# Generalized Minimax Programming with  Nondifferentiable (G, β)-Invexity

Authors: D. H. Yuan; X. L. Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We consider the generalized minimax programming problem (P) in which functions are locally Lipschitz (G, β)-invex.
Not only G-sufficient but also G-necessary optimality conditions are established for problem (P).
With G-necessary optimality conditions and (G, β)-invexity on hand, we construct dual problem (DI) for the
primal one (P) and prove duality results between problems (P) and (DI). These results extend several known results to a wider class of programs.


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## Body

## 1. Introduction

Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality. Based on convexity assumptions, nonlinear programming problems can be solved efficiently. There have been many attempts to weaken the convexity assumptions in order to treat many practical problems. Therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature [1]. One of these concepts, invexity, was introduced by Hanson in [2]. Hanson has shown that invexity has a common property in mathematical programming with convexity that Karush-Kuhn-Tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. Ben-Israel and Mond [3] introduced the concept of preinvex functions which is a special case of invexity.Recently, Antczak extended further invexity toG-invexity [4] for scalar differentiable functions and introduced new necessary optimality conditions for differentiable mathematical programming problem. Antczak also applied the introduced G-invexity notion to develop sufficient optimality conditions and new duality results for differentiable mathematical programming problems. Furthermore, in the natural way, Antczak’s definition of G-invexity was also extended to the case of differentiable vector-valued functions. In [5], Antczak defined vector G-invex (G-incave) functions with respect to η and applied this vector G-invexity to develop optimality conditions for differentiable multiobjective programming problems with both inequality and equality constraints. He also established the so-called G-Karush-Kuhn-Tucker necessary optimality conditions for differentiable vector optimization problems under the Kuhn-Tucker constraint qualification [5]. With this vector G-invexity concept, Antczak proved new duality results for nonlinear differentiable multiobjective programming problems [6]. A number of new vector duality problems such as G-Mond-Weir, G-Wolfe, and G-mixed dual vector problems to the primal one were also defined in [6].In the last few years, many concepts of generalized convexity, which include(p,r)-invexity [7], (F,ρ)-convexity [8], (F,α,ρ,d)-convexity [9], (C,α,ρ,d)-convexity [10], (ϕ,ρ)-invexity [11], V-r-invexity [12], and their extensions, have been introduced and applied to different mathematical programming problems. In particular, they have also been applied to deal with minimax programming; see [13–17] for details. However, we have not found a paper which deals with generalized minimax programming problem (P) under G-invexity or its generalizations assumptions.Note that the functionG∘f may not be differentiable even if the function G is differentiable. Yuan et al. [18] introduced the (Gf,βf)-invexity concept for the locally Lipschtiz function f. This (Gf,βf)-invexity extended Antczak’s G-invexity concept to the nonsmooth case. In this paper, we deal with nondifferentiable generalized minimax programming problem (P) with the vector (G,β)-invexity proposed in [18]. Here, the generalized minimax programming problem (P) is presented as follows:
(P)minsupy∈Yϕ(x,y)subjecttogj(x)≤0,j=1,…,m,
where Y is a compact subset of ℝp, ϕ(·,·):ℝn×ℝp→ℝ, and gj(·):ℝn→ℝ(j∈M). Let EP be the set of feasible solutions of problem (P); in other words, EP={x∈ℝn∣gj(x)≤0,j∈M}. For convenience, let us define the following sets for every x∈E:
(1)J(x)={j∈M∣gj(x)=0},Y(x)={y∈Y∣ϕ(x,y)=supz∈Yϕ(x,z)}.The rest of the paper is organized as follows. In Section2, we present concepts in regards to nondifferentiable vector (G,β)-invexity. In Section 3, we present not only G-sufficient but also G-necessary optimality conditions for problem (P). When the G-necessary optimality conditions and the (G,β)-invexity concept are utilized, dual problem (DI) is formulated for the primal one (P) and duality results between them are presented in Section 4.


## 2. Notations and Preliminaries

In this section, we provide some definitions and results that we will use in the sequel. The following convention for equalities and inequalities will be used throughout the paper. For anyx=(x1,x2,…,xn)T, y=(y1,y2,…,yn)T, we define the following:
(2)x>yifandonlyifxi>yi,fori=1,2,…,n,x≧yifandonlyifxi≥yi,fori=1,2,…,n,x⩾yifandonlyifxi≥yi,fori=1,2,…,n,butx≠y,x≯yisthenegationofx>y,x⩾yisthenegationofx⩾y.
Let ℝ+n={x∈ℝn|x≧0}, ℝ˙+n={x∈ℝn|x>0} and X be a subset of ℝn. For our convenience, denote Q:={1,2,…,q}, Q*:={1,2,…,q*}, K:={1,2,…,k}, and M:={1,2,…,m}. Further, we recall some definitions and a lemma.Definition 1 (see [19]).
Letd∈ℝn, X be a nonempty set of ℝn and f:X→ℝ. If
(3)f0(x;d)∶=limy→xμ↓0sup1μ(f(y+μd)-f(y))
exists, then f0(x;d) is called the Clarke derivative of f at x in the direction d. If this limit superior exists for all d∈ℝn, then f is called the Clarke differentiable at x. The set
(4)∂f(x)={ζ∣f0(x;d)≥〈ζ,d〉,∀d∈ℝn}
is called the Clarke subdifferential of f at x.Note that if a given functionf is locally Lipschitz, then the Clarke subdifferential ∂f(x) exists.Lemma 2 (see [18]).
Letψ be a real-valued Lipschitz continuous function defined on X and denote the image of X under ψ by Iψ(X); let φ:Iψ(X)→ℝ be a differentiable function such that φ′(γ) is continuous on Iψ(X) and φ′(γ)≥0 for each γ∈Iψ(X). Then the chain rule
(5)(φ∘ψ)0(x,d)=φ′(ψ(x))ψ0(x,d),
holds for each d∈ℝn. Therefore,
(6)∂(φ∘ψ)(x)=φ′(ψ(x))∂(ψ)(x).Definition 3.
Letf=(f1,…,fk) be a vector-valued locally Lipschitz function defined on a nonempty set X⊂ℝn. Consider the functions η:X×X→ℝn, Gfi:Ifi(X)→ℝ, and βif:X×X→ℝ+ for i∈K. Moreover, Gfi is strictly increasing on its domain Ifi(X) for each i∈K. If
(7)Gfi∘fi(x)-Gfi∘fi(u)≥(>)βif(x,u)Gfi′(fi(u))〈ζi,η(x,u)〉,∀ζi∈∂fi(u),
holds for all x∈X(x≠u) and i∈K, then f is said to be a (strictly) nondifferentiable vector (Gf,βf)-invex at u on X (with respect to η) (or shortly, (Gf,βf)-invex at u on X), where Gf=(Gf1,…,Gfk) and β:=(β1f,β2f,…,βkf). If f is a (strictly) nondifferentiable vector (Gf,βf)-invex at u on X(with respect to η) for all u∈X, then f is a (strictly) nondifferential vector (Gf,βf)-invex on X(with respect to η).


## 3. Optimality Conditions

In this section, we firstly establish theG-necessary optimality conditions for problem (P) involving functions which are locally Lipschitz with respect to the variable x. For this purpose, we will need some additional assumptions with respect to problem (P).Condition 4.
Assume the following: (a) the setY is compact;
(b)ϕ(x,y) and ∂ϕx(x,y) are upper semicontinuous at (x,y);
(c)ϕ(x,y) is locally Lipschitz in x and this Lipschitz continuity is uniform for y in Y;
(d)ϕ(x,y) is regular in x; that is, ϕx∘(x,y;·)=ϕx′(x,y;·);
(e)gj,j∈M are regular and locally Lipschitz at x0.Condition 5.
For eachη=(η1,…,ηm)∈ℝm satisfying the conditions
(8)ηj=0,∀j∈M∖J(x*),ηj≥0,∀j∈J(x*),
the following implication holds:
(9)zj*∈∂gj(x*)(∀j∈M),∑j=1m‍ηjzj*=0⇒ηj=0,j∈M.We will also use the following auxiliary programming problem(G-P):
(G-P)minsupy∈YGϕ∘ϕ(x,y)s.t.Gg∘g(x)∶=(Gg1∘g1(x),Gg2∘g2(x),…,Ggm∘gm(x))≦Gg(0),
where Gg(0):=(Gg1(0),Gg2(0),…,Ggm(0)). We denote by EG-P={x∈ℝn∣Gg∘g(x)≦Gg(0)}, J′(x-):={j∈M:Ggj∘gj(x-)=Ggj(0)}. If function Ggj is strictly increasing on Igj(X) for each j∈M, then EP=EG-P and J(x-)=J′(x-). So, we represent the set of all feasible solutions and the set of constraint active indices for either (P) or (G-P) by the notations E and J(x-), respectively.The following necessary optimality conditions are presented in [20].Theorem 6 (necessary optimality conditions).
Letx* be an optimal solution of (P). One also assumes that Conditions 4 and 5 hold. Then there exist positive integer q* and vectors yi∈Y(x*) together with scalars λi*(i∈Q*) and μj*(j∈M) such that
(10)0∈∑i=1q*‍λi*∂xϕ(x*,yi)+∑j=1m‍μj*∂gj(x*),μj*gj(x*)=0,μj*≥0,j∈M,∑i=1q*‍λi*=1,λi*>0,i∈Q*.
Furthermore, if α is the number of nonzero λi* and β is the number of nonzero μj*, then
(11)1≤α+β≤n+1.Making use of Theorem6, we can derive the following G-necessary conditions theorem for problem (P), see Theorem 7, here we require the scalars λi*(i=1,…,q*) to be positive.Theorem 7 (G-necessary optimality conditions).
Let problem(P) satisfy Conditions 4 and 5; let x* be an optimal solution of problem (P). Assume that Gϕ is both continuously differentiable and strictly increasing on Iϕ(X,Y). If Ggj is both continuously differentiable and strictly increasing on Igj(X) with Ggj′(gj(x*))>0 for each j∈M, then there exist positive integer q*(1≤q*≤n+1) and vectors yi∈Y(x*) together with scalars λi*(i∈Q*) and μj*(j∈M) such that
(12)0∈∑i=1q*‍λi*Gϕ′(ϕ(x*,yi))∂xϕ(x*,yi)+∑j=1m‍μj*Ggj′(gj(x*))∂gj(x*),(13)μj*(Ggj∘gj(x*)-Ggj(0))=0,μj*≥0,j∈M,(14)∑i=1q*‍λi*=1,λi*≥0,i∈Q*.Proof.
Sincex* is an optimal solution to problem (P), it is easy to see that x* is an optimal solution to problem (G-P). Consider problem (G-P), it is easy to check that problem (G-P) satisfies the assumptions of Theorem 6. Therefore, we choose yi∈Y(x*) and i∈Q* with q*≤n+1, such that they satisfy Theorem 6.
Now, for eachyi, we consider the scalar programming (G-Pyi) as follows:
(G-Pyi)minGϕ∘ϕ(x,yi)subjecttoGgj∘gj(x)≤0,j∈M.
It is easy to see that x* is an optimal solution to problem (G-Pyi). Thus, there exist λi>0 and μji≥0 for j∈M such that
(15)0∈λi∂x(Gϕ∘ϕ)(x*,yi)+∑j=1m‍μji*∂(Ggj∘gj)(x*),(16)μji(Ggj∘gj(x*)-Ggj(0))=0,μji≥0,j∈M.
So, we obtain from (15) that
(17)0∈∑i=1q*‍λi∂x(Gϕ∘ϕ)(x*,yi)+∑j=1m‍(∑i=1q*‍μji)∂(Ggj∘gj)(x*),
or
(18)0∈∑i=1q*‍λi*(Gϕ∘ϕ)(x*,yi)+∑j=1m‍μj*∂(Ggj∘gj)(x*),
where λi*=λi/∑j=1q*‍λj, μj*=∑i=1q*‍μji/∑i=1q*λi‍. By Lemma 2, we have
(19)∂x(Gϕ∘ϕ)(x*,yi)=Gϕ′(ϕ(x*,yi))∂xϕ(x*,yi),i∈Q*,∂(Ggj∘gj)(x*)=Ggj′(gj(x*))∂gj(x*),j∈M.
Now, from (18), we can deduce the required results.Next, we deriveG-sufficient optimality conditions for problem (P) under the assumption of (G,β)-invexity proposed in [18].Theorem 8 (G-sufficient optimality conditions).
Let(x*,μ*,ν*,q*,λ*,y-) satisfy conditions (12)–(14), where ν*=ϕ(x*,y1)=⋯=ϕ(x*,yq*); let Gϕ be both continuously differentiable and strictly increasing on Iϕ(X,Y); let Ggj be both continuously differentiable and strictly increasing on Igj(X) for each j∈M. Assume that ϕ(·,yi) is (Gϕ,βiϕ)-invex at x* on E for each i∈Q* and gj is (Gjg,βjg)-invex at x* on E for each j∈M. Then, x* is an optimal solution to (P).Proof.
Suppose, contrary to the result, thatx* is not an optimal solution for problem (P). Hence, there exists x0∈E such that
(20)supy∈Yϕ(x0,y)<ϕ(x*,yi),i∈Q*.
By the monotonicity of Gϕ, we have
(21)Gϕ∘ϕ(x0,yi)<Gϕ∘ϕ(x*,yi),i∈Q*.
Employing (13), (14), and the fact that
(22)Ggj∘gj(x0)≤Ggj(0)=Ggj∘gj(x*),j∈J(x*),
we can write the following statement
(23)∑i=1q*λi*(Gϕ∘ϕ(x0,yi)-Gϕ∘ϕ(x*,yi))βiϕ(x0,x*)+∑j=1mμj*(Ggj∘gj(x0)-Ggj∘gj(x*))βjg(x0,x*)<0.
By the generalized invexity assumptions of ϕ(·,yi) and gj, we have
(24)Gϕ∘ϕ(x0,yi)-Gϕ∘ϕ(x*,yi)≥(>)βiϕ(x0,x*)Gϕ′(ϕ(x*,yi))〈ξiϕ,η(x0,x*)〉,∀ξiϕ∈∂xϕ(x*,yi),i∈Q*,Ggj∘gj(x0)-Ggj∘gj(x*)≥(>)βjg(x0,x*)Ggj′(gj(x*))〈ξjg,η(x0,x*)〉,∀ξjg∈∂gj(x*),j∈M.
Employing (24) to (23), we have
(25)∑i=1q*‍λi*Gϕ′(ϕ(x*,yi))〈ξiϕ,η(x0,x*)〉+∑j=1m‍μj*Ggj′(gj(x*))〈ξjg,η(x0,x*)〉<0
or
(26)〈∑i=1q*‍λi*Gϕ′(ϕ(x*,yi))ξiϕ+∑j=1m‍μj*Ggj′(gj(x*))ξjg,η(x0,x*)〉<0,
which implies that
(27)0∈∑i=1q*‍λi*Gϕ′(ϕ(x*,yi))∂xϕ(x*,yi)+∑j=1m‍μj*Ggj′(gj(x*))∂gj(x*).
This is a contradiction to condition (12).Example 9.
LetY=[0,1]. Define
(28)ϕ(x,y)={ex+2y,y≥xe2x+y,x<y,g(x)=|x-32|-12.
Then, ϕ(·,y) is (log,1)-invex at x=1 for each y∈Y, g is 1-invex at x=1, and
(29)f(x):=supy∈Yϕ(x,y)={ex+2,y≥xe2x+1,x<y.
Since
(30)log(ϕ(x,y))={x+2y,y≥x2x+y,x<y,
then
(31)∂xlog(ϕ(x,y))={1,y≥x[1,2],x=y2,x<y.
Consider x0=1. Since Y(x0)={1}, then we can assume that q=1. Therefore,
(32)0∈λGϕ′(ϕ(1,1))∂xϕ(1,1)-μgj′(1)=λ[1,2]-μ,
where  λ=μ=1. Now, from Theorem 8, we can say that x0=1 is an optimal solution to (P).


## 4. Duality

Making use of the optimality conditions of the preceding section, we present dual problem(DI) to the primal one (P) and establish G-weak, G-strong, and G-strict converse duality theorems. For convenience, we use the following notations:
(33)K(x)={∫(q,λ,y-)∈N×ℝ˙+q×ℝpq∣1≤q≤n+1,λ=(λ1,…,λq)∈ℝ˙+qwith∑i=1q‍λi=1,y-=(y1,…yq)withyi∈Y(x),i=1,…q∫}.H1(q,λ,y-) denotes the set of all triplets (z,μ,ν)∈ℝn×ℝ+m×ℝ+ satisfying
(34)0∈∑i=1q‍λiGϕ′(ϕ(z,yi))∂zϕ(z,yi)+∑j=1m‍μjGgj′(gj(z))∂gj(z),(35)ϕ(z,yi)≥ν,i=1,2,…,q,(36)μjgj(z)≥0,j∈M,(37)yi∈Y(z),(q,λ,y-)∈K(z).Our dual problem(DI) can be stated as follows:
(DI)max(q,λ,y-)∈K(z)sup(z,μ,ν)∈H1(q,λ,y-)ν.
Note that if H1(q,λ,y-) is empty for some triplet (q,λ,y-)∈K(z), then define sup(z,μ,ν)∈H1(q,λ,y-)ν=-∞.Theorem 1010 (G-weak duality).
Letx and (z,μ,ν,q,λ,y-) be (P)-feasible and (DI)-feasible, respectively; let Gϕ be both continuously differentiable and strictly increasing on Iϕ(X,Y); let Ggj be both continuously differentiable and strictly increasing on Igj(X) for each j∈M. If ϕ(·,yi) is (Gϕ,βiϕ)-invex at z for each i∈Q and gj is (Ggj,βjg)-invex at z for each j∈M, then
(38)supy∈Yϕ(x,y)⩾ν.Proof.
Suppose to the contrary thatsupy∈Yϕ(x,y)<ν. Therefore, we obtain
(39)ϕ(x,y)<ν≤ϕ(z,yi),yi∈Y(z),∀y∈Y,i∈Q.
Thus, we obtain from the monotonicity assumption of Gϕ that
(40)Gϕ∘ϕ(x,yi)<Gϕ∘ϕ(z,yi),yi∈Y(z),i∈Q.
Again, we obtain from the monotonicity assumption of Ggj and the fact
(41)gj(x)≤0,μjgj(z)≥0,μj≥0,j∈M
that
(42)Ggj∘gj(x)≤Ggj∘gj(z),j∈M.
Hence,
(43)∑i=1q‍λiGϕ∘ϕ(x,yi)-Gϕ∘ϕ(z,yi)βiϕ(x,z)+∑j=1m‍μjGgj∘gj(x)-Ggj∘gj(z)βjg(x,z)<0.
Similar to the proof of Theorem 8, by (43) and the generalized invexity assumptions of ϕ(·,yi) and gj, we have
(44)〈∑i=1q‍λiGϕ′(ϕ(z,yi))ξiϕ+∑j=1m‍μjGgj′(gj(z))ξjg,η(x,z)〉<0,
which follows that
(45)0∈∑i=1q‍λiGϕ′(ϕ(z,yi))∂xϕ(z,yi)+∑j=1m‍μjGgj′(gj(z))∂gj(z).
Thus, we have a contradiction to (34). So supy∈Yϕ(x,y)⩾ν.Theorem 11 (G-strong duality).
Let problem(P) satisfy Conditions 4 and 5; let x* be an optimal solution of problem (P). Suppose that Gϕ is both continuously differentiable and strictly increasing on Iϕ(X,Y) and Ggj is both continuously differentiable and strictly increasing on Igj(X) with Ggj′(gj(x*))>0 for each j∈M. If the hypothesis of Theorem 10 holds for all (DI)-feasible points (z,μ,ν,q,λ,y-), then there exist (q*,λ*,y-*)∈K(z) and (x*,μ*,ν*)∈H1(q*,λ*,y-*) such that (q*,λ*,y-*,x*,μ*,ν*) is a (DI) optimal solution, and the two problems (P) and (DI) have the same optimal values.Proof.
By Theorem7, there exists ν*=ϕ(x*,yi*),i=1,…,q*, satisfying the requirements specified in the theorem, such that (q*,λ*,y-*,x*,μ*,ν*) is a (DI) feasible solution, then the optimality of this feasible solution for (DI) follows from Theorem 10.Theorem 12 (G-strict converse duality).
Letx- and (z,μ,ν,q,λ,y-) be optimal solutions for (P) and (DI), respectively. Suppose that Gϕ is both continuously differentiable and strictly increasing on Iϕ(X,Y) and Ggj is both continuously differentiable and strictly increasing on Igj(X) for each j∈M. If ϕ(·,yi) is (Gϕ,βiϕ)-invex at z for each i∈Q and gj is (Ggj,βjg)-invex at z for each j∈M, then x-=z; that is, z is a (P)-optimal solution and supy∈Yϕ(x-,y)=ν.Proof.
Suppose to the contrary thatx-≠z. Similar to the arguments as in the proof of Theorem 8, there exist ξiϕ∈ϕ(z,yi) and ξjg∈gj(z) such that
(46)0=〈∑i=1q‍λiGϕ′(ϕ(z,yi))ξiϕ+∑j=1m‍μjGgj′(gj(z))ξjg,η(x-,z)〉<∑i=1q‍λiGϕ∘ϕ(x-,yi)-Gϕ∘ϕ(z,yi)βiϕ(x-,z)+∑j=1m‍μjGgj∘gj(x-)-Ggj∘gj(z)βjg(x-,z),∑j=1m‍μjGgj∘gj(x-)-Ggj∘gj(z)βjg(x-,z)≤0.
Therefore,
(47)∑i=1q‍λiGϕ∘ϕ(x-,yi)-Gϕ∘ϕ(z,yi)βiϕ(x-,z)>0.
From the above inequality, we can conclude that there exists i0∈Q, such that
(48)Gϕ∘ϕ(x-,yi0)-Gϕ∘ϕ(z,yi0)>0
or
(49)ϕ(x-,yi0)>ϕ(z,yi0).
It follows that
(50)supy∈Yϕ(x-,y)≥ϕ(x-,yi0)>ϕ(z,yi0)>ν.
On the other hand, we know from Theorem 10 that
(51)supy∈Yϕ(x-,y)=ν.
This contradicts to (50).


## 5. Conclusion

In this paper, we have discussed the applications of(G,β)-invexity for a class of nonsmooth minimax programming problem (P). Firstly, we established G-necessary optimality conditions for problem (P). Under the nondifferential (G,β)-invexity assumptions, we have also derived the sufficiency of the G-necessary optimality conditions for the same problem. Further, we have constructed a dual model (DI) and derived G-duality results between problems (P) and (DI). Note that many researchers are interested in dealing with the minimax programming under generalized invexity assumptions; see [1, 10, 11, 14–17]. However, we have not found results for minimax programming problems under the G-invexity or its extension assumptions. Hence, this work extends the applications of G-invexity to the generalized minimax programming as well as to the nonsmooth case.



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