Convex optimization, which studies the problem of minimizing convex functions over convex sets, plays important roles in many branches of applied mathematics. The foremost reason is that; it is very suitable to extremum problems. For instance, some necessary conditions for the existence of a minimum also become sufficient in the in terms of convexity. And convex optimization can be a smooth or a non-smooth convex optimization. Since the concept of convexity does not satisfy some mathematical models, various generalizations of convexity such as quasi-convexity and pseudo-convexity, which retain some important properties of convexity and equally provide a better representation of reality, were introduced in the literature to fill these gaps.

While the quasi-convexity property of a function guarantees the convexity of their sublevel sets, the pseudo-convexity property implies that the critical points are minimizers [1] . One of the features of convexity of functions is the relationship it has with the monotonicity of some maps. For example, a differentiable function is said to be convex if and only if its gradient is a monotone map. In non-smooth analysis, the generalized convexity of functions can be equally characterized in terms of the generalized monotonicity of their related operators [2] .

The concepts of pseudo-convexity, traced to [3] , within his research on analytical functions and independently introduced into the field of optimization by [4] , have many applications in mathematical programming and economic problems [5] [6] [7] . And pseudo-monotonicity, introduced by [8] as a generalization of monotone operators, has been used to describe a property of consumer’s demand correspondence [9] . Although the simplest class of pseudo-monotone operators consists of gradients of pseudo-convex functions, there are some monotone operators that are not sub-differentials [9] . And generalized monotonicity of maps is frequently used in complementarity problems, equilibrium problems and variational inequalities [10] .

Let X be a real Banach space with norm ‖   .   ‖ , X * be its topological dual and 〈 x * , x 〉 be the duality pairing between x ∈ X and x * ∈ X * . We denote the closed segment [ x , y ] = { λ x + ( 1 − λ ) y : λ ∈ [ 0 , 1 ] } for x , y ∈ X , and define ( x , y ] , [ x , y ) and ( x , y ) similarly.

Definition 2.1 [11] Let f : X → ℝ ∪ { + ∞ } be an extended real valued function, the effective domain is defined by

dom ( f ) = { x ∈ X : f ( x ) < + ∞ } .

Definition 2.2 [12] A function f : X → ℝ ∪ { + ∞ } is said to be lower semi-continuous at x ∈ X if and only if: ∀ λ ∈ ℝ , such that λ < f ( x ) , ∃ V ⊂ U ( x ) : λ < f ( y ) ∀ y ∈ V .

Definition 2.3 [7] [13] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be quasi-convex, if for any x , y ∈ X and z ∈ [ x , y ] we have

f ( z ) ≤ max { f ( x ) , f ( y ) } . (1)

Definition 2.4 [7] [13] [14] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be strictly quasi-convex, if the inequality (1) is strict when x ≠ y .

Definition 2.5 [2] Let T : X → X * be a multivalued operator with domain D ( T ) = { x ∈ X : T ( x ) ≠ ∅ } . T is said to be quasi-monotone if for any x , y ∈ X , x * ∈ T and y * ∈ T ( y ) , we have

〈 x * , y − x 〉 > 0 ⇒ 〈 y * , y − x 〉 ≥ 0.

Definition 2.6 [2] Let T : X → X * be a multivalued operator with domain D ( T ) = { x ∈ X : T ( x ) ≠ ∅ } . T is said to be pseudo-monotone if for any x , y ∈ X , x * ∈ T and y * ∈ T ( y ) , we have

〈 x * , y − x 〉 ≥ 0 ⇒ 〈 y * , y − x 〉 ≥ 0. (2)

Definition 2.7 [7] Let T : X → X * be a multivalued operator with domain D ( T ) = { x ∈ X : T ( x ) ≠ ∅ } . T is said to be strictly pseudo-monotone if for any different two points x , y ∈ X , x * ∈ T and y * ∈ T ( y ) , we have

〈 x * , y − x 〉 ≥ 0 ⇒ 〈 y * , y − x 〉 > 0. (3)

Definition 2.8 [15] An operator ∂ that associates to any lower semi-continuous function f : X → ℝ ∪ { + ∞ } and a point x ∈ X a subset ∂ f ( x ) of X * is a sub-differential if it satisfies the following properties:

1) ∂ f ( x ) = { x * ∈ X * : 〈 x * , y − x 〉 + f ( x ) ≤ f ( y ) , ∀ y ∈ X } , whenever f is convex;

2) 0 ∈ ∂ f ( x ) , whenever x ∈ dom   f is a local minimum of f;

3) ∂ ( f + g ) ( x ) ⊂ ∂ f ( x ) + ∂ g ( x ) , whenever g is a real a real-valued convex continuous function which is ∂ -differentiable at x.

Where g-differentiable at x means that both ∂ g ( x ) and ∂ ( − g ) ( x ) are non-empty. We say that f is ∂ -differentiable at x when ∂ f ( x ) is non-empty while ∂ f ( x ) are called the sub-gradients of f at x.

Definition 2.9 [15] The Clarke-Rockafellar generalized directional derivative of f at x 0 ∈ dom ( f ) in the direction d ∈ X is given by

f ↑ ( x 0 , d ) = sup ε > 0 lim sup x → f x 0 λ ↘ 0 inf d ′ ∈ B ε ( d ) f ( x + λ d ′ ) − f ( x ) λ , (4)

where B ε ( d ) = { d ′ ∈ X : ‖ d ′ − d ‖ < ε } , λ ↘ 0 indicates the fact that λ > 0 and λ → 0 , and x → f x 0 means that both x → x 0 and f ( x ) → f ( x 0 ) ;

While,

Definition 2.10 [15] The Clarke-Rockafellar sub-differential of f at x 0 is defined by

∂ f ( x 0 ) = { x * ∈ X * : ( x * , d ) ≤ f ↑ ( x 0 , d ) , ∀ d ∈ X } ; (5)

if x 0 ∈ X \ dom ( f ) , then

∂ f ( x 0 ) = ∅ , [7] .

Definition 2.11 [2] [7] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be quasi-convex (with respect to Clarke-Rockerfeller Sub-differentials) if for any x , y ∈ X ,

∃ x * ∈ ∂ f ( x ) : 〈 x * , y − x 〉 > 0 ⇒ ∀ z ∈ [ x , y ] , f ( z ) ≤ f ( y ) . (6)

Definition 2.12 [7] [16] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be pseudo-convex (with respect to Clarke-Rockerfeller Subdifferentials) if for any x , y ∈ X :

∃ x * ∈ ∂ f ( x ) : 〈 x * , y − x 〉 ≥ 0 ⇒ f ( x ) ≤ f ( y ) . (7)

Definition 2.13 [2] [7] [14] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be strictly pseudo-convex (with respect to Clarke-Rockerfeller Subdifferentials) if for any two different points x , y ∈ X :

∃ x * ∈ ∂ f ( x ) : 〈 x * , y − x 〉 ≥ 0 ⇒ f ( x ) < f ( y ) , when x ≠ y . (8)

Definition 2.14 [7] A lower semi-continuous function f : X → ℝ ∪ { + ∞ } is said to be radially continuous if for all x , y ∈ X , f is continuous on [ x , y ] .

Definition 2.15 [7] A function f : X → ℝ ∪ { + ∞ } is said to be radially non-constant if for all x , y ∈ X , with x ≠ y , f ≡ constant on [ x , y ] .

Definition 2.16 A sub-differential operator ∂ f : X → X * is said to said to be quasi-monotone if for any x , y ∈ X , x * ∈ ∂ f ( x ) and y * ∈ ∂ f ( y ) , we have

〈 x * , y − x 〉 > 0 ⇒ 〈 y * , y − x 〉 ≥ 0. (9)

Definition 2.17 A sub-differential operator ∂ f : X → X * is said to said to be quasi-monotone if for any x , y ∈ X , x * ∈ ∂ f ( x ) and y * ∈ ∂ f ( y ) , we have

〈 x * , y − x 〉 ≥ 0 ⇒ 〈 y * , y − x 〉 ≥ 0. (10)

Theorem 2.1. (Approximate mean value inequality). Let f : X → ℝ ∪ { + ∞ } be a Clarke-Rockafellar sub-differentiable lower semi-continuous (l.s.c.) function on a Banach space X. Let a , b ∈ X with a ∈ dom   f and a ≠ b . Let ρ ∈ ℝ be such that ρ ≤ f ( b ) . Then, there exist c ∈ [ a , b ) and x n → f C and x n * ∈ ∂ f ( x n ) such that

1) lim inf n → + ∞ 〈 x n * , c − x n 〉 ≥ 0 ;

2) lim inf n → + ∞ 〈 x n * , b − a 〉 ≥ ρ − f ( a ) .

Proof. [15] .

Lemma 2.2. Let f : X → ℝ ∪ { + ∞ } be a Clarke-Rockafeller sub-differentiable lower semi-continuous (l.s.c.) function on a Banach space X. Let a , b ∈ X with f ( a ) < f ( b ) . Then, there exist c ∈ [ a , b ) , and two sequences c n → c , and c n * ∈ ∂ f ( c n ) with

〈 c n * , x − c n 〉 > 0 for every x = c + λ ( b − a ) with λ > 0 .

Proof. By Theorem 2.1, there exists an x 0 ∈ [ a , b ) and a sequence x n → f C and x n * ∈ ∂ f ( x n ) verifying

lim inf n → + ∞ 〈 x n * , c − x n 〉 ≥ 0 and lim inf n → + ∞ 〈 x n * , b − a 〉 > 0 . (11)

Putting x = c + λ ( b − a ) with λ > 0 it holds

〈 x n * , x − x n 〉 = 〈 x n * , c − x n 〉 + λ 〈 x n * , b − a 〉 > 0 (12)

for n very large. ■

We consider the relationship between pseudo-convexity and quasi-convexity.

Theorem 2.3. Let f : X → ℝ ∪ { + ∞ } be a lower semi-continuous (l.s.c.) Clarke-Rockafeller subdifferentiable function on a Banach space X. Then, f is quasi-convex if and only if ∂ f is quasi-monotone.

Proof. We show that if f is not quasi-convex, then ∂ f is not quasi-monotone.

Suppose that there exist some x , y , z in X with z ∈ [ x , y ] and f ( z ) > max { f ( x ) , f ( y ) } . According to Lemma 2.2 applied with a = x and b = z , there exists a sequence y n ∈ dom   ∂ f and y n * ∈ ∂ f ( y n ) such that

y n → y ¯ ∈ [ x , z ] , y ¯ ≠ z and 〈 y n * , y − y n 〉 > 0 . (13)

Let 0 < λ ≤ 1 be such that z = y ¯ + λ ( y − y ¯ ) and set z n = y n + λ ( y − y n ) , so that z n → z . Since f is lower semi-continuous, we may pick n ∈ ℕ very large with f ( z n ) > f ( y ) . Apply Lemma 2.2 again with a = y and b = z n to find sequences x k ∈ dom   ∂ f , x k * ∈ ∂ f ( x k ) such that

x k → x ¯ ∈ [ y , z n ] , x ¯ ≠ z n and 〈 x k * , y n − x k 〉 > 0 . (14)

In particular, x ¯ ≠ y n and

〈 y n * , x ¯ − y n 〉 = ‖ x ¯ − y n ‖ ‖ y − y n ‖ 〈 y n * , y − y n 〉 > 0 ; (15)

hence, 〈 y n * , x k − y n 〉 > 0 for k sufficiently large. But 〈 y n * , y n − x k 〉 > 0 , showing that ∂ f is not quasi-monotone.

Conversely, we suppose that f is quasi-convex and show that ∂ f is quasi-monotone. Let x * ∈ ∂ f ( x ) and y * ∈ ∂ f ( y ) with 〈 x * , y − x 〉 > 0 . We need to verify that f ↑ ( y , x − y ) ≤ 0 . We fix ε > 0 and ω ∈ ( 0 , ε ) such that 〈 x * , v − x 〉 > 0 for all v ∈ B ω ( y ) .

We fix v ∈ B ω ( y ) . Since f ↑ ( y , x − y ) > 0 we can find ε ′ ∈ ( 0 , ε − ω ) , u ∈ B ε ′ ( x ) and t ∈ ( 0 , 1 ) such that f ( u + t ( v − u ) ) > f ( u ) . From the quasi-convexity of f we deduce that f ( u ) < f ( v ) , whence,

f ( v + λ ( u − v ) ) ≤ f ( v ) for all λ ∈ ( 0 , 1 ) ,

so that

inf μ ∈ B ε ( x − y ) f ( v + λ μ ) − f ( v ) λ ≤ f ( v + λ ( u − v ) ) − f ( v ) λ ≤ 0 for all λ ∈ ( 0 , 1 ) .

Combining the inequalities and for any ε > 0 there exists ω > 0 such that

sup v ∈ B ω ( y ) λ ∈ ( 0 , 1 ) [ inf μ ∈ B ε ( x − y ) f ( v + λ μ ) − f ( v ) λ ] ≤ 0 ,

which shows that f ↑ ( y , x − y ) ≤ 0 . ■

Theorem 3.1. Let f : X → ℝ ∪ { + ∞ } be a lower semi-continuous (l.s.c.) function on a Banach space X such that f Clarke-Rockafeller suddifferentiable. Consider the following assertions:

(i) f is pseudoconvex.

(ii) f is quasiconvex and ( 0 ∈ ∂ f ( x ) ⇒ x is a global minimum of f).

Then, (i) implies (ii). And (ii) implies (i) if f is radially continuous.

Proof. (i) ⇒ (ii). We want to prove that f is quasiconvex. Suppose to the contrary that for some x , y ∈ X , z ∈ ( x , y ) we have f ( z ) > max { f ( x ) , f ( y ) } . Since f is lower semicontinuous, we can find some ε > 0 such that f ( z ′ ) > max { f ( x ) , f ( y ) } , for all z ′ ∈ B ε ( z ) . Since z cannot be a local nor global minimizers, there exist some v ∈ B ε ( z ) such that f ( v ) < f ( z ) . From Lemma 2.2, there exist u n → u ∈ [ v , z ) and u n * ∈ ∂ f ( u n * ) such that

〈 u n * , z − u n 〉 > 0 .

But since z ∈ ( x , y ) , either of the following must hold

〈 u n * , x − u n 〉 > 0 or 〈 u n * , y − u n 〉 > 0 .

Therefore,

f ( u n ) ≤ max { f ( x ) , f ( y ) } .

which is a contradiction.

(ii) ⇒ (i). Let x ∈ dom   ∂ f , y ∈ X , and x * ∈ ∂ f ( x ) such that 〈 x * , y − x 〉 ≥ 0 . If 0 ∈ ∂ f ( x ) , then x is a global minimum of f and f ( x ) ≤ f ( y ) in particular. Otherwise, [ 0 ∉ ∂ f ( x ) ] , there exist d ∈ X such that 〈 x * , d 〉 > 0 . We define a sequence { y n } by

y n = y + ( 1 2 n ‖ d ‖ ) d .

For every n ∈ ℕ , the point y n satisfies

y n ∈ B 1 / n ( y ) ,

〈 x * , y n − x 〉 = 〈 x * , y n − y 〉 + 〈 x * , y − x 〉 ≥ ( 1 2 n ‖ d ‖ ) 〈 x * , d 〉 > 0.

Using (7), we obtain that, for every n, f ( y n ) ≥ f ( x ) and by radial continuity of f, f ( y ) ≥ f ( x ) . ■

Theorem 3.2. Let f : X → ℝ ∪ { + ∞ } be a lower semi-continuous (l.s.c.) Clarke-Rockafeller sub-differentiable function. Consider the following assertions:

(i) f is pseudo-convex.

(ii) ∂ f is pseudo-monotone

Then, (i) implies (ii). And (ii) implies (i) if f is radially continuous.

Proof. (i) ⇒ (ii). Suppose x * ∈ ∂ f ( x ) such that 〈 x * , y − x 〉 ≥ 0 . By Theorem 3.1, f is quasi-convex. By Theorem 2.3, we conclude that ∂ f is quasi-monotone. Hence, 〈 y * , y − x 〉 ≥ 0 , for all y * ∈ ∂ f ( y ) . Suppose to the contrary that for some y * ∈ ∂ f ( y ) , we have 〈 y * , y − x 〉 = 0 . From (7), we obtain f ( x ) ≥ f ( y ) .

However, since f ↑ ( x , y − x ) > 0 , there exist ε > 0 , such that for some x n → x , λ n ↘ 0 and for all y ′ ∈ B ε ( y ) , we have f ( x n + t n ( y ′ − x n ) ) > f ( x n ) . By the quasiconvexity of f, it implies that f ( y ′ ) > f ( x n ) for every y ′ ∈ B ε ( y ) . In particular, f ( y ) > f ( x ) because f is lower semicontinuous. Thus, f ( y ′ ) ≥ f ( y ) . This shows that y is a local minimum and also a global minimum, which is a contradiction since we can have that f ( y ) > f ( x n ) .

(ii) ⇒ (i). Using Theorem, we prove that f is pseudoconvex. Since ∂ f is pseudomonotone, ∂ f is quasimonotone. By Theorem 3.1, f is quasi-convex. On the other hand, if x is not a minimizer of f, there exists y ∈ X such that f ( y ) < f ( x ) . Using Lemma 2.2, we find u ∈ dom   ∂ f and u * ∈ ∂ f ( u ) such that 〈 u * , x − u 〉 > 0 and by the pseudo-monotonicity of ∂ f , 〈 x * , x − u 〉 > 0 for every x * ∈ ∂ f ( x ) . Hence, 0 does not ∂ f ( x ) . Consequently, f satisfies condition 0 ∈ ∂ f ( x ) , which implies that x is a global minimum of f, which completes the proof. ■

Theorem 3.3. Let f : X → ℝ ∪ { + ∞ } be a lower semi-continuous (l.s.c.) Clarke-Rockafeller subdifferentiable function on a Banach space X. Consider the following assertions:

(i) f is strictly pseudoconvex.

(ii) f is strictly quasiconvex and ( 0 ∈ ∂ f ( x ) ⇒ x is a global minimum of f),

Then, (i) implies (ii). And (ii) implies (i) if f is radially continuous.

Proof. (i) ⇒ (ii). We want to prove that f is strictly quasiconvex. Let f be a strictly pseudo-convex function, then by Theorem 3.1, the function f is quasiconvex and satisfies the optimality condition

0 ∈ ∂ f ( x ) ⇒ (x is a global minimum of f).

Since f is quasiconvex, then according to [13] , it suffices to prove that f is radially non-constant. Assume by contradiction that there exists a closed segment [ x , y ] with x ≠ y where with f is constant. Let z ∈ ( x , y ) and apply the strict pseudo-convexityproperty to x and z, then

f ( z ) = f ( x ) ⇒ ( ∀ z * ∈ ∂ f ( z ) : 〈 z * , x − z 〉 < 0 ) .

Using the same argument for z and y we obtain

f ( z ) = f ( y ) ⇒ ( ∀ z * ∈ ∂ f ( z ) : 〈 z * , y − z 〉 < 0 ) .

Since ∂ f ( z ) is nonempty, it follows that for all z * ∈ ∂ f ( z ) , 〈 z * , x − y 〉 < 0 and 〈 z * , x − y 〉 > 0 ), which is a contradiction.

(ii) ⇒ (i). Assume that f satisfies condition ii) and f is radially continuous. Then by Theorem 3.1, f is pseudoconvex. We prove that f is pseudo-convex. Suppose by contradiction that there exist x ≠ y in X and x * ∈ ∂ f ( x ) such that

〈 x * , y − x 〉 ≥ 0 and f ( x ) ≥ f ( y ) .

Then, it follows by pseudo-convexity property that

∀ z ∈ [ x , y ] , f ( z ) = f ( x ) .

Since f is quasi-convex, then we have

∀ z ∈ [ x , y ] , f ( z ) ≥ f ( x ) ≥ f ( y ) .

So f is not radially non-constant on X (since f is constant on [ x , y ] ) which contradicts the fact f is strictly quasi-convex.

Theorem 3.4. Let f : X → ℝ ∪ { + ∞ } be a lower semi-continuous (l.s.c.) function such that f is radially Clarke-Rockafeller differentiable. Consider the following assertions:

(i) f is strictly pseudo-convex.

(ii) ∂ f is strictly pseudomonotone

Then, (i) implies (ii). And (ii) implies (i) if f is radially continuous.

Proof. (i) ⇒ (ii). Suppose that f is strictly pseudoconvex. We want to prove that ∂ f is strictly pseudomonotone. Suppose to the contrary that there exist two distinct points x , y ∈ X , x * ∈ ∂ f ( x ) and y * ∈ ∂ f ( y ) such that

〈 x * , y − x 〉 ≥ 0 and 〈 y * , y − x 〉 ≤ 0 .

Since f is strictly pseudoconvex, we have that

f ( x ) < f ( y ) and f ( y ) < f ( x ) .

Which is a contradiction. Therefore, ∂ f is strictly pseudomonotone.

(ii) ⇒ (i). Suppose that f satisfies condition (ii) and f is radially continuous. We want to prove that f is strictly pseudoconvex. Suppose to the contrary that there exist two distinct points x , y ∈ X , and x * ∈ ∂ f ( x ) such that

〈 x * , y − x 〉 ≥ 0 and f ( x ) ≥ f ( y ) .

Then,

〈 x * , z − x 〉 ≥ 0 for all z ∈ [ x , y ] . (16)

By theorem 3.2, f is quasiconvex. Consequently, f must be constant on [ x , y ] . Contrarily, from (15) and the strict monotonicity of ∂ f ( x ) , we have

〈 x * , z − x 〉 > 0 , ∀ z ∈ ( x , y ) and ∀ z * ∈ ∂ f ( z ) . (17)

Pick z 0 ∈ ( x , y ) such that ∂ f ( z 0 ) ≠ ∅ (such a z 0 exists since f is a radially Clarke-Rockafeller subdifferentiable function). Choose any z 0 * ∈ ∂ f ( z 0 ) . Then, 〈 z 0 * , z 0 − x 〉 > 0 . Therefore, 〈 z 0 * , y − z 0 〉 > 0 . Consequently, there exist ε > 0 such that

〈 z 0 * , y ′ − z 0 〉 > 0 for all y ′ ∈ B ε ( y ) .

By the pseudo-convexity of f, it follows that y is a global minimum of f. Hence, z 0 is also a global minimum of f. Thus, 0 ∈ ∂ f ( z 0 ) and this is a contradiction with (17).

We extended the relationships between convex functions and corresponding monotone maps to pseudo-convexity and the corresponding pseudo-monotonicity of their sub-differentiable maps. We characterized the lower semi-continuous Clarke-Rockafeller sub-differentiable pseudo-convex functions by the corresponding monotonicity of their Clarke-Rockafeller sub-differentials ∂ f , and have shown that if a lower semi-continuous Clarke-Rockafeller sub-differentiable function f : X → ℝ ∪ { + ∞ } is radially continuous, then f is pseudo-convex if and only if the sub-differential map ∂ f is pseudo-monotone.

The authors declare no conflicts of interest regarding the publication of this paper.

Offia, A., Osisiogu, U., Efor, T., Oyakhire, F., Ekhator, M., Nkume, F. and Aloke, S. (2023) Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces. Open Journal of Optimization, 12, 99-108. https://doi.org/10.4236/ojop.2023.123007