0 inty U Y + + +  . By Definition 2.2, without loss of generality, we may assume that U is convex, balanced, and absorbing. From the assumption of Y + -presubconvexlikeness, 0  , 3 x D  ,  bounded function u, and  > 0 such that 1 2 1 2 3 1 2 1 2 (1 ) ( ) ( ) ( ) , (1 ) (1 ) t t f x f x f x u Y t t t t         + − +  − + + − + − 6 Int. J. Anal. Appl. (2023), 21:63 Therefore, 3 3 ( )y f x  such that 1 2 1 2 1 1 2 2 01 2 1 2 1 2 1 2 1 2 01 2 1 2 1 2 1 2 1 2 0 1 2 3 1 (1 ) (1 ) (1 ) (1 ) ( (1 ) )[ ] (1 ) (1 ) (1 ) ( (1 ) )[ ( ) ( )] (1 ) (1 ) ( (1 ) )[ ( ) ] ( ( v v t y t y y y t t t t y y y t t t t t t t t f x f x y t t t t t t f x u Y y t                            + + + + + + + − = + − + + − − = + − + + + − + − −  + − + + + − + −  + − − + + = + 0 2 3 1 2 1 ) ) ( ) ( (1 ) )( ) .t f x t t Y u y     + + − + + − − + Since U is convex, balanced, and absorbing, by Definition 2.2, we may take 0  small enough such that 1 2 ( (1 ) )t t u U  − + −  . Therefore 0 0 1 2 ( (1 ) ) intt t u y y U Y   + + + − + − +  +  . And then 0 1 2 1 2 ( (1 ) ) ( (1 ) ) int int .t t Y t t u y Y Y Y     + + + + + + − − + − +  +  Therefore 1 2 0 1 2 3 1 2 0 (1 ) ( (1 ) ) ( ) ( (1 ) )( ) ( ( ) int ). t v v t t f x t t Y u y tf D Y         + +  + + −  + − + + − − +  + Hence 0 ( ( ) int ) t tf D Y  + + is a convex set. The sufficiency. Assume that 0 ( ( ) int ) t tf D Y  + + is convex. From Lemma 1.1 and Proposition 2.1,  intu Y +  such that for all 1 2 ,x x D  ,  ]1,0[ , 0  , 3 x D  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y    + + + −  + . The given intu Y +  can be consider as a bounded function. By Propositions 1 and 2 one has Theorem 2.1. 7 Int. J. Anal. Appl. (2023), 21:63 Theorem 2.1 Let Y be a locally convex topological space and D X be a nonempty set, and Y + a convex cone in Y. A set-valued map f : D Y→ is generalized Y + -subconvexlike on D if and only if f is Y + -presubconvexlike on D. 3. Optimal Conditions We consider the following optimization problem with set-valued maps: (VP) Y+ -min f (x) s.t. ( ) ( ) 0 i i g x Z + −  , 1, 2, ,i m=  0 ( ), 1, 2, j h x j = , n x D where f : X Y→ , : i i g X Z→ , jj WXh →: are set-valued maps, Zi+ is a closed convex cone in Zi and D is a nonempty subset of X. For a set-valued map f : X Y→ , we denote by ( ) ( ) x D f D f x  = . We now explain the kind of optimality we consider here. Let F be the feasible set of (VP), i.e. : { : ( ) ( ) , 1, 2, , ;0 ( ), 1, 2, , }. i i j F x D g x Z i m h x j n + =  −  =   =  We are looking for a weakly efficient solution of (VP) defined as follows. Definition 3.1 (Weakly Efficient Solution) A point x F is said to be a weakly efficient solution of (VP) with a weakly efficient value ( )y f x if for every x F , there exists no ( )y f x satisfying y y . Consider the set-valued optimization problem (VP). From now on we assume that Y+, Zi+ are pointed convex cones with nonempty interior of int Y + , int i Z + , respectively. The following three assumptions will be used in this paper. (A1) Generalized Convexity Assumption. There exist 0 intu Y +  , int i i u Z +  such that for all 1 2 ,x x D , 0  , [0,1]  ,, there exist 3 x D , 0( 1, 2, , ), 0( 1, 2, , ) i j i m t j n  =   =  such that 0 1 2 0 3 1 2 3 1 2 3 ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) i i i i i i j j j j u f x f x f x Y u g x g x g x Z h x h x t h x           + + + + −  + + + −  + + −  (A2) Interior Point Assumption. 8 Int. J. Anal. Appl. (2023), 21:63 int ( ) j h D  , (j = 1, 2, …, n). (A3) Finite Dimension Assumption. Wj (j = 1, 2, …, n) are finite dimensional spaces. Similar to the proof of Propositions 2.1 or 2.2, one has Proposition 3.1. Proposition 3.1 Assumption (A1) is satisfied if and only if the following set is convex: 1 1 0 : {( , , ) : , , 0, . . ( ) int , ( ) int , ( )}. m n i i j j i j i i i i j j j B y z w Y Z W x D t s t y f x Y z g x Z w t h x    = = + + =        +  +  Proposition 3.2 (Alternative Theorem) Assume that the assumption (A1) and either (A2) or (A3) are satisfied. Consider the following generalized inequality-equality systems: [System 1] , . . ( ) ( int ) , ( ) ( ) , 0 ( ). i i j x D s t f x Y g x Z h x + +   −  −   [System 2] * * * 1 1 1 1 ( , , ) ( ) \ {0}, . . ( ( )) ( ( )) ( ( )) 0. m n i i j j m n i i j ji j Y Z W s t x D f x g x h x       + = + = = =       + +   Then if System 1 has no solution x, then System 2 has a solution ( , , )   . If System 2 has a solution ( , , )   with 0  , then System 1 has no solution. Proof. Suppose that System 1 has no solution, then 0 B . Since (A1) holds, the set B is convex. By assumption, 1 m i i Z =  is infinite dimensional and (A2) holds (which is equivalent to saying that int B  ) or 1 n j j W =  is finite dimensional. Therefore by the separation theorem, nonzero vector * * * 1 1 ( , , ) m n i i j j Y Z W   = =    such that 0 0 0 1 1 ( ) ( ) ( ) 0 m n i i i i j j ji j y y z z t w     = = + + + +   for all 0 0 , ( ), ( ), ( ), int , int , 0, 0. i i j j i i i j x D y f x z g x w h x y Y z Z t + +         Since intY+, intZi+ are convex cones, we have 0 0 0 0 1 1 ( ) ( ) ( ) 0 m n i i i i i j j ji j y s y z s z t w     = = + + + +   For all 0 0 , ( ), ( ), ( ), int , int , 0, 0, 0( 0,1, 2, ). i i j j i i i j i x D y f x z g x w h x y Y z Z t s i m + +          =  9 Int. J. Anal. Appl. (2023), 21:63 Taking ),,2,1,0(0,0,0 mist iji =→→→ , we obtain 0 0 ( ) 0, inty y Y +    , and consequently 0 0 ( ) 0, inty y Y clY cl Y + + +     = , where clY + is the topological closure of the set Y + . Similarly, we have ( ) 0, i i i i z z Z +    , and hence * Y +  , * i i Z +  . Let 1( 1, 2, , ), 1( 1, 2, , ) i j i m t j n = =  = =  and take 0( 0,1, 2, ) i s i m→ =  , we have 1 1 ( ) ( ) ( ) 0 m n i i j ji j y z w   = = + +   For , ( ), ( ), ( ) i i j j x D y f x z g x w h x    . Hence, System 2 has a solution ( , , )   . Conversely, suppose that System 2 has a solution ( , , )   with 0  . If System 1 has a solution x D  , there would exist ( ), ( ), ( ) i i j j y f x z g x w h x   such that int , , 0 i i j y Y z Z w + + − − = . Thus, ( ) 0, ( ) 0, 0 i i j j y z w    = , i.e., 0)()()( 11 ++  == n j jj m i ii wzy  . which is a contradiction and hence System 1 does not have a solution. Theorem 3.1 [Fritz John Type Necessary Optimality Condition] Assume that the generalized convexity assumption (A1) is satisfied and either (A2) or (A3) holds. If x F is a weakly efficient solution of (VP) with ( )y f x , nonzero vector * * * 1 1 ( , , ) m n i i j j Y Z W   = =    such that 1 1 1 ( ) min[ ( ( )) ( ( )) ( ( ))] min ( ( )) 0, m n i i j ji jx D m i ii y f x g x h x g x      = = = = + + =    where ( )1 1 min ( ( )) : min ( ) i i m m i i z g x i ii i g x z  = = =  . Proof. Since x F is a weakly efficient solution of (VP) with ( )y f x , by definition the following system 10 Int. J. Anal. Appl. (2023), 21:63 , ( ( ) ) ( int ) , ( ) ( ) , 0 ( ) i i j x D f x y Y g x Z h x + +  − −   −    has no solution. By Proposition 2.2, there exists a nonzero vector * * * 1 1 ( , , ) m n i i j j Y Z W   = =    such that x D  there holds 1 1 ( ( ) ) ( ( )) ( ( )) 0 m n i i j ji j f x y g x h x   = = − + +   . Since x F , there exists ( ) i i z g x such that i i z Z + − . For such i z , it follows * i i Z +  that ( ) 0 i i z  . On the other hand, taking x x= we get 1 1 ( ( ) ) ( ) ( ( )) 0 m n i i j ji j f x y z h x   = = − + +   , and noticing that ( )y f x and 0 ( ) j h x we obtain 1 ( ) 0 m i ii z =  , and hence ( ) 0 i i z = . Since 1 1 ( ) ( ) (0) ( ) m n i i ji j y z y    = = + + =  , taking x x= again we get 1 1 ( ( ) ) ( ( )) ( ( )) 0 m n i i j ji j f x y g x h x   = = − + +   . Noticing that ( )y f x and 0 ( ) j h x , we obtain 1 ( ( )) 0 m i ii g x =  . We have shown previously that there exists ( ) i i z g x such that ( ) 0 i i z = . Therefore 1 min ( ( )) 0 m i ii g x = = . Theorem 3.2 (Sufficient Optimality Condition) Let x F and ( )y f x . If there exists a * * * 1 1 ( , , ) m n i i j j Y Z W   = =    with 0  such that 1 1 ( ) min[ ( ( )) ( ( )) ( ( ))] m n i i j ji jx D y f x g x h x    = =  + +  , then x is a weakly efficient solution of (VP) with ( )y f x . Proof. By contradiction, we assume that x F is not a weakly efficient solution of (VP) with ( )y f x . Then by definition,  0 x F and  0 0 ( )y f x such that 0 inty y Y + −  , which implies 11 Int. J. Anal. Appl. (2023), 21:63 that 0 ( ) 0y y −  . Since 0 x F , 0 0 ( ) j h x and  0 0 ( ) i i z g x such that 0 i i z Z + − , and hence 0 ( ) 0 i i z  . Consequently, 0 0 1 1 ( ) ( ) (0) 0 m n i i ji j y y z   = = − + +   . Hence x is a weakly efficient solution of (VP) with ( )y f x . From Theorem 3.2 and 3.3 one has Theorem 3.3. Theorem 3.3 (Strong Duality) Suppose all assumptions in Theorem 3.1 hold and there is no nonzero vector ( , ) m n R R  +   satisfying the system: 1 1 min[ ( ) ( )] 0 ( ) 0. m n i i j ji jx D i i g x h x g x    = = + = =   Let x be a solution of problem (P). Then the strong duality holds. That is, 1 1( ) 0, ( ) 0, 0 ( ) min ( ) max min[ ( ) ( ) ( )]. m n i i j ji jg x h x x D x D f x f x f x g x h x    = = =   = = + +  4. Applications to Single-Valued Optimization Problems Consider the optimization problem: (P) min f (x) s.t. ( ) 0 i g x  (i = 1, 2, …, m) ( ) 0, ( 1, 2, j h x j= = ,n) x D where f, gi, hj: X R→ are functions and D is a nonempty subset of X. Applying Theorem 3.1 to the above single-valued optimization problem we have the following Fritz John type necessary optimality condition. Theorem 4.1 Let x be an optimal solution of (P). Suppose the following generalized convexity assumption holds:  0,( 0,1, 2, , )iu i m =  such that 1 2,x x D  , 0  , [0,1]  ,  3 x D ,  0, ( 1, 2, , ) i i m  =  ,  ,0jt ( 1, 2, , )j n=  there holds 0 1 2 0 3 1 2 3 1 2 3 ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) i i i i i j j j j u f x f x f x R u g x g x g x R h x h x t h x           + + + + −  + + + −  + + − = ( 1, 2, ,i m=  ; 1, 2, , )j n=  . 12 Int. J. Anal. Appl. (2023), 21:63 Then, nonzero vector ( , , ) m n R R R   + +    such that 1 1 1 ( ) min[ ( ) ( ) ( )] min ( ) 0. m n i i j ji jx D m i ii f x f x g x h x g x      = = = = + + =    We now study some cases where the generalized convexity holds and consequently the Fritz John condition in the above theorem holds. Theorem 4.2 Let x be an optimal solution of (P). Suppose one of the following set of assumptions hold. (I) All functions gi are nonnegative on the set D and n = 0 (i.e. there is no equality constraints). (II) All functions f, gi are nonnegative on the set D and n = 1. Then, non-zero vector ( , , ) m n R R R   + +    such that 1 1 1 ( ) min[ ( ) ( ) ( )] min ( ) 0. m n i i j ji jx D m i ii f x f x g x h x g x      = = = = + + =    Proof. From Theorem 4.1, it suffices to prove that the generalized convexity assumption holds. First assume that assumption (I) holds. Let 1 2 ,x x D , [0,1]  . Case 1: 1 2 ( ) ( )f x f x .Then 1 2 1 2 1 ( ) (1 ) ( ) ( ) (1 )( ( ) ( )) 0.f x f x f x f x f x  + − − = − −  Let 3 1 x x= . Then 1 2 3 ( ) (1 ) ( ) ( ) .f x f x f x R  + + −  + (1) Since g is nonnegative on set D, for small enough (0, ]  one has 1 2 1 1 2 ( ) (1 ) ( ) ( ) ( ) ( ) (1 ) ( ) 0.g x g x g x g x g x     + − − = − + −  That is, 1 2 3 ( ) (1 ) ( ) ( ) m g x g x g x R   + + −  + . (2) Case 2: 1 2 ( ) ( )f x f x . In this case by choosing 3 2 x x= similarly as in case 1 we can prove (1) and (2). Hence the generalized convexity assumption holds. Now assume that assumption (II) holds. Let 1 2 ,x x D and [0,1]  . If h( 2 x ) = 0 then 1 2 1 ( ) (1 ) ( ) ( )h x h x h x  + − = . 13 Int. J. Anal. Appl. (2023), 21:63 Let 3 1 x x= . Then since f, gi are nonnegative, similarly as in (I) one can find a small enough 0 0  and 1 0  such that 1 2 0 3 ( ) (1 ) ( ) ( ) .f x f x f x R   + + −  + (3) 1 2 1 3 ( ) (1 ) ( ) ( ) . m g x g x g x R   + + −  + (4) Otherwise if h( 2 x )  0, then one can find 2 0  such that 1 2 2 2 ( ) (1 ) ( ) ( )h x h x h x  + − = . Let 3 2 x x= . Then since f, gi are nonnegative, similarly as in (I) one can find a small enough 0 0  and 1 0  such that (3) and (4) hold. Hence the generalized convexity assumption holds. Theorem 4.3 (Kuhn-Tucker Type Necessary Optimality Condition) Let x be an optimal solution of (P). Suppose all assumptions in Theorem 4.2 hold and there is no nonzero vector ( , ) m n R R  +   satisfying the system: 1 1( ) min [ ( ) ( )] 0 ( ) 0. m n i i j ji jx D U x i i g x h x g x    = = + = =   where )(xU is a neighbourhood of x , then,  nm RR  +),(  such that 1 1( ) ( ) min [ ( ) ( ) ( )] ( ) 0. m n i i j ji jx D U x i i f x f x g x h x g x    = = = + + =   5. Conclusion Remark Yang, Yang and Chen [1] defined the following generalized subconvexlike functions. ([1] introduced subconvexlikeness for vector-valued optimization). Let Y be a topological vector space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is said to be generalized Y + -subconvexlike on D if intu Y +   , such that 1 2 ,x x D  , 0  , [0,1]  , 3 x D  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y    + + + −  + . (5) And Zeng [2] introduced the presubconvexlikeness as follows. 14 Int. J. Anal. Appl. (2023), 21:63 Let Y be a topological vector space and D X be a nonempty set and Y + be a convex cone in Y. A set-valued map f : D Y→ is said to be Y + -presubconvexlike on D if bounded set-valued map u: D Y→ such that 1 2 ,x x D  , [0,1]  , 3 x D  , 0  , 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x Y    + + + −  + . (6) The inclusions (5) and (6) may be written as 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x   + + − , by the partial order induced by the convex cone Y + . In this paper, we proved that the above two generalized convexities are equivalent. And then, we worked with nonconvex set-valued optimization problems and attained some optimality conditions. Our Fritz John Type Necessary Optimality Condition (Theorem 3.1) and Kuhn-Tucker Type Necessary Optimality Condition (Theorem 4.3) extend the classic results in Clarke [14]. Our Proposition 3.1 are modifications of the alternative theorems in [15, 16]. 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