IHJPAS. 36(1)2023 

355 
 This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

Quasi Semi and Pseudo Semi (𝑝, 𝐸)-Convexity in Non-Linear Optimization 
Programming 

 

 

 

 

 

 

 

 

 
 

 

 

Abstract  

     The class of quasi semi (𝑝, 𝐸)-convex functions and pseudo semi (𝑝, 𝐸)-convex functions are 

presented in this paper by combining the class of 𝑝-convex functions with the class of quasi semi 

𝐸-convex functions and pseudo semi 𝐸-convex functions, respectively. Various non-trivial 

examples are introduced to illustrate the new functions and show their relationships with (𝑝, 𝐸)-

convex functions recently introduced in the literature. Different general properties and 

characteristics of this class of functions are established. In addition, some optimality properties of 

generalized non-linear optimization problems are discussed. In this generalized optimization 

problems, we used, as the objective function, quasi semi (𝑝, 𝐸)-convex (respectively, strictly quasi 

semi (𝑝, 𝐸)-convex functions and pseudo semi (𝑝, 𝐸)-convex functions), and the constraint set is 

(𝑝, 𝐸)-convex set. 

AMS Subject Classification: 46N10, 47N10, 90C48, 90C90, 49K27 

Keywords: (𝑝, 𝐸)-convex set, (𝑝, 𝐸)-convex function, quasi semi (𝑝, 𝐸)-convex function, pseudo 

semi (𝑝, 𝐸)-convex function 

1. Introduction and Preliminaries 

     Generalized convexity has drawn the attention of many researchers in recent years due to its 

vast applications in different areas, especially in optimization and applied sciences (see e.g., [1]-

[16]). One of the well-known generalizations of convex sets and convex functions is the class of 

so-called 𝐸-convex sets and 𝐸-convex functions introduced by [1] where mapping 𝐸: 𝑅𝑛 → 𝑅𝑛 is 

employed in this type of the generalized convexity. Due to some erroneous appeared in Youness's 

first paper, a new class of 𝐸-functions called semi 𝐸-convex functions, is introduced by [2], and 

its properties is studied [3]. This class also includes quasi-semi 𝐸-convex and pseudo semi 𝐸-

doi.org/10.30526/36.1. 2928 

Article history: Received 18 July 2022, Accepted 31 August 2022, Published in January 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Revan Imad Hazim 

Department of Mathematics , College of 

Education for Pure Sciences,Ibn Al –

Haitham/ University of Baghdad- Iraq. 

van.emad1203a@ihcoedu.uobaghdad.edu.iqri 

 

Saba Naser Majeed 

Department of Mathematics , College of 

Education for Pure Sciences,Ibn Al –

Haitham/ University of Baghdad- Iraq. 

saba.n.m@ihcoedu.uobaghdad.edu.iq 

 

 

 
 

https://creativecommons.org/licenses/by/4.0/
mailto:rivan.emad1203a@ihcoedu.uobaghdad.edu.iq
mailto:saba.n.m@ihcoedu.uobaghdad.edu.iq2
mailto:saba.n.m@ihcoedu.uobaghdad.edu.iq2


IHJPAS. 36(1)2023 
 

356 
 

convex functions. Youness motivated other researchers to extend some concepts from convex 

analysis into 𝐸-convexity and apply this concept to optimization problems (see, [4], [5], [6], and 

references therein). Another important recent generalization of convex sets and functions is 𝑝-

convex sets [7] and 𝑝-convex functions [8], affecting the actual number 𝑝 ∈ (0,1]. Very recently, 

[9] presented the class of (𝑝, 𝐸 )-convex sets and (𝑝, 𝐸)-convex functions by combining 𝐸-convex 

sets (respectively, 𝐸-convex functions) with 𝑝-convex sets (respectively, 𝑝-convex functions).  

        Inspired by the above research works and due to the importance of studying non-convex 

functions close to the convex in some sense, the class of quasi semi (𝑝, 𝐸)-convex functions and 

pseudo semi (𝑝, 𝐸)-convex functions is introduced by combining  𝑝-convex functions with quasi 

semi 𝐸-convex and pseudo semi 𝐸-convex functions, respectively.  These non-convex functions 

enrich the study of many real-life problems which are non-convex in nature by modeling them as 

optimization problems that are close to convex problems. The paper is presented as follows. The 

rest of this section contains preliminary material that makes this work self-contained. In section 2, 

the definitions of quasi semi (𝑝, 𝐸)-convex and pseudo semi (𝑝, 𝐸)-convex functions are 

presented, and various examples and relations related to the new functions with (𝑝, 𝐸)-convex 

functions are provided. In section 3, we provide different properties of quasi semi (𝑝, 𝐸)-convex 

and pseudo semi (𝑝, 𝐸)-convex functions. Section 4 is specified to study some optimality 

properties of non-linear optimization problems in which the objective function is quasi semi 

(𝑝, 𝐸)-convex or pseudo semi (𝑝, 𝐸)-convex functions and the constraint set is (𝑝, 𝐸)-convex set. 

     In all the definitions and results throughout this paper, let 𝑝 ∈ (0,1] and 𝑅𝑛 is the 𝑛-dimensional 

Euclidean space. Assume that 𝐴 is a non-empty subset of 𝑅𝑛, 𝑓: 𝐴 ⊆ 𝑅𝑛 ⟶ 𝑅  be a function, and 

𝐸 : 𝑅𝑛  → 𝑅𝑛 is a given mapping. Let us now recall the concepts related to 𝐸-convex set 

(respectively, 𝐸-convex function) and 𝑝-convex set and function. 

Definition 1.1. [1], [7] For any 𝑥, 𝑦 ∈ 𝐴, 𝑟, 𝑠 ∈ [0,1], and 𝑝 ∈ (0,1] such that 𝑟𝑝 + 𝑠𝑝 = 1. The 

set 𝐴 is named as 

1.  𝐸-convex if  𝑟𝐸(𝑥) + (1 − 𝑟)𝐸(𝑦) ∈ 𝐴.  

2. 𝑝-convex if 𝑟𝑥 + 𝑠𝑦 ∈ 𝐴.   

Definition 1.2. [1], [2], [8] For any 𝑥, 𝑦 ∈ 𝐴, 𝑟, 𝑠 ∈ [0,1], and 𝑝 ∈ (0,1] such that 𝑟𝑝 + 𝑠𝑝 = 1. 

The function 𝑓 is named as 

1.  𝐸-𝑐𝑜𝑛𝑣𝑒𝑥 if 𝐴 is 𝐸-convex set and 

 𝑓(𝑟𝐸(𝑥) + (1 − 𝑟)𝐸(𝑦)) ≤ 𝑟𝑓(𝐸(𝑥)) + (1 − 𝑟)𝑓(𝐸(𝑦)).  

2. Quasi semi 𝐸-𝑐𝑜𝑛𝑣𝑒𝑥  if 𝐴 is 𝐸-convex set and 

𝑓(𝑟𝐸(𝑥) + (1 − 𝑟)𝐸(𝑦)) ≤ max {𝑓(𝑥), 𝑓(𝑦)}.  

3. Pseudo semi 𝐸-convex on 𝐸-convex set 𝐴 if there exists a strictly positive function 𝑏: 𝑅𝑛 ×

𝑅𝑛 ⟶ 𝑅 such that if 𝑓(𝑥) < 𝑓(𝑦)  then 

       𝑓(𝑟𝐸(𝑥) + (1 − 𝑟)𝐸(𝑦)) ≤  𝑓(𝑦) + 𝑟(𝑟 − 1)𝑏(𝑥, 𝑦), 

           for   0 < 𝑟 < 1.     

4. 𝑝-convex if 𝐴 is 𝑝-𝑐𝑜𝑛𝑣𝑒𝑥 set and 𝑓 (𝑟𝑥 + s𝑦 ) ≤ 𝑟𝑓(𝑥) + 𝑠𝑓(𝑦).       

      Very recently, Hazim and Majeed [9] have extended the concepts of 𝐸-convexity and 𝑝-

convexity defined above to (𝑝, 𝐸)-convexity as follows. 

Definition 1.3. [9] The set 𝐴 is called (𝑝, 𝐸)-convex set if for all 𝑥, 𝑦 ∈ 𝐴 and for all 𝑟, 𝑠 ∈

[0,1], 𝑝 ∈ (0,1] such that 𝑟𝑝 + 𝑠𝑝 = 1 we have 𝑟𝐸(𝑥) + 𝑠𝐸(𝑦) ∈ 𝐴. 



IHJPAS. 36(1)2023 
 

357 
 

Definition 1.4. [9] For any 𝑥, 𝑦 ∈ 𝐴, 𝑟, 𝑠 ∈ [0,1], and 𝑝 ∈ (0,1] such that 𝑟𝑝 + 𝑠𝑝 = 1. The 

function 𝑓 is named as (𝑝, 𝐸)-𝑐𝑜𝑛𝑣𝑒𝑥 if 𝐴 is (𝑝, 𝐸)-convex and 

𝑓(𝑟𝐸(𝑥) + 𝑠𝐸(𝑦)) ≤ 𝑟𝑓(𝐸(𝑥)) + 𝑠𝑓(𝐸(𝑦)).   

Remark 1.5. From the definition of (𝑝, 𝐸)-convexity, one observes that 

i. In Definition 1.3, if 𝑝 = 1, the definition of 𝐸-convex set is obtained. Also, if 𝐸 = 𝐼 

(identity mapping), then 𝐴 is 𝑝-convex set;  

ii. Likewise, from Definition 1.4, if 𝑝 = 1 we have 𝑓 is 𝐸-convex function and when 𝐸 = 𝐼, 

then the definition of 𝑓 is 𝑝-convex function is obtained.  

 

     For the rest of the paper, the next remark is needed. 

Remark 1.6.  

1. The mapping 𝐸(𝑥) will be written as 𝐸𝑥. 

2. The set 𝐴 is (𝑝, 𝐸) convex set.  

2. Quasi Semi and Pseudo Semi (𝑝, 𝐸)-Convex Functions  

In this section, a new class of functions, which includes quasi semi (𝑝, 𝐸)-convex and pseudo semi 

(𝑝, 𝐸)-convex functions, is introduced. This class generalizes each of quasi semi 𝐸-convex and 

pseudo semi 𝐸-convex functions [2]. Some properties and related examples are established for this 

class.  

 

Definition 2.1.   The function 𝑓 is named as  

i. Quasi semi (𝑝, 𝐸)-convex on 𝐴 if  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ {𝑓(𝑥), 𝑓(𝑦)}  , 

and 𝑓 is strictly quasi semi (𝑝, 𝐸)-convex if   

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) < {𝑓(𝑥), 𝑓(𝑦)}  , 

where 𝑟, 𝑠 ∈ (0,1). 

ii. Pseudo semi (𝑝, 𝐸)-convex if there exist a strictly positive function 𝑏: 𝑅𝑛 × 𝑅𝑛 → 𝑅 such 

that if 𝑓(𝑥) < 𝑓(𝑦) 𝑡ℎ𝑒𝑛 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦)  ≤ 𝑓(𝑦) + (−𝑟𝑠) 𝑏(𝑥, 𝑦),       

for all 𝑟, 𝑠 ∈ (0,1). 

Remark 2.2. In Definition 2.1(i), if 𝑝 = 1 then 𝑓 turned to be quasi semi 𝐸-convex. Likewise, 𝑓 

in Definition 2.1(ii) becomes pseudo semi 𝐸-convex function. 

 

      Quasi semi and pseudo semi (𝑝, 𝐸)-convex functions are not necessarily (𝑝, 𝐸)-convex 

function as the following example shows. 

Example 2.3. Let 𝑓, 𝐸: 𝑅 → 𝑅  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡     𝑓(𝑥) =  {
−3     𝑖𝑓 𝑥 = 0
1      𝑖𝑓 𝑥 ≠ 0

      

                                                         

                                                                  and    𝐸𝑥 =  {
0      𝑖𝑓 𝑥 = 0
4     𝑖𝑓 𝑥 ≠ 0 .

 



IHJPAS. 36(1)2023 
 

358 
 

Let 𝑥, 𝑦 ∈ 𝑅, 𝑝 ∈ (0,1] and 𝑟, 𝑠 ∈ [0,1] 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  𝑟𝑝 +  𝑠𝑝 = 1. First, we show that 𝑓 is quasi 

semi (𝑝, 𝐸)-convex and pseudo semi (𝑝, 𝐸)-convex function. For showing 𝑓 is quasi semi (𝑝, 𝐸)-

convex, we consider three cases: 

Case 1: If 𝑥 = 𝑦 = 0 , we get 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(0) =  −3 = {𝑓(𝑥), 𝑓(𝑦)} . 

Case 2: If 𝑥 ≠ 0, 𝑦 ≠ 0, we get  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(4𝑟 + 4𝑠) =  1 =  {𝑓(𝑥), 𝑓(𝑦)}.  

Case 3: If 𝑥 = 0 , 𝑦 ≠ 0 , we get 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(4𝑠) = {
−3      if 𝑠 = 0
1        if 𝑠 ≠ 0

                        (1) 

         

                                          ≤ 𝑚𝑎𝑥 {𝑓(𝑥), 𝑓(𝑦)} = {−3,1} = 1  . 

In all cases, 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} , and hence 𝑓 is quasi semi (𝑝, 𝐸)-convex. From 

case 3, we get 𝑓(𝑥) = −3 < 1 = 𝑓(𝑦). Thus, from (1), one can choose a strictly positive function 

𝑏(𝑥, 𝑦) such that  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) ≤ 𝑓(𝑦) = 1. 

Thus, 𝑓 is pseudo semi (𝑝, 𝐸)-convex function. Finally, to show that 𝑓 is not (
1

2
, 𝐸)-convex, take 

𝑥 ≠ 0, 𝑦 ≠ 0 𝑎𝑛𝑑 𝑝 =
1

2
  with 𝑠 = 𝑟 =

1

4
. Then, 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 1 > 𝑟𝑓(𝐸𝑥) + 𝑠𝑓(𝐸𝑦) =

𝑟𝑓(4) + 𝑠𝑓(4) = 𝑟 + 𝑠 =
1

2
  as required.  

 

        The next example provides (𝑝, 𝐸)-convex function which is neither quasi semi (𝑝, 𝐸)-convex 

nor pseudo semi (𝑝, 𝐸)-convex.  

 

Example 2.4. Let 𝐴 = [−5, −∞) × [−5, −∞) ⊆  𝑅2 and 𝐸: 𝑅2→ 𝑅2 such that 

 

  𝐸(𝑥1, 𝑥2) = {
((𝑥1 + 1)

2  , (𝑥2 + 1)
2)    if  𝑥1, 𝑥2 < 0

(0,0)                                          o. w.
 

Define 𝑓: 𝑅2 → 𝑅  such that 𝑓(𝑥1, 𝑥2) =  {
𝑥1+𝑥2

3
     if 𝑥1, 𝑥2 < 0

         0                   o. w.    
 

      

 

First, we show that 𝐴 is (𝑝, 𝐸)-convex set. Let 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2)  ∈ 𝐴. If 𝑥1, 𝑥2, 𝑦1, 𝑦2 <

0 then 

 𝑟𝐸(𝑥1, 𝑥2) + 𝑠𝐸(𝑦1, 𝑦2) = ( 𝑟(𝑥1 + 1)
2 + 𝑠(𝑦1 + 1)

2, 𝑟(𝑥2 + 1)
2 + 𝑠(𝑦2 + 1)

2) ∈ [0, +∞) ×

[0, +∞) ⊆ 𝐴. Similarly, if 𝐸𝑥 = 𝐸𝑦 = (0,0) then 𝑟𝐸(𝑥1, 𝑥2) + 𝑠𝐸(𝑦1, 𝑦2) = (0,0) ∈ 𝐴. Thus, 𝐴 

is (𝑝, 𝐸)-convex set. To show that 𝑓 is  (𝑝, 𝐸)-convex function on 𝐴, let 𝑥 = (𝑥1, 𝑥2), 𝑦 =

(𝑦1, 𝑦2) ∈ 𝐴. If 𝑥1, 𝑥2, 𝑦1, 𝑦2 < 0, then  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 0 = 𝑟𝑓(𝐸𝑥) + 𝑠𝑓(𝐸𝑦). 

Similarly, 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑟𝑓(𝐸𝑥) + 𝑠𝑓(𝐸𝑦)     ∀𝑥, 𝑦 ∈ 𝐴. Hence, 𝑓 is  (𝑝, 𝐸)-convex function 

as required. Now, take 𝑥 = (
−1

2
,

−1

2
), 𝑦 = (

−1

4
,

−1

4
), 𝑟 = 𝑠 =

1

4
 and 𝑝 =

1

2
. Then, 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) =

𝑓 (𝑟 (
1

4
,

1

4
) + 𝑠 (

9

16
,

9

16
)) =  0  >  max {−

1

3
 , −  

1

6
}     = −  

1

6
 . 

 



IHJPAS. 36(1)2023 
 

359 
 

Also, 𝑓(𝑥) < 𝑓(𝑦) and  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 0 > 𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) = −
1

6
+ (−

1

16
) 𝑏(𝑥, 𝑦), 

for a strictly positive function 𝑏(𝑥, 𝑦). Hence, 𝑓 is neither quasi semi (
1

2
, 𝐸)-convex nor pseudo 

semi (
1

2
, 𝐸)-convex. 

 

   The relation between pseudo semi (𝑝, 𝐸)- convex function and quasi semi (𝑝, 𝐸)-convex 

functions are given in the next proposition and example. 

 

Proposition 2.5. Every pseudo semi (𝑝, 𝐸)- convex function on 𝐴 is quasi semi (𝑝, 𝐸)- convex. 

Proof. Let 𝑥, 𝑦 ∈ 𝐴 such that 𝑓(𝑥) < 𝑓(𝑦). Since 𝑓 is pseudo semi (𝑝, 𝐸)- convex, then we have 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) ≤ 𝑓(𝑦) = 𝑚𝑎𝑥 {𝑓(𝑥), 𝑓(𝑦)}.  ■ 

 

Example 2.6. Let 𝐴 = { 𝑥 = (𝑥1, … , 𝑥𝑛 ) ∈ 𝑅
𝑛: ∑ |𝑥𝑖 |

1

2
𝑛
𝑖=1 ≤  1 }, and 𝐸: 𝑅

𝑛→ 𝑅𝑛 such that 

𝐸(𝑥) = 𝐸(𝑥1, … , 𝑥𝑛 ) = (
𝑥1

2
, … ,

𝑥𝑛

2
 ) for all 𝑥 ∈ 𝑅𝑛. Define 𝑓: 𝑅𝑛 → 𝑅  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 

 

𝑓(𝑥) =  { 
−1       if 𝑥𝑖 = 0  ∀𝑖 = 1, … , 𝑛
0                           o. w.            

 

From [9, Example 2.4], the set 𝐴 is  (
1

2
, 𝐸)-convex. Next, we show that 𝑓 is quasi semi (

1

2
, 𝐸)-

convex function on 𝐴. To this end, let 𝑥, 𝑦 ∈ 𝐴 and we consider three cases: 

Case 1: If 𝑥𝑖 = 𝑦𝑖 = 0   ∀𝑖 = 1, … , 𝑛, then 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(0, … ,0) =  −1 = {𝑓(𝑥), 𝑓(𝑦)} . 

 

Case 2: If 𝑥𝑖 ≠ 0   ∀𝑖 = 1, … , 𝑛 and  𝑦𝑖 ≠ 0  for some 𝑖 = 1, … , 𝑛. 

 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = {
−1    if 𝑟

𝑥𝑖
2

+ 𝑠
𝑦𝑖
2

= 0  ∀𝑖 = 1, … , 𝑛

0                           o. w.            
 

                           ≤ 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} = 0 . 

 Case 3: If 𝑥𝑖 ≠ 0 and 𝑦𝑖 = 0 for some 𝑖 = 1, … , 𝑛. 

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 0 =  𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} . 

From all cases, we have 𝑓 is quasi semi (
1

2
, 𝐸)-convex function on  𝐴. To show 𝑓 is not pseudo 

semi (
1

2
, 𝐸)-convex function on 𝐴, 𝑡ake 𝑥 = (0, … ,1), 𝑦 = (0, … ,0)  such that 𝑓(𝑥) < 𝑓(𝑦). Let 

𝑟 = 𝑠 =
1

4
  then there exist strictly positive function 𝑏(𝑥, 𝑦) = 3 such that  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(𝑟𝐸(0, … ,1), +𝑠𝐸(0, … ,0)) = 𝑓 (0, … ,
1

2
𝑠)  = 0 

                                                                                                        >  𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) 

                                                                                                     = −1 −
3

16
= −

19

16
 

Hence, 𝑓is not pseudo semi (𝑝, 𝐸)-convex.  

 



IHJPAS. 36(1)2023 
 

360 
 

Proposition 2.7. The function 𝑓 is quasi semi (𝑝, 𝐸)-convex on 𝐴 if and only if the level set 𝐾𝑛 =

{𝑥 ∈ 𝐴:  𝑓(𝑥) ≤ 𝑛} is (𝑝, 𝐸)-convex set for all 𝑛 ∈ 𝑅. 

Proof. Let 𝑓 is quasi semi (𝑝, 𝐸)-convex on (𝑝, 𝐸)-convex set 𝐴. Then, for any 𝑥, 𝑦 ∈ 𝐾𝑛, we have  

𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐴 , 𝑓(𝑥) ≤ 𝑛,  𝑓(𝑦) ≤ 𝑛, and  

𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)}   ≤ 𝑛. It follows that 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐾𝑛. Conversely suppose 

that  𝐾𝑛 is (𝑝, 𝐸)-convex for all 𝑛 ∈ 𝑅. Let 𝑛 = 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} . Since 𝐴 is (𝑝, 𝐸)-convex 

set  then 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐴 and  𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑛 = 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} . Hence, 𝑓 is quasi semi 

(𝑝, 𝐸)-convex on 𝐴. ■ 

 

Proposition 2.8. If 𝑓 is pseudo semi (𝑝, 𝐸)-convex function on 𝐴 then the level set 𝐾𝑛 is (𝑝, 𝐸)-

convex set. 

Proof. let 𝑥, 𝑦 ∈ 𝐾𝑛. We show that   𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐾𝑛. Now, since 𝑓 is pseudo semi (𝑝, 𝐸)-convex. 

Then, we have a strictly positive function 𝑏(𝑥, 𝑦) such that if 𝑓(𝑥) < 𝑓(𝑦) then  𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤

𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) ≤ 𝑓(𝑦). Hence, 𝐾𝛼 is (𝑝, 𝐸)-convex set. ■ 

 

     The converse of the proceeding proposition does not satisfy as it is clarified in the next example. 

Example 2.9. Let 𝑓, 𝐸: 𝑅 → 𝑅 such that  

 

𝑓(𝑥) = {
1            if 𝑥 ∈ [0, ∞)

−1         if 𝑥 ∈ [−∞, 0)
   and    𝐸(𝑥) = { 

𝑥2        if       𝑥 ≥ 0
−𝑥2    if    𝑥 < 0

 

 

Then, for any 𝑛 ∈ 𝑅, the level set  

𝐾𝑛 = {
[−∞, 0)   if      𝑛 ∈ [0,1)

ℝ          if       𝑛 ≥ 1 
   

 

To show that 𝐾𝑛 is (𝑝, 𝐸)-convex set, we consider the following cases:  

 

Case 1: If 𝐾𝑛= 𝑅 (i.e., 𝑛 ≥ 1) then, 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐾𝑛  for all 𝑥, 𝑦 ∈ 𝐾𝑛. Hence, 𝐾𝑛 is (𝑝, 𝐸)-convex 

set. 

Case 2: If 𝐾𝑛= [−∞, 0) for 𝑛 ∈ [0,1). Then, 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓(−𝑟𝑥
2 − 𝑠𝑦2) = −1 ≤ 𝑛  for all  

𝑥, 𝑦 ∈ 𝐾𝑛. Thus, 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐾𝑛 which yields the (𝑝, 𝐸)-convexity of 𝐾𝑛.  

From both cases, we obtain the (𝑝, 𝐸)-convexity of 𝐾𝑛. To confirm that 𝑓 is not pseudo semi 

(𝑝, 𝐸)-convex function. Let 𝑥 = −1, 𝑦 = 1, 𝑟 = 𝑠 =
1

4
, and 𝑝 =

1

2
 then 𝑓(𝑥) < 𝑓(𝑦)  and there 

exists 𝑏(𝑥, 𝑦) = 3 > 0 such that 

 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) = 𝑓 (−
1

4
𝑥2 +

1

4
𝑦2) = 𝑓(0)  = 1 

                                                               

                                                             >  𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦) 

                                                                                      

                                                             = 𝑓(1) + (−
1

16
) (3) =

13

16
 

                                                                                       



IHJPAS. 36(1)2023 
 

361 
 

Hence, 𝑓 is not a pseudo semi (𝑝, 𝐸)-convex function. 

 

3. Some Properties of Quasi Semi and Pseudo Semi (𝑝, 𝐸)-Convex Functions 

     In this section, we discuss some properties of quasi semi (𝑝, 𝐸)-convex and pseudo semi (𝑝, 𝐸)-

convex functions. We start first by showing that the increasing quasi semi (𝑝, 𝐸)-convex functions 

(respectively, strictly increasing pseudo quasi semi (𝑝, 𝐸)-convex) functions defined on 𝐴 ⊆ 𝑅 are 

closed under addition and nonnegative scalar multiplication. 

 

Proposition 3.1. Let 𝑓, 𝑔: 𝐴 ⊆ 𝑅 ⟶ 𝑅  are two increasing quasi semi (𝑝, 𝐸)-convex functions on 

𝐴. Then, 𝛼𝑓 + 𝛽𝑔 is increasing quasi semi (𝑝, 𝐸)-convex function for all 𝛼, 𝛽 ≥ 0  . 

Proof. Let 𝑥, 𝑦 ∈ 𝐴 then either 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥. If 𝑥 ≤ 𝑦 and 𝑓 and 𝑔 are increasing functions, 

then 𝑓(𝑥) ≤ 𝑓(𝑦) and 𝑔(𝑥) ≤ 𝑔(𝑦) which yield  

             𝑚𝑎𝑥 {(𝛼𝑓 + 𝛽𝑔)(𝑥), (𝛼𝑓 + 𝛽𝑔)(𝑦)} = (𝛼𝑓 + 𝛽𝑔)(𝑦)                            (2)                              

Hence, 𝛼𝑓 + 𝛽𝑔 is increasing function. Let  𝑧 = 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐴. Then 

(𝛼𝑓 + 𝛽𝑔 )(𝑧) = 𝛼𝑓(𝑧) + 𝛽𝑔(𝑧) ≤  𝛼 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)}  + 𝛽 𝑚𝑎𝑥{𝑔(𝑥), 𝑔(𝑦)}  

                                                        = 𝛼 𝑓(𝑦) + 𝛽𝑔(𝑦) =  (𝛼𝑓 + 𝛽𝑔)(𝑦) 

                        = 𝑚𝑎𝑥{(𝛼𝑓 + 𝛽𝑔)(𝑥), (𝛼𝑓 + 𝛽𝑔)(𝑦)} , 

where the last conclusion follows from (2). Hence, 𝛼𝑓 + 𝛽𝑔 is increasing quasi semi (𝑝, 𝐸)-

convex function. If  𝑦 ≤ 𝑥 , we proceed similarly to obtain the required conclusion. ■ 

Proposition 3.2. Let 𝑓, 𝑔: 𝐴 ⊆ 𝑅 ⟶ 𝑅 are strictly increasing pseudo semi (𝑝, 𝐸)-convex on 

𝐴.Then, for all 𝛼, 𝛽 ≥ 0, 𝛼𝑓 + 𝛽𝑔 is strictly increasing pseudo semi (𝑝, 𝐸)-convex on 𝐴. 

 

Proof. From the definition of 𝑓 and 𝑔 we have, if 𝑓(𝑥) < 𝑓(𝑦) and 𝑔(𝑥) < 𝑔(𝑦) then then there 

exist 𝑏1, 𝑏2: 𝑅
𝑛 × 𝑅𝑛 → 𝑅 such that  

                            (𝛼𝑓 + 𝛽𝑔)(𝑥) < (𝛼𝑓 + 𝛽𝑔)(𝑦),                             (3)  

Thus, 𝛼𝑓 + 𝛽𝑔 is strictly increasing. Let 𝑧 = 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐴 then      

 𝑓(𝑧) ≤ 𝑓(𝑦) + (−𝑟𝑠) 𝑏1(𝑥, 𝑦)  and 𝑔(𝑧) ≤ 𝑔(𝑦) + (−𝑟𝑠) 𝑏2(𝑥, 𝑦). 

Now, (𝛼𝑓 + 𝛽𝑔)(𝑧) =  𝛼 𝑓( 𝑧) + 𝛽𝑔(𝑧) 

                                  ≤  𝛼 (𝑓(𝑦) + (−𝑟𝑠)𝑏1(𝑥, 𝑦)) +  𝛽(𝑔(𝑦) + (−𝑟𝑠)𝑏2(𝑥, 𝑦)) 

                                   = (𝛼𝑓 + 𝛽𝑔)(𝑦) + (−𝑟𝑠)[𝑏1(𝑥, 𝑦) + 𝑏2(𝑥, 𝑦)] 

                                    = (𝛼𝑓 + 𝛽𝑔)(𝑦) + (−𝑟𝑠) 𝑏(𝑥, 𝑦),                                      (4) 

where 𝑏(𝑥, 𝑦) = 𝑏1(𝑥, 𝑦) + 𝑏2(𝑥, 𝑦). Since 𝑏1 and 𝑏2 are strictly positive functions, then 𝑏(𝑥, 𝑦) 

is strictly positive. From (3) and (4), we obtain the required conclusion. ■   

 

          Next, we show the supremum property of an arbitrary non-empty finite collection of quasi 

semi (𝑝, 𝐸)-convex functions. 

Proposition 3.3. Let 𝑓𝑖 : 𝑅 ⟶ 𝑅 be bounded from above increasing quasi semi (𝑝, 𝐸)-convex  

functions for each 𝑖 ∈ 𝛬 = {1, … , 𝑛}. Define, 𝑓: 𝑅𝑛 ⟶ 𝑅  such that 𝑓 = 𝑠𝑢𝑝𝑖∈𝛬 𝑓𝑖 . Then 𝑓 is quasi 

semi (𝑝, 𝐸)-convex. 

 

Proof. Let 𝑥, 𝑦 ∈ 𝑅 such that 𝑥 ≤ 𝑦 and 𝑓𝑖  is quasi semi (𝑝, 𝐸)-convex for each 𝑖 ∈ 𝛬 = {1, … , 𝑛}. 

Then, 𝑓𝑖 (𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ {𝑓𝑖 (𝑥), 𝑓𝑖 (𝑦)}  = 𝑓𝑖 (𝑦)  for each 𝑖 ∈ 𝛬. Applying the supremum for the 

both sides of the above inequality respectively, we get  

𝑠𝑢𝑝𝑖∈𝛬 𝑓𝑖 (𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑠𝑢𝑝𝑖∈𝛬 {{𝑓𝑖 (𝑥), 𝑓𝑖 (𝑦)} },  



IHJPAS. 36(1)2023 
 

362 
 

                                    𝑓((𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑠𝑢𝑝𝑖∈𝛬 𝑓𝑖 (𝑦) = 𝑓(𝑦) = 𝑚𝑎𝑥 {𝑓(𝑥), 𝑓(𝑦)}.                                                                                                           

 

The last inequalities yields 𝑓 is quasi semi (𝑝, 𝐸)-convex. If  𝑦 ≤ 𝑥 , we proceed similarly to obtain 

the required conclusion      ■ 

  

        Two composite properties are held for quasi semi (respectively, pseudo semi) (𝑝, 𝐸)-convex 

functions as shown next. 

 

Proposition 3.4. Let 𝑓:𝐴 ⊆ 𝑅𝑛 ⟶ 𝑅  and 𝐺: 𝑅 ⟶ 𝑅 is an increasing function. Then 

i. If  𝑓  is  quasi semi (𝑝, 𝐸)-𝑐𝑜𝑛𝑣𝑒𝑥 on 𝐴,  then 𝑔𝑜𝑓 ∶ 𝐴 → 𝑅 is quasi semi (𝑝, 𝐸)-convex 

function. 

ii. If  𝑓  is  pseudo semi (𝑝, 𝐸)-𝑐𝑜𝑛𝑣𝑒𝑥 on 𝐴 and 𝐺 is sublinear and strictly positive, then 𝑔𝑜𝑓 ∶

𝐴 → 𝑅 is pseudo semi (𝑝, 𝐸)-convex function. 

Proof. Let us show (i). Let 𝑥, 𝑦 ∈ 𝐴  and 𝑓 is quasi semi (𝑝, 𝐸)-convex on 𝐴, then 𝑟𝐸𝑥 + 𝑠𝐸𝑦 ∈ 𝐴 

and 𝑓(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)} . Since 𝐺 is an increasing function then, 𝐺(𝑓(𝑟𝐸𝑥 +

𝑠𝐸𝑦) ≤ 𝑚𝑎𝑥{𝑓(𝑥), 𝑓(𝑦)}) . That is,  

(𝐺𝑜𝑓)(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝑚𝑎𝑥{𝐺(𝑓(𝑥)), 𝐺(𝑓(𝑦))}  = 𝑚𝑎𝑥{ (𝐺𝑜𝑓)(𝑥), (𝐺𝑜𝑓)(𝑦} . 

 Hence, 𝐺𝑜𝑓 is quasi semi (𝑝, 𝐸)-convex on 𝐴. For proving (ii), if 𝑓(𝑥) < 𝑓(𝑦) then 𝑓(𝑟𝐸𝑥 +

𝑠𝐸𝑦) ≤ 𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦). Since 𝐺 is an increasing function, then, using the last expression, 

if (𝐺𝑜𝑓)(𝑥) < (𝐺𝑜𝑓)(𝑦) we get  

(𝐺𝑜𝑓)(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ 𝐺[𝑓(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦)]. 

From the assumption, 𝐺 is a sublinear mapping. Thus, the last inequality yields, 

(𝐺𝑜𝑓)(𝑟𝐸𝑥 + 𝑠𝐸𝑦) ≤ (𝐺𝑜𝑓)(𝑦) + (−𝑟𝑠)(𝐺𝑜𝑏)(𝑥, 𝑦). 

Since 𝐺 and 𝑏 are strictly positive functions, then (𝐺𝑜𝑏)(𝑥, 𝑦) is strictly positive. Hence, we obtain 

the required conclusion.  ■ 

 

Proposition 3.5. Let 𝑔: 𝐴 ⊆  𝑅𝑛 → 𝑅𝑛 be a linear mapping such that 𝐸𝑜𝑔 = 𝑔𝑜𝐸. Assume also 

that 𝑓: 𝑉 ⊆  𝑅𝑛 → 𝑅 such that  𝑉 = 𝑔(𝐴). Then 

i. If 𝑓 quasi semi (𝑝, 𝐸)-convex function, then 𝑓𝑜𝑔: 𝐴 → 𝑅 is quasi semi (𝑝, 𝐸)-convex 

function. 

ii. If 𝑓 pseudo semi (𝑝, 𝐸)-convex function, then 𝑓𝑜𝑔: 𝐴 → 𝑅 is pseudo semi (𝑝, 𝐸)-convex 

function. 

Proof. Let 𝑥, 𝑦 ∈ 𝐴 then 𝑟𝐸𝑥 +  𝑠 𝐸𝑦 ∈ 𝐴. For proving (i), we need to show that  

(𝑓𝑜𝑔)(𝑟𝐸𝑥 +  𝑠 𝐸𝑦) ≤ 𝑚𝑎𝑥{𝑓(𝑔(𝑥)) , 𝑓(𝑔(𝑦))} .  

Now, from the linearity of 𝑔 and the fact that 𝐸𝑜𝑔 = 𝑔𝑜𝐸, we have 

         (𝑓𝑜𝑔)(𝑟𝐸𝑥 +  𝑠 𝐸𝑦) = 𝑓(𝑟 𝑔(𝐸𝑥) + 𝑠 𝑔(𝐸𝑦))  

                                            = 𝑓(𝑟 (𝑔𝑜𝐸)(𝑥) + 𝑠(𝑔𝑜𝐸)(𝑦)) 

                                            = 𝑓(𝑟(𝐸𝑜𝑔)(𝑥) + 𝑠(𝐸𝑜𝑔)(𝑦)) 

                                            = 𝑓 (𝑟 (𝐸(𝑔(𝑥))) + 𝑠 (𝐸(𝑔(𝑦)))).                         (5) 

Note that, since 𝑥, 𝑦 ∈ 𝐴 then 𝑔(𝑥), 𝑔(𝑦) ∈ 𝑉 = 𝑔(𝐴).  Also, we have 𝑉 = 𝑔(𝐴) is (𝑝, 𝐸)-

convex set (see [9, Proposition 2.11]) thus  

                                      𝑟 (𝐸(𝑔(𝑥))) + 𝑠 (𝐸(𝑔(𝑦))) ∈ 𝑉.                                (6)                                



IHJPAS. 36(1)2023 
 

363 
 

From (5)-(6) and the fact that 𝑓 is quasi semi (𝑝, 𝐸)-convex function, we get  

(𝑓𝑜𝑔)(𝑟𝐸𝑥 +  𝑠 𝐸𝑦)  ≤ 𝑚𝑎𝑥{𝑓(𝑔(𝑥)), 𝑓(𝑔(𝑦))} = 𝑚𝑎𝑥{(𝑓𝑜𝑔)(𝑥), (𝑓𝑜𝑔)(𝑦)}  . 

Hence, 𝑓𝑜𝑔 is quasi semi (𝑝, 𝐸)-convex. Let us prove (ii), from the assumption 𝑥, 𝑦 ∈ 𝐴, 

𝑔(𝑥), 𝑔(𝑦) ∈ 𝑉 = 𝑔(𝐴), and 𝑓 is pseudo semi (𝑝, 𝐸)-convex function. Hence,  

          𝑓(𝑔(𝑥)) < 𝑓(𝑔(𝑦))    (7)                                                                                                            

Again, we follow the steps of the proof of (i) to obtain the equality (5). Namely, 

(𝑓𝑜𝑔)(𝑟𝐸𝑥 +  𝑠 𝐸𝑦) =  𝑓 (𝑟 (𝐸(𝑔(𝑥))) + 𝑠 (𝐸(𝑔(𝑦)))) 

From definition of 𝑓 there exist strictly positive 𝑏: 𝑅𝑛 × 𝑅𝑛  → 𝑅 such that the right-hand side 

expression above yields  

                                      ≤ 𝑓(𝑔(𝑦)) + (−𝑟𝑠) 𝑏(𝑥, 𝑦) = (𝑓𝑜𝑔)(𝑦) + (−𝑟𝑠)𝑏(𝑥, 𝑦)           (8) 

Thus, from (7) and (8),  𝑓𝑜𝑔 is pseudo semi (𝑝, 𝐸)-convex function.  ■ 

        

 

4. Applications to Non-Linear Optimization Programming                                                     

     In this section, a non-linear optimization programming problem denoted by (𝑁𝐿𝑃) and is 

defined as 

𝑓(𝑥)  

                                                              subject to 𝑥 ∈ 𝐴, 

where 𝐴 is (𝑝, 𝐸)-convex. The set of all optimal solutions (or global minimum) of problem (𝑁𝐿𝑃)  

is defined as 𝑎𝑟𝑔𝑚𝑖𝑛𝐴 𝑓= {𝑥
∗ ∈ 𝐴: 𝑓(𝑥 ∗) ≤ 𝑓(𝑥) for all ∈ 𝐴 }. A point 𝑥 ∗ is called a local 

minimizer for problem (𝑁𝐿𝑃) if there exists 𝛿 > 0 such that 𝑓(𝑥∗) ≤ 𝑓(𝑥) for all 𝑥 ∈ 𝐵(𝑥∗, 𝛿) ∩

𝐴 where 𝐵 𝛿 (𝑥
∗) is an open ball. 

 

    The following optimality properties are satisfied under different conditions for the objective 

function 𝑓 and the mapping 𝐸. 

 

Proposition 4.1. Let 𝑓 is pseudo semi (𝑝, 𝐸)-convex function. Then, every local minimum  𝑥 ∗ =

𝐸(𝑥 ∗) ∈ 𝐸(𝐴) of problem (𝑁𝐿𝑃) is a global minimum. 

Proof. Suppose 𝑥∗ = 𝐸(𝑥 ∗) is not global minimum, then there exists 𝑢 ∈ 𝐴 with 𝑓(𝑢) < 𝑓(𝑥∗) =

𝑓(𝐸𝑥 ∗).  From the assumptions on 𝑓, we have for all 𝑟, 𝑠 ∈ [0,1] with  𝑟𝑝 + 𝑠𝑝 =

1 (𝑖. 𝑒. , 𝑠 = (1 − 𝑟𝑝)
1

𝑝 ), if 𝑓(𝑢) < 𝑓(𝑥 ∗).  then , we have 𝑓(𝑟𝐸𝑢 + 𝑠𝐸𝑥 ∗) ≤ 𝑓(𝑥∗) +

(−𝑟𝑠)𝑏(𝑥∗, 𝑢). Now, if 𝑓(𝑥∗) ≤ 0 or 𝑓(𝑥∗) ≥ 0. then we have  

         𝑓(𝑟𝐸𝑢 + 𝑠𝐸𝑥 ∗) ≤ 𝑓(𝑥∗) + (−𝑟𝑠)𝑏(𝑥∗, 𝑢) ≤ 𝑓(𝑥 ∗).                  (9) 

Now, for sufficiently small, 𝑟 ∈ (0,1]) then 𝑟𝐸𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗ will be close enough to 𝑥 ∗. i.e., 

there exists 𝛿 > 0 such that 𝑟𝐸𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗ ∈ 𝐵 𝛿 (𝑥
∗) ∩ 𝐴. From the local minimality of 𝑥 ∗, 

one obtains 𝑓(𝑥∗) ≤ 𝑓(𝑟𝐸𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗) which contradicts (9). Thus, 𝑥 ∗ is a global minimum.

  ■ 

  

Proposition 4.2. let 𝑓 is quasi semi (𝑝, 𝐸)-convex function. Then every local minimum  𝑥∗ =

𝐸(𝑥 ∗) ∈ 𝐸(𝐴) of problem (𝑁𝐿𝑃) is a global minimum. 

 



IHJPAS. 36(1)2023 
 

364 
 

Proof. Suppose 𝑥∗ = 𝐸(𝑥 ∗) is not global minimum, then there exists 𝑢 ∈ 𝐴 with 𝑓(𝑢) < 𝑓(𝑥∗) =

𝑓(𝐸𝑥 ∗).  From the assumptions on 𝑓, we have for all 𝑟, 𝑠 ∈ [0,1] with  𝑟𝑝 + 𝑠𝑝 =

1 (𝑖. 𝑒. , 𝑠 = (1 − 𝑟𝑝)
1

𝑝 ). Since 𝑓(𝑢) < 𝑓(𝑥∗) and 𝑓 is quasi semi (𝑝, 𝐸)-convex then we have  

 

𝑓(𝑟𝐸𝑢 + 𝑠𝐸𝑥 ∗) ≤ 𝑚𝑎𝑥{𝑓(𝑢), 𝑓(𝑥∗)}  =  𝑓(𝑥∗)                                                   (10) 

Now, for sufficiently small, 𝑟 ∈ (0,1]) then 𝑟𝐸𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗ will be close enough to 𝑥 ∗. i.e., 

there exists 𝛿 > 0 such that 𝑟 𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗ ∈ 𝐵 𝛿 (𝑥
∗) ∩ 𝐴.  From the local minimality of 𝑥 ∗, 

we have  𝑓(𝑥∗) ≤ 𝑓(𝑟𝐸𝑢 + (1 − 𝑟𝑝)
1

𝑝𝑥∗) which contradicts (10). Thus, 𝑥 ∗ is a global minimum.     

■ 

 

Remark 4.3. The conclusions of Propositions 4.1 and 4.2 do not hold if the objective function 𝑓 

is not quasi semi (𝑝, 𝐸)-convex (respectively, not pseudo semi (𝑝, 𝐸)-convex) function as the 

following example confirms. 

 

Example 4.4. Consider the optimization problem 

 

𝑓(𝑥, 𝑦)  

such that (𝑥, 𝑦) ∈ 𝐴, 

where 𝐴 = {(𝑥, 𝑦) ∈ 𝑅2 ∶  |𝑥|
1

2 +  |𝑦|
1

2  ≤ 4} and 𝑓: 𝑅2 → 𝑅 such that  

 

𝑓(𝑥, 𝑦) = {
(𝑦 − 1)2                                  − 2 ≤ 𝑥 ≤ 2

(𝑦 − 1)2 − (2 − 𝑥)2                       o. w.     
 

Define 𝐸: 𝑅2 →  𝑅2 as 𝐸(𝑥. 𝑦) = (0, 𝑦). Then 𝐴 is (
1

2
, 𝐸)-convex set, 𝑓 is not quasi semi (

1

2
, 𝐸)-

 convex and not pseudo semi (
1

2
, 𝐸)- convex on 𝐴. To show 𝐴 is (

1

2
, 𝐸)- convex set, let 

(𝑥1, 𝑦1),(𝑥2, 𝑦2) ∈ 𝐴. Then, |𝑥1|
1

2 + |𝑦1|
1

2 ≤ 4 and |𝑥2|
1

2 +  |𝑦2|
1

2  ≤ 4. Now,  

𝑟𝐸(𝑥1, 𝑦1) + 𝑠𝐸(𝑥2, 𝑦2) = (0, 𝑟𝑦2 + 𝑠𝑦2). Note that  

|0|
1

2 +  |𝑟𝑦1 + 𝑠𝑦2|
1

2 ≤ 𝑟|𝑦1|
1

2 + s|𝑦2|
1

2 ≤ 4(𝑟 + 𝑠) ≤  4. Hence, 𝐴 is (
1

2
, 𝐸)-convex set. Next, we 

show that 𝑓 is not quasi semi (
1

2
, 𝐸)- convex on 𝐴. Let (2,1), (−2,1) ∈ 𝐴, and 𝑟 = 𝑠 =

1

4
 , 𝑝 =

1

2
. 

Then  

𝑓 (
1

4
 𝐸(2,1) +

1

4
 𝐸(−2,1)) = 𝑓 (0,

1

2
) = 

1

4
> 𝑚𝑎𝑥{𝑓(2,1), 𝑓(−2,1)}  = 0. 

Also, 𝑓 is not pseudo semi (
1

2
, 𝐸)- convex on 𝐴. To this end, let 𝑥 = (2,

1

2
) , 𝑦 = (2,1) ∈ 𝐴 and 

𝑟 = 𝑠 =
1

4
 , 𝑝 =

1

2
 such that 𝑓(𝑥) =  

1

4
< 𝑓(𝑦) = 0. Then,  

𝑓 (
1

4
 𝐸 (2,

1

2
) +

1

4
 𝐸(2,1)) = (

1

8
+

1

4
− 1)

2

=
25

64
> 𝑓(2,1) −

1

16
𝑏(𝑥, 𝑦)   

                                                                             = −
1

16
𝑏(𝑥, 𝑦), 

 for any strictly positive function 𝑏(𝑥, 𝑦).  Thus, 𝑓 is not pseudo semi (
1

2
, 𝐸)- convex on 𝐴 as 

claimed. Now, take 𝑥0 = (0,1) ∈ 𝐴 such that 𝐸(0,1) = (0,1) and 𝑓(0,1) = 0 ≤ 𝑓(𝑥, 𝑦) for all 



IHJPAS. 36(1)2023 
 

365 
 

(𝑥, 𝑦) ∈ 𝐴 ∩ [−2,2] × 𝑅. Hence, 𝑥0 = (0,1) is a local minimum. However, the conclusion of 

Propositions 4.1 and 4.2 does not satisfy, i.e., 𝑥0 is not a global minimum. Indeed, take = (3,1) ∈

𝐴 . then  

𝑓(3,1) = (1 − 1)2 − (2 − 3)2 = −1 < 𝑓(𝑥0) = 0. 

 

Proposition 4.5. Let 𝑓 is strictly quasi semi (𝑝, 𝐸)- convex on 𝐴. Then, the global minimum of 

problem (𝑁𝐿𝑃)  is singleton.  

Proof. Let 𝑥 ∗, 𝑦∗ be two different global minima of  (𝑁𝐿𝑃) then 𝑓(𝑥∗) = 𝑓(𝑦∗) ≤ 𝑓(𝑥) for any 

𝑥 ∈ 𝐴.  From the assumptions on 𝑓 and 𝐴, we have 𝑟𝐸𝑥∗ + 𝑠𝐸𝑦∗ ∈ 𝐴 and 

𝑓(𝑟𝐸𝑥 ∗ + 𝑠𝐸𝑦∗) < 𝑚𝑎𝑥{𝑓(𝑥∗), 𝑓(𝑦∗)}  = 𝑓(𝑥∗). 

The above inequality yields that 𝑟𝐸𝑥 ∗ + 𝑠𝐸𝑦∗ is a global minimum which is a contradiction. 

Hence, there is a unique global minimum.  ■   

 

Proposition 4.6. Let 𝑓 is quasi semi (𝑝, 𝐸)- convex on 𝐴. Then, the set of global minima of 

problem (𝑁𝐿𝑃) is (𝑝, 𝐸)- convex. 

Proof. Let 𝑥1
∗, 𝑥2

∗ ∈ 𝑎𝑟𝑔𝑚𝑖𝑛𝐴𝑓 = {𝑥
∗ ∈ 𝐴: 𝑓(𝑥∗) ≤ 𝑓(𝑥)    ∀𝑥 ∈ 𝐴} the set of global minima of 

problem (𝑁𝐿𝑃) we must prove 𝑟𝐸𝑥1
∗ + 𝑠𝐸𝑥2

∗ ∈ 𝑎𝑟𝑔𝑚𝑖𝑛𝐴𝑓. since 𝑓 is quasi semi (𝑝, 𝐸)-convex 

on the (𝑝, 𝐸)-convex set 𝐴, we have 𝑟𝐸𝑥1
∗ + 𝑠𝐸𝑥2

∗ ∈ 𝐴 and for each 𝑥 ∈ 𝐴, 

𝑓(𝑟𝐸𝑥1
∗ + 𝑠𝐸𝑥2

∗) ≤ 𝑚𝑎𝑥{𝑓(𝑥1
∗), 𝑓(𝑥2

∗)}  ≤ 𝑓(𝑥). Therefore, 𝑟𝐸𝑥1
∗ + 𝑠𝐸𝑥2

∗ ∈ 𝑎𝑟𝑔𝑚𝑖𝑛𝐴𝑓 as 

required.  ■ 

5.Conclusion 

     In this paper, new generalized convex functions (quasi semi (𝑝, 𝐸)-convex, and pseudo semi 

(𝑝, 𝐸)-convex functions) are defined, and their various general and optimality properties are 

studied. These functions are a combination of 𝑝-convex and 𝐸-convex functions introduced in the 

literature. Different examples are established to illustrate these functions and to confirm some 

properties proved throughout the work.  

 

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