Higher order strongly general convex functions and variational inequalities

1.   Introduction

It is well known that the convex sets and convex functions had played crucial important part in the developments of the pure and applied sciences and are continue to inspire novel and innovative applications. Convex functions have been extended and generalized in various directions in recent years. Mohsen et al ^[1] introduced the concept of higher order strongly convex functions and studied their properties. These results can be viewed as a significant refinement of the results of Lin and Fukushima ^[2]. Higher order strongly convex functions include the strongly convex functions, which were introduced and studied by Polyak ^[3]. Karmardian ^[4] used the strongly convex functions to discuss the unique existence of a solution of the nonlinear complementarity problems. Awan et al ^[5,6] have derived Hermite-Hadamard type inequalities for various classes of strongly convex functions, which provide upper and lower estimate for the integrand. For the applications of strongly convex functions in optimization, variational inequalities and other branches of pure and applied sciences, see ^{[7,8,9,10,11,12,13,14,15]} and the references therein. It is known that a set may not be convex set. However, a set can be made convex set with respect to an arbitrary function. Motivated by this fact, Noor ^[15] introduced the concept of general convex sets involving an arbitrary function. It has been shown that the minimum of a differentiable general convex function on the general convex set can be characterized by the general variational inequalities. Cristescu at al^[16,17] have investigated algebraic and topological properties of the $g$-convex sets defined by Noor ^[15] in order to deduce their shape. They are a subclass of star-shaped sets, which have Youness ^[18] type convexity. A representation theorem based on extremal points is given for the class of bounded $g$-convex sets. Examples showing that this convexity is a frequent property in connection with a wild range of applications are given. Noor ^[15] has shown that the optimality conditions of the differentiable general convex functions can be characterized a class of variational inequalities called general variational inequality, the origin of which can be traced back to Stampacchia ^[19]. For the formulation, applications, numerical methods, sensitivity analysis and other aspects of general variational inequalities, see ^{[10,11,12,13,14,18,19,20,21,22]} and the references therein. These facts and observations motivated us to consider higher order strongly convex functions with respect to an arbitrary function, which is the main motivation of this paper. Several new concepts of monotonicity are introduced. We establish the relationship between these classes and derive some new results under some mild conditions. We have shown that the minimum of a differentiable higher order strongly convex functions on the general convex set can be characterized by a class of variational inequality. This results inspired us to consider the higher order strongly general variational inequalities. Due to the inherent nonlinearity, the projection method and its variant form can not be used to suggest the iterative methods for solving these nonlinear variational. To overcome these draw backs, we use the technique of the auxiliary principle ^{[12,14,20,21]} to suggest an implicit method for solving general variational inequalities. Convergence analysis of the proposed method is investigated under pseudo-monotonicity, which is a weaker condition than monotonicity. Parallelogram laws for Banach spaces, which are due to Bynum ^[23] and Chen et al ^[24,25], can be obtained from these definitions, are discussed in Section 5. Some new special cases are discussed, which can be viewed itself novel and interesting applications of the higher order strongly convex functions. It is expected that the ideas and techniques of this paper may stimulate further research in this field.

2.   Formulations and basic facts

Let $K$ be a nonempty closed set in a real Hilbert space $H$. We denote by $\langle \cdot, \cdot\rangle$ and $\parallel\cdot\parallel$ be the inner product and norm, respectively.

Definition 2.1. ^[10,17,18] A set $K$ in $H$ is said to be a convex set, if

Definition 2.2. A function $F$ is said to be convex function, if t

It is well known that $u\in K$ of a differential convex functions $F$ is equivalent to finding $u\in K$ such that

which is called the variational inequality, introduced and studied by Stampacchia^[19]. Variational inequalities can be regarded as a novel and significant extension of variational principles.

We would like to mention that in many important applications the underlying the set may not be a convex set. To overcome this drawback, the underlying set can be made convex set with respect to an arbitrary function, which is called a general convex set.

Definition 2.3. ^[15]. The set $K_g$ in $H$ is said to be a general convex set, if there exists a function $g$ such that

We now discuss some special cases of the general convex sets.

(I). If $g = I,$ the identity operator, then general convex set reduces to the classical convex set. Clearly every convex set is a general convex set, but the converse is not true. Cristescu et al^[16] discussed various applications of the general convex sets related to the necessity of adjusting investment or development projects out of environmental or social reasons. For example, the easiest manner of constructing this kind of convex sets comes from the problem of modernizing the railway transport system. Shape properties of the general convex sets with respect to a projection are investigated.

(II). If $g(v) = mv, \quad m\in (0, 1),$ then the general convex set becomes the $m$-convex set, which is mainly due to Toader^[26].

Definition 2.4. ^[26] The set $K_m$ is said to be $m$-convex set, if

For the sake of simplicity, we always assume that $\forall u, v \in H: u, g(v) \in K_g,$ unless otherwise specified.

Definition 2.5. A function $F$ is said to be general convex($g$-convex) function, if there exists an arbitrary non-negative function $g,$ such that

The general convex functions were introduced by Noor ^[15]. Noor ^[15] proved that the minimum $u \in H: g(u) \in K_g$ of the differentiable general convex functions $F$ can be characterized by the class of variational inequalities of the type:

which is known as general variational inequalities. For the applications of the general variational inequalities in various branches of pure and applied sciences, see ^{[10,12,13,14,15,19,20,21,22]} and the references therein.

We now introduce some new classes of higher order strongly general convex functions and higher order strongly affine general convex functions.

Definition 2.6. A function $F$ on the convex set $K_g$ is said to be higher order strongly convex with respect to an arbitrary function $g,$ if there exists a constant $\mu > 0,$ such that

A function $F$ is said to higher order strongly concave with respect to an arbitrary function $g,$ if and only if, $-F$ is higher order strongly general convex with respect to an arbitrary function $g.$

If $t = \frac{1}{2},$ then

The function $F$ is said to be higher order strongly general $J$-convex function.

We now discuss some special cases.

I. If $g = I,$ the identity operator, then $K_g = K,$ the convex set and Definition 2.6 reduces to:

Definition 2.7. A function $F$ on the convex set $K$ is said to be higher order strongly convex function, if there exists a constant $\mu > 0,$ such that

which were introduced and studied by Mohsen et al ^[1]. It has been shown that higher order strongly convex functions can be viewed as a significant refinement of the concept introduced by Lin and Fukushima ^[2]. For the applications of higher order strongly convex functions, see ^[27,28,29] and the references therein.

II. If $g(v) = mv, \quad m\in (0, 1),$ the identity operator, then $K_g = K_,$ the $m$-convex set and Definition 2.6 reduces to:

Definition 2.8. A function $F$ on the convex set $K$ is said to be higher order strongly $m$-convex function, if there exists a constant $\mu > 0,$ such that

which appears to be new ones.

III. If $g(v) = mv, \quad m\in (0, 1)$ and $F(mv) = mF(v),$ then $K_g = K_,$ the $m$-convex set and Definition 2.8 reduces to:

Definition 2.9. A function $F$ on the convex set $K$ is said to be higher order higher order strongly $m$-convex function in the sense of Toader ^[26], if there exists a constant $\mu > 0,$ such that

IV. If $p = 2,$ then the higher order strongly general convex function becomes strongly convex functions, that is,

For the properties of the strongly convex functions in variational inequalities and equilibrium problems, see Noor ^[13,14,15].

We would like to mention that for suitable and appropriate choice of the arbitrary function $g$ and $p,$ we can obtain some new and previous known classes of convex functions. This show that the higher order strongly general convex functions are unifying ones.

Definition 2.10. A function $F$ on the convex set $K_g$ is said to be a higher order strongly affine general convex function with respect to an arbitrary function $g,$ if there exists a constant $\mu > 0,$ such that

Note that, if a functions is both higher order strongly general convex and higher order strongly general concave, then it is higher order strongly affine general convex function.

Definition 2.11. A function $F$ is called higher order strongly quadratic equation with respect to an arbitrary function $g,$ if there exists a constant $\mu > 0,$ such that

This function $F$ is also called higher order strongly general affine $J$-convex function.

Definition 2.12. A function $F$ on the convex set $K_g$ is said to be higher order strongly quasi convex with respect to an arbitrary function $g,$ if there exists a constant $\mu > 0$ such that

Definition 2.13. A function $F$ on the convex set $K_g$ is said to be higher order strongly log-convex with respect to an arbitrary function $g,$ if there exists a constant $\mu > 0$ such that

where $F(\cdot) > 0.$

From the above definitions, we have

This shows that every higher order strongly general log-convex function is a higher order strongly general convex function and every higher order strongly general convex function is a higher order strongly general quasi-convex function. However, the converse is not true. See also ^{[27,28,29,30,31,32]} for the applications of strongly functions convex and their variant forms.

Definition 2.14. An operator $T:K\rightarrow H$ is said to be:

1. higher order strongly general monotone, if and only if, there exists a constant $\alpha > 0$ such that

2. higher order strongly general pseudomonotone, if and only if, there exists a constant $\nu > 0$ such that

3. higher order strongly general relaxed pseudomonotone, if and only if, there exists a constant $\mu > 0$ such that

Definition 2.15. A differentiable function $F$ on the convex set $K$ is said to be higher order strongly general pseudo convex function, if and only if, if there exists a constant $\mu > 0$ such that

3.   Main results

In this section, we consider some basic properties of higher order strongly convex functions.

Theorem 3.1. Let $F$ be a differentiable function on the convex set $K.$ Then the function $F$ is higher order strongly general convex function, if and only if,

Proof. Let $F$ be a higher order strongly general convex function on the general convex set $K_g.$ Then

which can be written as

Taking the limit in the above inequality as $t\rightarrow 0,$ we have

which is the required result (3.1).

Conversely, let (3.1) hold. Then, $\forall u, g(v)\in K_g, t\in [0, 1],$ $v_t = u+t(g(v)-u)\in K_g,$ we have

In a similar way, we have

Multiplying (3.2) by $t$ and (3.3) by $(1-t)$ and adding the resultant, we have

showing that $F$ is a higher order strongly general convex function.

Theorem 3.2. Let $F$ be a differentiable higher order strongly convex function on the general convex set $K_g.$ Then $F'(.)$ is a higher order strongly general monotone operator.

Proof. Let $F$ be a higher order strongly general convex function on the general convex set $K_g.$ Then, from Theorem 3.1. We have

Changing the role of $u$ and $v$ in (3.4), we have

Adding (3.4) and (3.5), we have

which shows that $F'(.)$ is a higher order strongly general monotone operator.

We remark that the converse of Theorem 3.2 is not true. However, we have the following result.

Theorem 3.3. If the differential operator $F^{\prime}(.)$ of a differentiable higher order strongly general convex function $F$ is a higher order strongly general monotone operator, then

Proof. Let $F'(.)$ be a higher order strongly general monotone operator. Then, from (3.6), we have

Since $K_g$ is a general convex set, $\forall u, g(v) \in K_g,$ $t\in [0, 1],$ $v_t = u+t(g(v)-u)\in K_g.$ Taking $v = v_t$ in (3.8), we have

which implies that

Consider the auxiliary function

from which, we have

Then, from (3.9), we have

Integrating (3.10) between 0 and 1, we have

Thus it follows that

which is the required (3.7).

We note that, if $p = 2,$ then Theorem 3.3 can be viewed as the converse of Theorem 3.2.

We now give a necessary condition for higher order strongly general pseudo-convex function.

Theorem 3.4. Let $F'(.)$ be a higher order strongly general relaxed pseudomonotone operator. Then $F$ is a higher order strongly general pseudo-convex function.

Proof. Let $F'$ be a higher order strongly general relaxed pseudomonotone operator. Then, $\forall u, g(v)\in K_g,$

implies that

Since $K_g$ is a general convex set, $\forall u, g(v) \in K_g, \quad t \in [0, 1],$ $v_t = u+t(g(v)-u)\in K_g.$

Taking $v = v_t$ in (3.11), we have

Consider the auxiliary function

which is differentiable, since $F$ is differentiable function. Then, using (3.12), we have

Integrating the above relation between 0 to 1, we have

that is,

showing that $F$ is a higher order strongly general pseudo-convex function.

Definition 3.1. A function $F$ is said to be sharply higher order strongly general pseudo convex, if there exists a constant $\mu > 0$ such that

Theorem 3.5. Let $F$ be a sharply higher order strongly general pseudo convex function on $K_g$ with a constant $\mu > 0.$ Then

Proof. Let $F$ be a sharply higher order strongly general pseudo convex function on $K_g$. Then

from which, we have

Taking limit in the above inequality, as $t \rightarrow 0,$ we have

the required result.

Definition 3.2. A function $F$ is said to be a pseudo convex function, if there exists a strictly positive bifunction $B(., .),$ such that

Theorem 3.6. If the function $F$ is higher order strongly general convex function such that $F(g(v)) < F(u),$ then the function $F$ is higher order strongly general pseudo convex.

Proof. Since $F(g(v)) < F(u)$ and $F$ is higher order strongly general convex function, then $\forall u, g(v) \in K_g, \quad t\in [0, 1],$ we have

where $B(u, g(v) = F(u) - F(g(v)) > 0.$ Consequently the function $F$ is higher order strongly general pseudo convex. the required result.

Theorem 3.7. Let $f$ be a higher order strongly general affine function. Then $F$ is a higher order strongly general convex function, if and only if, $g = F-f$ is a general convex function.

Proof. Let $f$ be a higher order strongly general affine function, Then

From the higher order strongly general convexity of $F,$ we have

From (3.19) and (3.14), we have

from which it follows that

which show that $G = F-f$ is a convex function.

We now discuss the optimality for the differentiable generalized strongly convex functions, which is the main motivation of our next result.

Theorem 3.8. Let $F$ be a differentiable higher order strongly general convex function with modulus $\mu > 0.$ If $u \in K_g$ is the minimum of the function $F,$ then

Proof. Let $u \in K_g$ be a minimum of the function $F.$ Then

Since $K_g$ is a general convex set, so, $\forall u, g(v) \in K_g, \quad t\in [0, 1],$

Taking $v = v_t$ in (3.17), we have

Since $F$ is differentiable higher order strongly convex function, so

from which, using (3.18), we have

the required result (3.16).

Remark: We would like to mention that, if

then $u \in K_g$ is the minimum of the function $F.$ The inequality of the type (3.19) is called the higher order strongly general variational inequality, which appears to be new one.

4.   Higher order general variational inequalities

In this section, we consider a more general variational inequality of which (3.19) is a special case.

For given two operators $T, g,$ we consider the problem of finding $u\in K$ for a constant $\mu$ such that

which is called the higher order strongly general variational inequality. It is clear that or $Tu = F^{\prime}(g(u)),$ problem (4.1) is exactly the general variational inequality (3.19). We also note that, if $\mu = 0,$ then (3.14) is equivalent to finding $u \in K,$ such that

which is known as the general variational inequality, which was introduced and studied by Noor ^[15] in 2008. For suitable and appropriate choice of the parameter $\mu$ and $p,$ one can obtain several new and known classes of variational inequalities, see ^{[13,15,20,21,22]}. We note that the projection method and its variant forms can be used to study the higher order strongly general variational inequalities (4.1) due to its inherent structure. This fact motivated us to consider the auxiliary principle technique, which is mainly due to Glowinski et al^[21] and Lions and Stampacchia ^[20] as developed by Noor ^[13,14]. We use this technique to suggest some iterative methods for solving the general variational inequalities (4.1).

For given $u\in K$ satisfying (4.1), consider the problem of finding $w\in K,$ such that

where $\rho > 0$ is a parameter. The problem (4.3) is called the auxiliary higher order strongly general variational inequality. It is clear that the relation (4.3) defines a mapping connecting the problems (4.1) and (4.3). We not that, if $w(u) = u,$ then $w$ is a solution of problem (4.1). This simple observation enables to suggest an iterative method for solving (4.1).

Algorithm 4.1. For given $u\in K,$ find the approximate solution $u_{n+1}$ by the scheme

The Algorithm 4.1 is known as the implicit method. Such type of methods have been studied extensively for various classes of variational inequalities. See ^[13,14] and the reference therein. If $\nu = 0,$ then Algorithm 4.1 reduces to:

Algorithm 4.2. For given $u_0\in K,$ find the approximate solution $u_{n+1}$ by the scheme

which appears to new ones even for solving the general variational inequalities (4.2).

In to study the convergence analysis of Algorithm 4.1, we need the following concept.

Definition 4.1. The operator $T$ is said to be pseudo $g$-monotone with respect to $\mu \|g(v)-u\|^p, p > 1,$ if

If $\mu = 0,$ then Definition 4.1 reduces to:

Definition 4.2. The operator $T$ is said to be pseudo $g$-monotone, if

which appears to be a new one.

We now study the convergence analysis of Algorithm 4.1.

Theorem 4.1. Let $u\in K$ be a solution of (4.1) and $u_{n+1}$ be the approximate solution obtained from Algorithm 4.1. If $T$ is a pseudo $g$- monotone operator, then

Proof. Let $u \in K$ be a solution of (4.1), then

implies that

Now taking $v = u_{n+1}$ in (4.6), we have

Taking $v = u$ in (4.4), we have

Combining (4.7) and (4.8), we have

Using the inequality

we obtain

the required result (4.5).

Theorem 4.2. Let the operator $T$ be a pseudo $g$-monotone. If $u_{n+1}$ be the approximate solution obtained from Algorithm 4.1 and $u \in K$ is the exact solution (4.1), then $\lim_{n\rightarrow \infty}u_n = u.$

Proof. Let $u\in K$ be a solution of (4.1). Then, from (4.5), it follows that the sequence $\{\|u - u_n\|\}$ is nonincreasing and consequently $\{u_n\}$ is bounded. From (4.5), we have

from which, it follows that

Let $\hat{u}$ be a cluster point of $\{u_n\}$ and the subsequence $\{u_{n_j}\}$ of the sequence $u_n$ converge to $\hat{u} \in H.$ Replacing $u_n$ by $u_{n_j}$ in (5.5), taking the limit $n_j \rightarrow 0$ and from (4.9), we have

This implies that $\hat{u} \in K$ K satisfies and

Thus it follows from the above inequality that the sequence $u_n$ has exactly one cluster point $\hat{u}$ and

In order to implement the implicit Algorithm 4.1, one uses the predictor-corrector technique. Consequently, Algorithm 4.1 is equivalent to the following iterative method for solving the general variational inequality (4.1).

Algorithm 4.3. For a given $u_0 \in K,$ find the approximate solution $u_{n+1}$ by the schemes

Algorithm 4.3 is called the two-step iterative method and appears to be a new one.

Using the auxiliary principle technique, on can suggest several iterative methods for solving the higher order strongly general variational inequalities and related optimization problems. We have only given a glimpse of the higher order strongly general variational inequalities. It is an interesting problem to explore the applications of such type variational inequalities in various fields of pure and applied sciences.

5.   Parallelogram laws

In this section, we discuss some characterizations of uniformly Banach spaces involving the notion of strongly general convexity, which can be viewed as novel application.

Setting $F(u) = \|u\|^p$ in Definition 2.6, we have

Taking $t = \frac{1}{2}$ in (5.1), we have

which implies that

which is known as the lower parallelogram for the Banach spaces. In a similar way, one can obtain the upper parallelogram law as

From Definition (2.10), we have

which is known as the general parallelogram law for the Banach spaces.

We now discuss important cases.

I. If $g(v) = mv, \quad m\in (0, 1),$ then we have a new $m$-parallelogram law as:

II. For $g = I,$ the identity operator, we obtain the parallelogram laws obtained by Xi ^[33] characterising the $p$-uniform convexity and $q$-uniform smoothness of a Banach space. See also Bynum ^[23] and Chen et al ^[24,25], who have studied the properties and applications of the parallelogram laws. For the applications of the parallelogram laws in Banach spaces in prediction theory and applied sciences, see ^[24,25] and the references therein.

6.   Conclusions

In this paper, we have introduced and studied a new class of convex functions, which is called higher order strongly general convex function. It is shown that several new classes of strongly convex functions can be obtained as special cases of these higher order strongly convex functions. We have studied the basic properties of these functions. We have also considered a new class of higher order strongly general variational inequalities. Using the auxiliary principle technique, an implicit iterative method is suggested for finding the approximate solution of higher order general variational inequalities. Using the pseudo-monotonicity of the operator, convergence criteria is discussed. Some special cases are considered as application of the main results. We have derived the parallelogram laws in Banach spaces, which have applications in prediction theory and stochastic analysis. The interested readers may explore the applications and other properties of the higher order strongly general convex functions in various fields of pure and applied sciences. This is an interesting direction of future research.

Acknowledgements

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments. The authors are grateful to the referees for their very useful and valuable comments.

Conflict of interest

The authors declare that they have no conflicts of interest.