Mixed Type Nondifferentiable Higher-Order Symmetric Duality Over Cones

Home , Duality (optimization), Strong duality, Weak duality

S S symmetry

Article Mixed Type Nondifferentiable Higher-Order Symmetric Duality over Cones

Izhar Ahmad 1,† , Khushboo Verma 2,† and Suliman Al-Homidan 1,*,†

1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia; [email protected] 2 Department of Applied Sciences and Humanities, Faculty of Engineering, University of Lucknow, New Campus, Lucknow 226 021, India; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work.

Received: 29 December 2019; Accepted: 1 February 2020; Published: 11 February 2020

Abstract: A new mixed type nondifferentiable higher-order symmetric dual programs over cones is formulated. As of now, in the literature, either Wolfe-type or Mond–Weir-type nondifferentiable symmetric duals have been studied. However, we present a uniﬁed dual model and discuss weak, strong, and converse duality theorems for such programs under higher-order F- convexity/higher-order F- pseudoconvexity. Self-duality is also discussed. Our dual programs and results generalize some dual formulations and results appeared in the literature. Two non-trivial examples are given to show the uniqueness of higher-order F- convex/higher-order F- pseudoconvex functions and existence of higher-order symmetric dual programs.

Keywords: higher-order nondifferentiable programming; symmetric duality; self duality higher-order F-convexity/higher-order F-pseudoconvexity; duality theorems

1. Introduction The study of higher-order duality has computational advantages over the ﬁrst-order duality when approximations are used, as it provides tighter bound for the value of the objective function. Mangasarian [1] formulated a class of higher-order duality in nonlinear problems. Later on, Mond and Zhang [2] obtained various higher-order duality results under higher-order invexity assumptions. Chen [3] discussed Mond–Weir-type higher-order symmetric duality involving F-convex functions, whereas Mishra [4] obtained Mond–Weir-type higher-order symmetric duality theorems under generalized invexity. Khurana [5] presented symmetric duality results for a Mond–Weir-type dual programs over arbitrary cones under cone-pseudoinvexity and strongly cone-pseudoinvexity. Later on, a higher-order Mond–Weir-type nondifferentiable multiobjective dual problem is formulated and established duality relations involving higher-order (F, α, ρ, d) type-I fuctions by Ahmad et al. [6]. In [7], Gupta and Jayswal obtained multiobjective higher-order symmetric duality results under higher-order K-preinvexity/K-pseudoinvexity. The theoretical and algorithmic concepts of mixed duality in nonlinear programming problems are interesting and useful. Under K-preinvexity and K-pseudoinvexity assumptions, Ahmad and Husain [8] formulated multiobjective mixed type symmetric dual programs over cones and proved duality results. Chandra et al. [9] studied mixed type symmetric duality results. Xu [10] proved duality theorems for two mixed type duals of a multiobjective programming problem. Yang et al. [11] presented mixed type symmetric duality for nondifferentiable nonlinear programming problems. By ignoring nonnegativity constraints of symmetric dual given in [12], Ahmad [13] presented multiobjective mixed type symmetric duality results. Recently, Verma et al. [14] formulated a

Symmetry 2020, 12, 274; doi:10.3390/sym12020274 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 274 2 of 18 higher-order nondifferentiable mixed symmetric dual model and duality results are studied under higher-order invexity/generalized invexity. In the present paper, a new mixed type nondifferentiable higher-order symmetric dual programs over cones are formulated. As of now, in the literature, either Wolfe-type or Mond–Weir-type nondifferentiable symmetric dual programs have been discussed. However, our model uniﬁes both dual programs. Under higher-order F- convexity/higher-order F- pseudoconvexity, appropriate duality theorems are proved. Self-duality is also discussed. Our study extends and generalizes the existing results appeared in [15–21]. Two non-trivial examples are given to show the uniqueness of higher-order F- convex/higher-order F- pseudoconvex functions and existence of higher-order symmetric dual programs.

2. Preliminaries We consider the following nonlinear programming problem.

(P) Minimize F(x) x ∈ X, where X ⊆ Rn and F : X → R.

2.1. Deﬁnitions (a) The support function s(x|E) of E is given by

s(x|E) = max{xTy : y ∈ E}.

The subdifferential of s(x|E) is deﬁned by

∂s(x|E) = {z ∈ E : zT x = s(x|E)}.

The normal cone for any convex set S ⊂ Rn at a point x ∈ S is deﬁned by

n T NS(x) = {y ∈ R : y (z − x) ≤ 0 for all z ∈ S}.

It is readily veriﬁed that for a compact convex set D, y is in ND(x) if and only if

s(y|D) = xTy.

∗ (b) Let C be a closed convex cone in Rn with nonempty interior. The positive polar cone C of C is given by ∗ C = {q ∈ Rn : xTq ≥ 0 for all x ∈ C}. (c) A function F : X × X × Rn 7→ R (where X ⊆ Rn) is sublinear in its third component, if for all x, y ∈ X :

n (i) F(x, y; b1 + b2) ≤ F(x, y : b1) + F(x, y : b2), for all b1, b2 ∈ R ; and (ii) F(x, y; αb) = αF(x, y; b), for all α ∈ R, for all b ∈ Rn.

(d)[3] A function φ : Rn 7→ R is called higher-order F− convex at u ∈ Rn with respect to g : Rn × Rn 7→ R, if for all (x, p) ∈ Rn × Rn,

T φ(x) − φ(u) − g(u, p) + p ∇pg(u, p) ≥ F(x, u; ∇xφ(u) + ∇pg(u, p)).

(e)[3] A function φ : Rn 7→ R is called higher-order F− pseudoconvex at u ∈ Rn with respect to g : Rn × Rn 7→ R, if for all (x, p) ∈ Rn × Rn, Symmetry 2020, 12, 274 3 of 18

T F(x, u; ∇xφ(u) + ∇pg(u, p)) ≥ 0 ⇒ φ(x) − φ(u) − g(u, p) + p ∇pg(u, p) ≥ 0.

The above deﬁnitions (d) and (e) are validated by the following example.

2.2. Example

−x 2 2 2 p Consider φ(x) = e + x , Fx,u(a) = |a|(x − u ), g(u, p) = u+1 and X = {x : x ≥ 1}.

Now, we show that the function φ(x) is a higher-order F- pseudoconvex at u = 1 and for all x ≥ 1.

2 2 F(x, u; ∇xφ(u) + ∇pg(u, p)) = |∇xφ(u) + ∇pg(u, p)|(x − u )

1 = | − e−u + 2u + |(x2 − u2), u + 1

For u = 1, the above expression reduces to

1 F(x, 1; ∇ φ(1) + ∇ g(1, p)) = | − e−1 + 2 + |(x2 − 1) ≥ 0, f or x ≥ 1, (1) x p 2 Now, p p φ(x) − φ(u) − g(u, p) + pT∇ g(u, p) = e−x + x2 − e−u − u2 − + . p u + 1 u + 1

At u = 1, the above equality becomes

1 φ(x) − φ(u) − g(u, p) + pT∇ g(u, p) = e−x + x2 − − 1 ≥ 0, f or x ≥ 1, (2) p e The Equations (1) and (2) show that function φ(x) is a higher-order F− pseudoconvex at u = 1 and for all x ≥ 1 (see Figures1–3).

Now, we determine whether φ(x) is a higher-order F− convex function.

T φ(x) − φ(u) − g(u, p) + p ∇pg(u, p) − F(x, u; ∇xφ(u) + ∇pg(u, p))

p p 1 = e−x + x2 − e−u − u2 − + − | − e−u + 2u + |(x2 − u2). u + 1 u + 1 u + 1

At u = 1 1 1 = e−x + x2 − − 1 − | − e−1 + 2 + |(x2 − 1) ≡ χ, (see Figure 4) e 2 0, ( f or x ≥ 2), which shows that the function φ(x) is not a higher-order F− convex (see Figure4) at u = 1 and x = 2. This concludes that every higher-order F− convex function is a higher-order F− pseudoconvex function, but the converse is not necessarily true. (See AppendixA for codes of Figures1–4). Symmetry 2020, 12, 274 4 of 18

Figure 1. Graph of Fx,u(.) against x and u.

Figure 2. Graph of Fx,u(.) against x.

Figure 3. Graph of φ against x.

Figure 4. Graph of χ against x.

3. Higher-Order Mixed Type Symmetric Duality over Cones

For N = {1, 2, 3, ..., n} and H = {1, 2, 3, ..., h}, let us assume A1 ⊂ N, B1 ⊂ H and A2 = N \ A1 and B2 = H \ B1, where |A1| denotes the number of elements in the set A1, also |A2|, |B1| and |B2| are deﬁned similarly. |A | Again if |A1| = 0, then |A2| = n and therefore R 1 is a zero-dimensional Euclidean space and | | R A2 is n-dimensional Euclidean space. | | | | Now x ∈ Rn if and only if x = {x1, x2}, i.e. x1 ∈ R A1 , x2 ∈ R A2 . Similarly, any y ∈ Rh if and | | | | only if y = {y1, y2}, i.e. y1 ∈ R B1 , y2 ∈ R B2 . Let Symmetry 2020, 12, 274 5 of 18

| | | | (i) f 1 : R A1 × R B1 → R,

| | | | (ii) f 2 : R A2 × R B2 → R,

| | | | | | (vi) g2 : R A2 × R B2 × R B2 → R be twice differentiable functions.

Now we formulate the following mixed type nondifferentiable higher-order symmetric dual programs over cones.

Primal Problem (MHP):

1 1 1 1 2 2 2 2 1 T 1 Minimize G(x, y, z, p) = f (x , y ) + s(x |E1) + f (x , y ) + s(x |E2) − (y ) z

1 1 1 1 2 2 2 2 1 T 1 1 1 1 + g (x , y , p ) + g (x , y , p ) − (p ) ∇p1 g (x , y , p )

2 T 2 2 2 2 2 T 2 2 2 2 2 2 2 − (p ) ∇p2 g (x , y , p ) − (y ) [∇y2 f (x , y ) + ∇p2 g (x , y , p )] subject to

1 1 1 1 1 1 1 1 ∗ −(∇y1 f (x , y ) − z + ∇p1 g (x , y , p )) ∈ C3 , (3)

2 2 2 2 2 2 2 2 ∗ −(∇y2 f (x , y ) − z + ∇p2 g (x , y , p )) ∈ C4 , (4)

1 T 1 1 1 1 1 1 1 1 (y ) [∇y1 f (x , y ) − z + ∇p1 g (x , y , p )] ≥ 0, (5)

1 T 1 1 1 1 1 1 1 1 (p ) [∇y1 f (x , y ) − z + ∇p1 g (x , y , p )] ≥ 0, (6)

2 T 2 2 2 2 2 2 2 2 (p ) [∇y2 f (x , y ) − z + ∇p2 g (x , y , p )] ≥ 0, (7)

1 2 2 x ∈ C1, x ∈ C2, y ≥ 0, (8) 1 2 z ∈ D1, z ∈ D2. (9) Dual Problem (MHD):

1 1 1 1 2 2 2 2 1 T 1 Maximize H(u, v, w, r) = f (u , v ) − S(v |D1) + f (u , v ) − S(v |D2) + (u ) w

1 1 1 1 2 2 2 2 1 T 1 1 1 1 + h (u , v , r ) + h (u , v , r ) − (r ) ∇r1 h (u , v , r )

2 T 2 2 2 2 2 T 2 2 2 2 2 2 2 − (r ) ∇r2 h (u , v , r ) − (u ) [∇u2 f (u , v ) + ∇r2 h (u , v , r )] subject to 1 1 1 1 1 1 1 1 ∗ (∇u1 f (u , v ) + w + ∇r1 h (u , v , r )) ∈ C1 , (10) 2 2 2 2 2 2 2 2 ∗ (∇u2 f (u , v ) + w + ∇r2 h (u , v , r )) ∈ C2 , (11) 1 T 1 1 1 1 1 1 1 1 (u ) [∇u1 f (u , v ) + w + ∇r1 h (u , v , r )] ≤ 0, (12) Symmetry 2020, 12, 274 6 of 18

1 T 1 1 1 1 1 1 1 1 (r ) [∇u1 f (u , v ) + w + ∇r1 h (u , v , r )] ≤ 0, (13) 2 T 2 2 2 2 2 2 2 2 (r ) [∇u2 f (u , v ) + w + ∇r2 h (u , v , r )] ≤ 0, (14) 1 2 2 v ∈ C3, v ∈ C4, u ≥ 0, (15) 1 2 w ∈ E1, w ∈ E2, (16) 1 |B | 2 |B | 1 |A | 2 |A | where, p ∈ R 1 , p ∈ R 2 , r ∈ R 1 , r ∈ R 2 and E1, E2, D1, and D2 are compact convex sets in | | | | | | | | R B1 , R B2 , R A1 , and R A2 , respectively.

4. Remark The above dual programs generalize several models that appeared in the literature can be seen below.

n h (A) If |B1| = 0, |A1| = 0, C1 = R + and C2 = R +, then the above programs reduce to non-differentiable Wolfe-type dual programs (see [18]).

n h (B) Let |B2| = 0, |A2| = 0, C1 = R + and C2 = R +. Then the above programs become non-differentiable Mond–Weir-type dual programs (see [18]).

n h (C) Let C1 = R + and C2 = R + in (MHP) and (MHD), respectively. Then we get the programs proposed by Verma et al. [14].

n h (D) Let C1 = R +, C2 = R +, E = 0 and D = 0 in (MHP) and (MHD), respectively. Then we get the programs discussed by Verma et al. [22].

5. Duality Theorems In this section, we establish the relations between primal problem (MHP) and dual problem (MHD).

Theorem 1. (Weak Duality) Let (x1, x2, y1, y2, z1, z2, p1, p2) be feasible for (MHP) and (u1, u2, v1, v2, w1, w2, r1, r2) be feasible for (MHD). Suppose that

(i) f 1(., v1) + (.)Tw1 is higher-order F− pseudo-convex function at u1 with respect to h1(u1, v1, r1),

(ii) − f 1(x1,.) + (.)Tz1 is higher-order F− pseudo-convex function at y1 with respect to −g1(x1, y1, p1),

(iii) f 2(., v2) + (.)Tw2 is higher-order F− convex function at u2 with respect to h2(u2, v2, r2),

(iv) − f 2(x2,.) + (.)Tz2 is higher-order F− convex function at y2 with respect to −g2(x2, y2, p2),

1 1 1 1 T 1 1 T 1 1 1 1 ∗ (v) F(x , u ; ψ ) + (u ) ψ + (r ) ψ ≥ 0, ∀ x , u ∈ C1, ψ ∈ C1 ,

1 1 2 1 T 2 1 T 2 1 1 2 ∗ (vi) F(v , y ; ψ ) + (y ) ψ + (p ) ψ ≥ 0, ∀ v , y ∈ C3, ψ ∈ C3 ,

2 2 3 2 T 3 2 T 3 2 2 3 ∗ (vii) F(x , u ; ψ ) + (u ) ψ + (r ) ψ ≥ 0, ∀ x , u ∈ C2, ψ ∈ C2 and

2 2 4 2 T 4 2 T 4 2 1 4 ∗ (viii) F(v , y ; ψ ) + (y ) ψ + (p ) ψ ≥ 0, ∀ v , y ∈ C4, ψ ∈ C4 . Then G(x1, x2, y1, y2, z1, z2, p1, p2) ≥ H(u1, u2, v1, v2, w1, w2, r1, r2). (17) Symmetry 2020, 12, 274 7 of 18

Proof. From hypotheses (iii) and (iv), we get

2 2 2 2 T 2 2 2 2 2 T 2 2 2 2 2 2 T 2 2 2 2 f (x , v ) + (x ) w − f (u , v ) − (u ) w − h (u , v , p ) + (p ) ∇p2 h (u , v , p )

2 2 2 2 2 2 2 2 2 2 ≥ F(x , u ; ∇x2 f (u , v ) + w + ∇r2 g (u , v , r ))(18)

and

2 2 2 2 T 2 2 2 2 2 T 2 2 2 2 2 2 T 2 2 2 2 f (x , y ) − (y ) z − f (x , v ) + (v ) z + g (x , y , p ) − (p ) ∇p2 g (x , y , p )

2 2 2 2 2 2 2 2 2 2 ≥ F(v , y ; −(∇y2 f (x , y ) − z + ∇p2 h (x , y , p )). (19)

Let 3 2 2 2 2 2 2 2 2 ∗ ψ = (∇u2 f (u , v ) + w + ∇r2 h (u , v , r )) ∈ C2 and 4 2 2 2 2 2 2 2 2 ∗ ψ = −(∇y2 f (x , y ) − z + ∇p2 g (x , y , p )) ∈ C4 . Now from hypotheses (vii) and (viii) , we get

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 F(x , u ; ∇x2 f (u , v ) + w + ∇r2 g (u , v , r )) + u [∇x2 f (u , v ) + w + ∇r2 g (u , v , r )] 2 2 2 2 2 2 2 2 2 ≥ −r [∇x2 f (u , v ) + w + ∇r2 g (u , v , r )] and

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 F(v , y ; −(∇y2 f (x , y ) − z + ∇p2 h (x , y , p ))) − y [∇y2 f (x , y ) − z + ∇p2 h (x , y , p )] 2 2 2 2 2 2 2 2 2 ≥ p [∇y2 f (x , y ) − z + ∇p2 h (x , y , p )]. The above inequalities along with inequalities (7) and (14) imply that

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 F(x , u ; ∇x2 f (u , v ) + w + ∇r2 g (u , v , r )) ≥ −u [∇x2 f (u , v ) + w + ∇r2 g (u , v , r )]

and

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 F(v , y ; −(∇y2 f (x , y ) − z + ∇p2 h (x , y , p ))) ≥ y [∇y2 f (x , y ) − z + ∇p2 h (x , y , p )].

The above inequalities along with Equations (18) and (19) give

2 2 2 2 T 2 2 2 2 2 T 2 2 2 2 2 2 T 2 2 2 2 f (x , v ) + (x ) w − f (u , v ) − (u ) w − h (u , v , p ) + (p ) ∇p2 h (u , v , p )

2 2 2 2 2 2 2 2 2 ≥ −u [∇x2 f (u , v ) + w + ∇r2 g (u , v , r )]

and

2 2 2 2 T 2 2 2 2 2 T 2 2 2 2 2 2 T 2 2 2 2 f (x , y ) − (y ) z − f (x , v ) + (v ) z + g (x , y , p ) − (p ) ∇p2 g (x , y , p )

2 2 2 2 2 2 2 2 2 ≥ y [∇y2 f (x , y ) − z + ∇p2 h (x , y , p )].

Now adding the above two inequalities, we obtain Symmetry 2020, 12, 274 8 of 18

2 2 2 2 T 2 2 T 2 2 2 2 2 2 T 2 2 2 2 2 2 2 2 f (x , y ) + (x ) w − (y ) z + g (x , y , p ) − (p ) ∇p2 g (x , y , p ) − y [∇y2 f (x , y )

2 2 2 2 2 −z + ∇p2 h (x , y , p )]

2 2 2 2 T 2 2 T 2 2 2 2 2 2 T 2 2 2 2 2 2 2 2 ≥ f (u , v ) + (u ) w − (v ) z + h (u , v , p ) − (p ) ∇p2 h (u , v , p ) − u [∇x2 f (u , v )

2 2 2 2 2 +w + ∇r2 g (u , v , r )].

2 T 2 2 2 T 2 2 Noting that (x ) w ≤ s(x |E2) and (v ) z ≤ s(v |D2), we have

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 f (x , y ) + s(x |E2) + g (x , y , p ) − y [∇y2 f (x , y ) + ∇p2 h (x , y , p )]

2 2 2 2 2 2 2 2 ≥ f (u , v ) − s(v |D2) + h (u , v , p )(20) 2 T 2 2 2 2 2 2 2 2 2 2 2 2 −(p ) ∇p2 h (u , v , p ) − u [∇x2 f (u , v ) + ∇r2 h (u , v , r )]. By taking 1 1 1 1 1 1 1 1 1 ∗ ψ = (∇u1 f (u , v ) + w + ∇r1 h (u , v , r )) ∈ C1 and 2 1 1 1 1 1 1 1 1 ∗ ψ = −(∇y1 f (x , y ) − z + ∇p1 g (x , y , p )) ∈ C3 , we have

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F(x , u ; ∇u1 f (u , v ) + w + ∇r1 g (u , v , r )) ≥ −(u + r )[∇u1 f (u , v ) + w + ∇r1 g (u , v , r )] (by the hypothesis (v)) and

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F(v , y ; −(∇y1 f (x , y ) − z + ∇p1 h (x , y , p ))) ≥ (y + p )[∇y1 f (x , y ) − z + ∇p1 h (x , y , p )] (by the hypothesis (vi)).

The above two inequalities along with inequalities (5), (6), (12), and (13) give

1 1 1 1 1 1 1 1 1 1 F(x , u ; ∇u1 f (u , v ) + w + ∇r1 g (u , v , r )) ≥ 0 and

1 1 1 1 1 1 1 1 1 1 F(v , y ; −(∇y1 f (x , y ) − z + ∇p1 h (x , y , p ))) ≥ 0, which by hypotheses (i) and (ii) imply

1 1 1 1 T 1 1 1 1 1 T 1 1 1 1 1 1 T 1 1 1 1 f (x , v ) + (x ) w − f (u , v ) − (v ) w − h (u , v , p ) + (p ) ∇p1 h (u , v , p ) ≥ 0 and

1 1 1 1 T 1 1 1 1 1 T 1 1 1 1 1 1 T 1 1 1 1 f (x , y ) + (v ) z − f (x , v ) − (y ) z + g (x , y , p ) − (p ) ∇p1 g (x , y , p ) ≥ 0.

Adding the above two inequalities, we obtain

1 1 1 1 T 1 1 T 1 1 1 1 1 1 T 1 1 1 1 f (x , y ) − (y ) z + (x ) w + g (x , y , p ) − (p ) ∇p1 g (x , y , p )

1 1 1 1 T 1 1 T 1 1 1 1 1 1 T 1 1 1 1 ≥ f (u , v ) + (v ) w − (v ) z + h (u , v , p ) − (p ) ∇p1 h (u , v , p ). Symmetry 2020, 12, 274 9 of 18

1 T 1 1 1 T 1 1 Using (x ) w ≤ s(x |E1) and (v ) z ≤ s(v |D1), we have

1 1 1 1 T 1 1 1 1 1 1 1 T 1 1 1 1 f (x , y ) − (y ) z + s(x |E1) + g (x , y , p ) − (p ) ∇p1 g (x , y , p )

1 1 1 1 T 1 1 1 1 1 1 T 1 1 1 1 ≥ f (u , v ) + (v ) w − s(v |D1) + h (u , v , p ) − (p ) ∇p1 h (u , v , p ). (21)

Combining inequalities (20) and (21), we get

G(x1, x2, y1, y2, z1, z2, p1, p2) ≥ H(u1, u2, v1, v2, w1, w2, r1, r2).

Theorem 2. (Strong Duality) Let (x¯1, x¯2, y¯1, y¯2, z¯1, z¯2, p¯1, p¯2) be an optimal solution of (MHP). Suppose that

1 1 1 1 2 2 2 2 (i) ∇p1 p1 g (x¯ , y¯ , p¯ ) is positive or negative deﬁnite matrix and ∇p2 p2 g (x¯ , y¯ , p¯ ) is negative deﬁnite matrix,

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 (ii) ∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ ) 6= 0 and ∇y2 f (x¯ , y¯ ) − z¯ + ∇p2 g (x¯ , y¯ , p¯ ) 6= 0,

1 T 1 1 1 1 1 1 1 1 1 (iii) (p¯ ) [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0 ⇒ p¯ = 0 and 2 2 2 2 2 2 2 2 2 2 2 2 2 2 y¯ [∇y2 g (x¯ , y¯ , p¯ ) − ∇p2 g (x¯ , y¯ , p¯ ) + ∇y2y2 f (x¯ , y¯ )p¯ ] = 0 ⇒ p¯ = 0,

1 1 1 1 1 1 1 1 1 1 1 1 (iv) g (x¯ , y¯ , 0) = h (x¯ , y¯ , 0), ∇x1 g (x¯ , y¯ , 0) = ∇r1 h (x¯ , y¯ , 0), 1 1 1 1 1 1 2 2 2 2 2 2 ∇y1 g (x¯ , y¯ , 0) = ∇p1 g (x¯ , y¯ , 0), g (x¯ , y¯ , 0) = h (x¯ , y¯ , 0) and 2 2 2 2 2 2 ∇x2 g (x¯ , y¯ , 0) = ∇r2 h (x¯ , y¯ , 0).

Then,

(I) (x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1 = 0, r¯2 = 0) is feasible for (MHD), and

(II) G(x¯1, x¯2, y¯1, y¯2, z¯1, z¯2, p¯1, p¯2) = H(x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1, r¯2).

Furthermore, if the hypotheses of Theorem1 are satisﬁed for all feasible solutions of (MHP) and (MHD), then (x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1 = 0, r¯2 = 0) is an optimal solution for (MHD).

Proof. As (x¯1, x¯2, y¯1, y¯2, z¯1, z¯2, p¯1, p¯2) is an optimal solution of (MHP), by the Fritz John necessary | | | | | | | | optimality conditions [23,24], there exist α, γ, δ1, δ2 ∈ R, β1 ∈ R B1 , β2, η ∈ R B2 , ξ1 ∈ R A1 , ξ2 ∈ R A2 such that the following conditions are satisﬁed at (x¯1, x¯2, y¯1, y¯2, z¯1, z¯2, p¯1, p¯2) :

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {α[∇x1 f (x¯ , y¯ ) + ξ + ∇x1 g (x¯ , y¯ , p¯ ) − ∇p1x1 g (x¯ , y¯ , p¯ )p¯ ] + [∇y1x1 f (x¯ , y¯ )

1 1 1 1 1 1 1 1 1 1 1 +∇p1x1 g (x¯ , y¯ , p¯ )](β − γy¯ − δ p¯ )}(x − x¯ ) ≥ 0, for all x ∈ C1, (22)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 {α[∇x2 f (x¯ , y¯ ) + ξ + ∇x2 g (x¯ , y¯ , p¯ )] + {∇p2x2 g (x¯ , y¯ , p¯ )}(β − αy¯ − αp¯

2 2 2 2 2 2 2 2 2 2 2 2 −δ p¯ ) + {∇y2x2 f (x¯ , y¯ )}(β − αy¯ − δ p¯ )}(x − x¯ ) ≥ 0, for all x ∈ C2, (23) Symmetry 2020, 12, 274 10 of 18

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α[∇y1 f (x¯ , y¯ ) − z¯ + ∇y1 g (x¯ , y¯ , p¯ ) − ∇p1y1 g (x¯ , y¯ , p¯ )p¯ ] + (∇y1y1 f (x¯ , y¯ ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 +∇p1y1 g (x¯ , y¯ , p¯ ))(β − γy¯ − δ p¯ ) − γ[∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0,

| | for all, y1 ∈ R B1 , (24)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 {∇p2y2 g (x¯ , y¯ , p¯ )}(β − αy¯ − αp¯ − δ p¯ ) + α[∇y2 g (x¯ , y¯ , p¯ ) − ∇p2 g (x¯ , y¯ , p¯ )]

2 2 2 2 2 2 2 2 |B2| +{∇y2y2 f (x¯ , y¯ )}(β − αy¯ − δ p¯ ) − η = 0 for all, y ∈ R , (25)

1 1 1 1 1 1 1 1 1 {∇p1 p1 g (x¯ , y¯ , p¯ )}(β − αp¯ − γy¯ − δ p¯ )

1 1 1 1 1 1 1 1 1 −δ [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0, (26)

2 2 2 2 2 2 2 2 2 {∇p2 p2 g (x¯ , y¯ , p¯ )}(β − αy¯ − αp¯ − δ p¯ )

2 2 2 2 2 2 2 2 2 −δ [∇y2 f (x¯ , y¯ ) − z¯ + ∇p2 g (x¯ , y¯ , p¯ )] = 0, (27)

1 1 1 1 1 1 1 1 1 β [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0, (28)

2 2 2 2 2 2 2 2 2 β [∇y2 f (x¯ , y¯ ) − z¯ + ∇p2 g (x¯ , y¯ , p¯ )] = 0, (29)

1 1 1 1 1 1 1 1 1 γy¯ [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0, (30)

1 1 1 1 1 1 1 1 1 1 δ p¯ [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0, (31)

2 2 2 2 2 2 2 2 2 2 δ p¯ [∇y2 f (x¯ , y¯ ) − z¯ + ∇p2 g (x¯ , y¯ , p¯ )] = 0, (32) ( − ) 1 + 1 − 1 1 ∈ ( 1) ( ) α γ y β δ p ND1 z¯ , 33 2 2 2 2 β − δ p ∈ ND2 (z¯ ), (34) 1 T 1 1 1 (ξ ) x¯ = S(x |E1), ξ ∈ C1, (35) 2 T 2 2 2 (ξ ) x¯ = S(x |E2), ξ ∈ C2, (36) ηy¯2 = 0, (37)

(α, β1, β2, γ, δ1, δ2, η) 6= 0, (38)

(α, β1, β2, γ, δ1, δ2, η) ≥ 0. (39)

Premultiplying Equations (26) and (27) by (β1 − αp¯1 − γy¯1 − δ1 p¯1) and (β2 − αp¯2 − αy¯2 − δ2 p¯2), respectively, and then using Equations (28)–(32), we get

1 1 1 1 1 T 1 1 1 1 1 1 1 1 1 (β − αp¯ − γy¯ − δ p¯ ) ∇p1 p1 g (x¯ , y¯ , p¯ )(β − αp¯ − γy¯ − δ p¯ ) = 0, (40) and

2 2 2 2 2 2 2 2 2 2 2 2 2 2 (β − αp¯ − αy¯ − δ p¯ )∇p2 p2 g (x¯ , y¯ , p¯ )(β − αp¯ − αy¯ − δ p¯ )

2 2 2 2 2 2 2 2 2 2 = −αδ y¯ [∇y2 f (x¯ , y¯ ) − z¯ + ∇p2 g (x¯ , y¯ , p¯ )]. (41)

Using hypothesis (i) in (40), we get

β1 = αp¯1 + γy¯1 + δ1 p¯1. (42) Symmetry 2020, 12, 274 11 of 18

Further, using inequalities (4), (8), and (39) in (41), we obtain

2 2 2 2 2 2 2 2 2 2 2 2 2 2 (β − αp¯ − αy¯ − δ p¯ )∇p2 p2 g (x¯ , y¯ , p¯ )(β − αp¯ − αy¯ − δ p¯ ) ≥ 0,

which on using hypothesis (i), we obtain

β2 = αp¯2 + αy¯2 + δ2 p¯2. (43)

From Equations (26), (27), along with hypothesis (ii), we obtain

δ1 = 0 (44) and δ2 = 0. (45)

Now suppose, α = 0. Then, Equations (43) and (45) give β2 = 0. The Equation (24) with hypothesis (ii) implies γ = 0, and Equation (42) yields β1 = 0. Therefore, Equation (25) implies η = 0. Thus (α, β1, β2, γ, δ1, δ2, η) = 0, which contradicts the Equation (38). Therefore,

α > 0. (46)

From Equations (28), (30) and (31), we have

1 1 1 1 T 1 1 1 1 1 1 1 1 (β − γy¯ − δ p¯ ) [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0, which along with (42), gives

1 T 1 1 1 1 1 1 1 1 α(p¯ ) [∇y1 f (x¯ , y¯ ) − z¯ + ∇p1 g (x¯ , y¯ , p¯ )] = 0. (47) Using hypothesis (iii), we get p¯1 = 0. (48)

Further, from Equations (25), (37), and (43), we obtain

2 2 2 2 2 2 2 2 2 2 2 2 2 αy¯ [∇y2 g (x¯ , y¯ , p¯ ) − ∇p2 g (x¯ , y¯ , p¯ ) + ∇y2y2 f (x¯ , y¯ )p¯ ] = 0, (49) which by hypothesis (iii) and Equation (46) implies

p¯2 = 0. (50)

Therefore, (42) and (43) reduce to β1 = γy¯1 (51) and β2 = αy¯2. (52)

Now from Equations (24), (42), and (48) with hypotheses (ii) and (iv), we get

α = γ > 0. (53)

Therefore, Equation (52) implies β2 y¯2 = ≥ 0. (54) α Symmetry 2020, 12, 274 12 of 18

Now, Equations (22) and (23) along with (42), (43), (50) and hypothesis (iv), we obtain

1 1 1 ¯1 1 1 1 1 1 1 1 α[∇x1 f (x¯ , y¯ ) + ξ + ∇r1 g (x¯ , y¯ , p¯ )](x − x¯ ) ≥ 0, for all x ∈ C1 (55) and 2 2 2 ¯2 2 2 2 2 2 2 2 α[∇x2 f (x¯ , y¯ ) + ξ + ∇r2 g (x¯ , y¯ , p¯ )](x − x¯ ) ≥ 0, for all x ∈ C2. (56) 1 1 1 Let x ∈ C1, then x¯ + x ∈ C1. Then above inequality (55) implies

1 T 1 1 1 ¯1 1 1 1 1 1 (x¯ ) [∇x1 f (x¯ , y¯ ) − ξ + ∇r1 g (x¯ , y¯ , p¯ )] ≥ 0, for all x ∈ C1. (57) Similarly, 2 T 2 2 2 ¯2 2 2 2 2 2 (x¯ ) [∇x2 f (x¯ , y¯ ) + ξ + ∇r2 g (x¯ , y¯ , p¯ )] ≥ 0, for all x ∈ C2. (58) Therefore,

1 1 1 ¯1 1 1 1 1 ∗ ∇x1 f (x¯ , y¯ ) + ξ + ∇r1 g (x¯ , y¯ , p¯ ) ∈ C1 (59)

2 2 2 ¯2 2 2 2 2 ∗ ∇x2 f (x¯ , y¯ ) + ξ + ∇r2 g (x¯ , y¯ , p¯ ) ∈ C2 . (60) If we take ξ¯1 = w¯ 1 and ξ¯2 = w¯ 2, then (x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1 = 0, r¯2 = 0) satisﬁes the dual constraints (10)–(15). Therefore, it is feasible solution for (MHD).

2 2 Now from Equations (34), (46), (50), and (52), we get y¯ ∈ ND2 (z¯ ). As D2 is a compact convex set | | in R A2 , 2 T 2 2 (y¯ ) z¯ = S(y¯ |D2). (61) Moreover, since β1 = γy¯1 and γ > 0 then from Equations (33), (48), and (53), we obtain y¯1 ∈ ( 1) |A1| ND1 z¯ . As D1 is a compact convex set in R ,

1 T 1 1 (y¯ ) z¯ = S(y¯ |D1). (62)

Therefore by Equations (35), (36), (48), (50), (61), (62), and hypothesis (iv), we get the equal objective functions value of (MHP) and (MHD) at (x¯1, x¯2, y¯1, y¯2, z¯1, z¯2, p¯1 = 0, p¯2 = 0) and (x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1 = 0, r¯2 = 0). From Theorem1, we get that (x¯1, x¯2, y¯1, y¯2, w¯ 1, w¯ 2, r¯1 = 0, r¯2 = 0) is an optimal solution for (MHD).

Theorem 3. (Converse Duality) Let (u¯1, u¯2, v¯1, v¯2, w¯ 1, w¯ 2, r¯1, r¯2) be an optimal solution of (MHD). Suppose that

1 1 1 1 2 2 2 2 (i) ∇r1r1 h (u¯ , v¯ , r¯ ) is positive or negative deﬁnite matrix and ∇r2r2 h (u¯ , v¯ , r¯ ) is negative deﬁnite matrix,

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 (ii) ∇v1 f (u¯ , v¯ ) + w¯ + ∇r1 h (u¯ , v¯ , r¯ ) 6= 0 and ∇v2 f (u¯ , v¯ ) + w¯ + ∇r2 g (u¯ , v¯ , r¯ ) 6= 0,

1 T 1 1 1 1 1 1 1 1 1 (iii) (r¯ ) [∇u1 f (u¯ , v¯ ) + w¯ + ∇r1 h (u¯ , v¯ , r¯ )] = 0 ⇒ r¯ = 0 and 2 2 2 2 2 2 2 2 2 2 2 2 2 2 u¯ [∇u2 h (u¯ , v¯ , r¯ ) − ∇r2 h (u¯ , v¯ , r¯ ) + ∇u2u2 f (u¯ , v¯ )r¯ ] = 0 ⇒ r¯ = 0,

1 1 1 1 1 1 1 1 1 1 1 1 (iv) h (u¯ , v¯ , 0) = g (u¯ , v¯ , 0), ∇u1 h (u¯ , v¯ , 0) = ∇p1 g (u¯ , v¯ , 0), 1 1 1 1 1 1 2 2 2 2 2 2 ∇v1 h (u¯ , v¯ , 0) = ∇r1 h (u¯ , v¯ , 0), h (u¯ , v¯ , 0) = g (u¯ , v¯ , 0) and 2 2 2 2 2 2 ∇u2 h (u¯ , v¯ , 0) = ∇p2 g (u¯ , v¯ , 0). Symmetry 2020, 12, 274 13 of 18

Then,

(I) (u¯1, u¯2, v¯1, v¯2, z¯1, z¯2, p¯1 = 0, p¯2 = 0) is feasible for (MHP), and

(II) G(u¯1, u¯2, v¯1, v¯2, z¯1, z¯2, p¯1, p¯2) = H(u¯1, u¯2, v¯1, v¯2, w¯ 1, w¯ 2, r¯1, r¯2).

Furthermore, if the hypotheses of Theorem1 are satisﬁed for all feasible solutions of (MHP) and (MHD), then (u¯1, u¯2, v¯1, v¯2, z¯1, z¯2, p¯1 = 0, p¯2 = 0) is an optimal solution for (MHP).

Proof. The proof follows on the line of Theorem2.

5.1. Self Duality In general, primal problem (MHP) and dual problem (MHD) are not self-duals without including 1 |A | |B | 2 |A | |B | restrictions on f , h and g. If we assume D1 = D2, E1 = E2, f : R 1 × R 1 → R, f : R 2 × R 2 → | | | | | | | | | | | | | | | | | | R, h1 : R A1 × R B1 × R A1 → R, h2 : R A2 × R B2 × R A2 → R, g1 : R A1 × R B1 × R B1 → R, | | | | | | g2 : R A2 × R B2 × R B2 → R are skew symmetric, that is,

f i(u1, v1) = − f i(v1, u1), i = 1, 2,

hi(u1, v1, r1) = −hi(v1, u1, r1), i = 1, 2,

C1 = C3 and C2 = C4. By rewriting the dual problem (MHD) as a minimization problem, we get

1 1 1 1 2 2 2 1 T 1 Minimize H(u, v, w, r) = −{ f (u , v ) − S(v |D1) + f2(u , v ) − S(v |D2) + (u ) w

1 1 1 1 2 2 2 1 T 1 1 1 1 2 T 2 2 2 2 + h (u , v , r ) + h2(u , v , r ) − (r ) ∇r1 h (u , v , r ) − (r ) ∇r2 h (u , v , r )

2 T 2 2 2 2 2 2 2 −(u ) [∇u2 f (u , v ) + ∇r2 h (u , v , r )]} subject to 1 1 1 1 1 1 1 1 ∗ ∇u1 f (u , v ) + w + ∇r1 h (u , v , r ) ∈ C1 , 2 2 2 2 2 2 2 2 ∗ ∇u2 f (u , v ) + w + ∇r2 h (u , v , r ) ∈ C2 , 1 T 1 1 1 1 1 1 1 1 (u ) [∇u1 f (u , v ) + w + ∇r1 h (u , v , r )] ≤ 0, 1 T 1 1 1 1 1 1 1 1 (r ) [∇u1 f (u , v ) + w + ∇r1 h (u , v , r )] ≤ 0, 2 T 2 2 2 2 2 2 2 2 (r ) [∇u2 f (u , v ) + w + ∇r2 h (u , v , r )] ≤ 0, 1 2 2 v ∈ C3, v ∈ C4, u ≥ 0, As f , h and g are skew symmetric, therefore

1 1 1 1 1 1 2 2 2 2 2 2 ∇u1 f (u , v ) = −∇u1 f (v , u ), ∇u1 f (u , v ) = −∇u1 f (v , u ),

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 ∇u1 h (u , v , r ) = −∇u1 h (v , u , r ), and ∇u1 h (u , v , r ) = −∇u1 h (v , u , r ).

Now the above problem becomes: Symmetry 2020, 12, 274 14 of 18

1 1 1 1 2 2 2 2 1 T 1 Minimize H(u, v, w, r) = f (v , u ) + S(v |D1) + f (v , u ) + S(v |D2) − (u ) w

1 1 1 1 2 2 2 2 1 T 1 1 1 1 +h (v , u , r ) + h (v , u , r − (r ) ∇r1 h (v , u , r )

2 T 2 2 2 2 2 T 2 2 2 2 2 2 2 −(r ) ∇r2 h (v , u , r ) − (u ) [∇u2 f (v , u ) + ∇r2 g (v , u , r )] subject to

1 1 1 1 1 1 1 1 ∗ −(∇u1 f (v , u ) + w + ∇r1 h (v , u , r )) ∈ C3 , 2 2 2 2 2 2 2 2 ∗ −(∇u2 f (v , u ) + w + ∇r2 h (v , u , r )) ∈ C4 , 1 T 1 1 1 1 1 1 1 1 (u ) [∇u1 f (v , u ) + w + ∇r1 h (v , u , r )] ≥ 0, 1 T 1 1 1 1 1 1 1 1 (r ) [∇u1 f (v , u ) + w + ∇r1 h (v , u , r )] ≥ 0, 2 T 2 2 2 2 2 2 2 2 (r ) [∇u2 f (v , u ) + w + ∇r2 h (v , u , r )] ≥ 0, 1 2 2 v ∈ C1, v ∈ C2, u ≥ 0, which is identical to (MHP), that is, the objective function and the constraint functions of (MHP) and (MHD) are identical. Thus, (MHP) is a self-dual.

Thus, if (x1, x2, y1, y2, w1, w2, p1, p2) is feasible for (MHP), then (y1, y2, x1, x2, w1, w2, p1, p2) is feasible for (MHD) and vice versa.

The weak and strong duality theorems are veriﬁed by the following example.

5.2. Example

2 2 2 2 x+|x| Let f (x, y) = (x1 + y1 + x1 + y1 + x2 + y2 + x2 + y2), s(x|E) = 2 , g(x, y, p) = 1 T 1 T 2 2 p (∇yy f (x, y)p), h(x, y, r) = 2 r (∇yy f (x, y)r) and C1 = C2 = C3 = C4 = R+. Then, the problems (MHP) and (MHD) are as follows.

Primal Problem (MHP):

Minimize G(x, y, z, p) =

2 2 2 2 x1+|x1| x2+|x2| 2 2 x1 + x2 + x1 + x2 + y1 + y2 + y1 + y2 + 2 + 2 − y1z1 − p1 − p2 − y2(2y2 + 1 + 2p2) subject to

(2y1 + 1 − z1 + 2p1) ≤ 0, (2y2 + 1 − z2 + 2p2) ≤ 0, 2 (2y1 + y1 − y1z1 + 2p1y1) ≥ 0, 2 (2p1y1 + p1 − p1z1 + 2p1) ≥ 0, 2 (2p2y2 + p2 − p2z2 + 2p2) ≥ 0, x, z, y2 ≥ 0.

Dual Problem (MHD):

Maximize H(u, v, w, r) =

2 2 2 2 v1+|v1| v2+|v2| 2 2 u1 + u2 + u1 + u2 + v1 + v2 + v1 + v2 − 2 − 2 + u1w1 − r1 − r2 − u2(2u2 + 1 + 2r2) Symmetry 2020, 12, 274 15 of 18 subject to

(2u1 + 1 + w1 + 2r1) ≥ 0, (2u2 + 1 + w2 + 2r2) ≥ 0, 2 (2u1 + u1 + u1w1 + 2r1u1) ≤ 0, 2 (2r1u1 + r1 + r1w1 + 2r1) ≤ 0, 2 (2r2u2 + r2 + r2w2 + 2r2) ≤ 0, v, w, u2 ≥ 0,

where

(i) f (x, y) : R2 × R2 −→ R,

(ii) g(x, y, p) : R2 × R2 × R2 −→ R and

(iii) h(x, y, r) : R2 × R2 × R2 −→ R.

! 0 Let x¯ = y¯ = u¯ = v¯ = w¯ = z¯ = p¯ = r¯ = . 0

Then, (x, y, z, p) is feasible for (MHP) and (u, v, w, r) is feasible for (MHD).

1 T 1 2 1 Now, f (., v) + (.) w = (u1 + u1 + u1w1) is a higher-order F- pseudo-convex function at u¯ = 0 1 1 T 1 2 with respect to h and − f (x,.) + (.) z = (−y1 − y1 + y1z1) is a higher-order F-pseudo-convex 1 1 2 T 2 2 function at y¯ = 0 with respect to −g . Now, f (., v) + (.) w = (u2 + u2 + u2w2) is a higher-order 2 2 2 T 2 2 F-convex function at u¯ = 0 with respect to h . Similarly, − f (x,.) + (.) z = (−y2 − y2 + y2z2) is a higher-order F-convex function at y¯2 = 0 with respect to −g2.

Also conditions (v) to (viii) of weak duality theorem (Theorem1) are satisﬁed for ! 0 x¯ = y¯ = u¯ = v¯ = w¯ = z¯ = p¯ = r¯ = . Therefore, weak duality theorem is satisﬁed. 0

Furthermore,

! 1 2 0 (i) ∇ 1 1 g (x¯, y¯, p¯) = is positive deﬁnite matrix p p 0 2

and ! 2 −2 0 ∇ 2 2 g (x¯, y¯, p¯) = is negative deﬁnite matrix, p p 0 −2 ! 1 1 1 2 2 2 1 (ii) {∇ 1 f (x¯, y¯) − z¯ + ∇ 1 g (x¯, y¯, p¯), ∇ 2 f (x¯, y¯) − z¯ + ∇ 2 g (x¯, y¯, p¯)} = , y p y p 1

1 T 1 1 1 1 (iii) (p¯ ) {∇y1 f (x¯, y¯) − z¯ + ∇p1 g (x¯, y¯, p¯)} = 0 =⇒ p¯ = 0 and 2 2 2 2 2 2 2 2 2 2 2 2 2 2 y¯ [∇y2 g (x¯ , y¯ , p¯ ) − ∇p2 g (x¯ , y¯ , p¯ ) + ∇y2y2 f (x¯ , y¯ )p¯ ] = 0 =⇒ p¯ = 0.

(As p¯ initially taken as zero).

i i i i 1 1 1 1 1 1 (iv) h (x¯, y¯, 0) = g (x¯, y¯, 0), ∇xh (x¯, y¯, 0) = ∇r g (x¯, y¯, 0), ∇y1 g (x¯ , y¯ , 0) = ∇p1 g (x¯ , y¯ , 0), for i = 1, 2. Symmetry 2020, 12, 274 16 of 18

Therefore, all the assumptions of strong duality theorem (Theorem2) are also satisﬁed. Now objective function value is equal to 0.

6. Special Cases

n h (i) Let C1 = R +, C2 = R +, |B1| = 0, |A1| = 0 , E1 = 0 and D1 = 0 in (MHP) and (MHD). Then, we get the programs proposed by Verma and Gulati [21].

n h (ii) If C1 = R +, C3 = R +, removed the higher-order and non-differentiable terms and omission of inequalities (6), (7), (13), and (14), then we get the model presented by Chandra et al. [9].

n h (iii) Let C1 = R + and C3 = R +, |A2| = 0, |B2| = 0, p = 0 and r = 0. Then, our programs are reduced to the programs presented in [25].

(iv) If C1 = −C1, C3 = −C3, |B2| = 0, |A2| = 0, E1 = 0, D1 = 0 and omission of inequalities (7) and (14), then we get the dual programs discussed in [26].

(v) If C2 = −C2, C4 = −C4, |B1| = 0, |A1| = 0, E2 = 0, D2 = 0 and omission of inequalities (6) and (13), then we get the dual programs studied in [26].

T (vi) If C2 = −C2, C4 = −C4, |B1| = 0, |A1| = 0, E2 = 0, D2 = 0, g(x, y, p) = (1/2)p ∇yy f (x, y)p T and h(u, v, r) = (1/2)r ∇xx f (u, v)r, and omission of inequalities (6) and (13), then we get the dual programs is equal to obtained in [27].

n h For C2 = R +, C4 = R +, and omission of inequalities (6) and (13), we get the following special cases (vii) to (viii):

(vii) If |B1| = 0 and |A1| = 0 , then we get the programs considered in [17].

T T (viii) Let |B1| = 0, |A1| = 0, g(x, y, p) = (1/2)p ∇yy f (x, y)p and h(u, v, r) = (1/2)r ∇xx f (u, v)r. Then, our programs become the programs presented in [16].

n h For C1 = R +, C3 = R +, and omission of inequalities (7) and (14), we get the following special cases (ix) to (xiii):

T T (ix) If g(x, y, p) = (1/2)p ∇yy f (x, y)p and h(u, v, r) = (1/2)r ∇xx f (u, v)r, then we get the programs obtained in [15].

T T (x) If |B2| = 0, |A2| = 0, g(x, y, p) = (1/2)p ∇yy f (x, y)p and h(u, v, r) = (1/2)r ∇xx f (u, v)r , then we get the dual programs derived in [19].

T (xi) If |B2| = 0, |A2| = 0, E1 = 0, D1 = 0, g(x, y, p) = (1/2)p ∇yy f (x, y)p and T h(u, v, r) = (1/2)r ∇xx f (u, v)r, then we get the dual programs studied in [4].

T T (xii) If |B2| = 0, |A2| = 0, g(x, y, p) = (1/2)p ∇yy f (x, y)p, h(u, v, r) = (1/2)r ∇xx f (u, v)r, E1 = 0 and D1 = 0, then we get the dual programs presented in [28].

(xiii) Let C1 = −C1, C3 = −C3, |A2| = 0, |B2| = 0, p = 0, r = 0, E1 = 0 and D1 = 0. Then, we obtain symmetric dual programs [29,30]. Symmetry 2020, 12, 274 17 of 18

7. Conclusions A new mixed type nondifferentiable higher-order symmetric dual programs over cones has been formulated. It is also shown the uniqueness of higher-order F-convexity/higher-order F-pseudoconvexity by a non-trivial example. Under these assumptions, the weak, strong, and converse duality theorems have been derived, and veriﬁed by an example. Self-duality has been also discussed. The presented programs and results generalize some existing duals and their corresponding theorems appeared in the literature. The symmetric duality between (MHP) and (MHD) can be extended for fractional and multiobjective programming problems over cones.

Author Contributions: All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript. Funding: Deanship of Research, King Fahd University of Petroleum and Minerals, Saudi Arabia, Project No. IN171012. Acknowledgments: The first and third authors would like to thank the King Fahd University of Petroleum and Minerals, Saudi Arabia to provide the financial support under the Internal Research Project no. IN171012. The authors are thankful to referees for their valuable suggestions which improved the results and presentation of this article. Conflicts of Interest: The authors declare no conflicts of interest.

Appendix A The Figures1–4 are plotted in Wolfram Mathematica. The codes are given below: Figure1: Plot3D[(−u2 + x2)(−e( − u) + 2u + 1/(1 + u)), x, 0, 5, u, 0, 5] Figure2: Plot[(−1 + x2)(−e( − 1) + 2 + 1/2), x, 0, 5, Frame → True, FrameLabel → x] Figure3: Plot[(e( − x) + x2 − 1/e − 1), x, 0, 5, Frame → True, FrameLabel → x] Figure4: Plot[e( − x) + x2 − 1/e − 1 − 2.1321205588(x2 − 1), x, 0, 4, Frame → True, FrameLabel → x].

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