We obtain a new Taylor's formula in terms of the k+1 order subdifferential of a Ck,1
function from Rn to Rm. As its applications in optimization problems, we build k+1 order sufficient optimality conditions of this kind of functions and k+1 order necessary conditions for strongly C-quasiconvex functions.

1. Introduction

For a function from Rn to R, Luc [1] studied the k+1 order subdifferential of it, established a Taylor-type formula in terms of such k+1 order subdifferential, and applied such Taylor-type formula to consider two-order optimality conditions in vector optimization and characterizations of quasiconvex functions. In vector optimization, notions of Pareto solution, weak Pareto solution, sharp minima and weak sharp minima are very important; see [2–14] and the references therein. Some authors have attained many necessary or sufficient optimality conditions in optimization problems. In particular, Zheng and Yang provided some results on sharp minima, and weak sharp minima for high-order smooth vector optimization problems in Banach spaces. By the tools of nonsmooth analysis, many optimality conditions were obtained; for examples, one can see [6, 7, 15, 16] and the references therein. Such optimality conditions play a key role in many issues of mathematical programming such as sensitivity analysis and error bounds.

Motivated by Luc [1] and Zheng and Yang [17], in this paper, we consider the k+1 order subdifferential and optimality conditions of a Ck,1 vector-valued function from Rn to Rm. We will first prove a new Taylor's formula in the terms of k+1 order subdifferential for Ck,1 functions from Rn to Rm, which is analogous to that for real-valued functions in [1]. Then, under the positive definiteness assumption of k+1 order subdifferential, we will use this formula to derive k+1 order optimality conditions of weak Pareto and Pareto solutions in the terms of k+1 order subdifferential for a Ck,1 function from Rn to Rm. Finally, we will define a kind of strongly C-quasiconvex functions and prove a necessary condition in the terms of (k+1)th order subdifferential for such kind of functions. Our results extend the corresponding results in [1] for Ck,1 functions from Rn to R to that for Ck,1 vector-valued functions from Rn to Rm and in [17] for functions in smooth setting to that in nonsmooth setting, respectively.

The outline of the paper is as follows. In the next section, we give some notions and preliminary results in vector optimization problems. In Section 3, we build our Taylor's formula in the terms of k+1 order subdifferential for a Ck,1 function from Rn to Rm. In Section 4, as applications in optimization problems, we establish some optimality conditions in terms of (k+1)th order subdifferential. In Section 5, we give a necessary condition in the terms of (k+1)th order subdifferential for a strongly C-quasiconvex vector-valued function.

2. Preliminaries

Let X,Y be Banach spaces, Y* the dual space of Y, C⊂Y a closed convex cone with int(C)≠∅, and C+ the dual cone of C; that is,
(1)C+={y*∈Y*:0≤〈y*,c〉∀c∈C}.
For y1,y2∈Y, we define y1<Cy2 and y1≤Cy2 if y2-y1∈int(C) and y2-y1∈C, respectively. Let A be a subset of Y and a∈A. Recall that (i) a is a weak Pareto point of A if there exists no point y∈A such that y<Ca; (ii) a is a Pareto point of A if there exists no point y∈A∖{a} such that y≤Ca; (iii) a is an ideal point of A if a≤Cy for all y∈A. Let WE(A,C), E(A,C), and I(A,C) denote the sets of all weak Pareto, Pareto, and ideal points of A, respectively. It is easy to verify that
(2)a∈WE(A,C)⟺(a-int(C))∩A=∅,a∈E(A,C)⟺(a-C)∩A={a},I(A,C)⊂E(A,C)⊂WE(A,C).

Let Xn:={(x1,…,xn):xi∈X,i=1,…,n} be equipped with the norm ∥(x1,…,xn)∥=∑i=1n∥xi∥.

Let Φ:Xn→Y be n-linear and symmetric mapping [17]; that is, for any s,t∈ℝ and x1,z1,x2,…,xn∈X,
(3)Φ(sx1+tz1,x2,…,xn)=sΦ(x1,x2,…,xn)+tΦ(z1,x2,…,xn),Φ(x1,…,xn)=Φ(xi1,…,xin),
where (i1,…,in) is an arbitrary permutation of (1,…,n). Let f:X→Y be a mapping. It is known that its derivative f(n)(x) is n-linear, symmetric, and continuous mapping if f is n-time smooth.

Let f be a function from Rn to Rm and C⊂Rm be a closed convex cone. Consider the following vector optimization problem
(4)C-minx∈Rnf(x).
A vector x-∈X is said to be a local weak Pareto (resp., Pareto and ideal) solution of (4) if there exists δ>0 such that f(x-) is a weak Pareto (resp., Pareto and ideal) point of f(B(x-,δ)), where B(x-,δ) denotes the open ball with center x- and radius δ. We say that x- is a sharp Pareto solution of (4) of order r if there exist η,δ∈(0,+∞) such that
(5)η∥x-x-∥≤[f(x)-f(x-)]+r,∀x∈B(x-,δ),
where [f(x)-f(x-)]+:=d(f(x)-f(x-),-C).

We denote by Ck,1,k>0, the class of k-time differentiable mappings from Rn to Rm whose kth order derivatives are locally Lipschitz mappings and by C0,1 the class of locally Lipschitz functions from Rn to Rm. By Rademacher's theorem (see [18]), for any f∈Ck,1, f(x)=(f1(x),…,fm(x)), its kth order derivative Dkf(x) is a function differentiable almost everywhere. The (k+1)th order subdifferential of f at x∈Rn is defined as “generalized Jacobian” of Dk+1f at x in Clarke's sense [18] as follows:
(6)∂k+1f(x):=co{limDk+1f(xi):xi→x,∂k+1f(x):=coDk+1f(xi)existsatxi}.
It is worth mentioning that each element in ∂k+1f(x) is a k+1 linear and symmetric mapping from (Rn)k+1 to Rm. For more details about ∂k+1f(x), we refer the reader to [18].

It is similar to the proof of Lemma 2.1 in [1], and one can verify the following chain rule.

Lemma 1.

Let x,u in Rn, g be a function from R to Rn defined by g(t)=x+tu for every t∈R, and let f be a Ck,1 function from Rn to Rm. Then,
(7)∂k+1(f∘g)(t)⊆∂k+1f(x+tu)(u,…,u).

3. A New Taylor's Formula in Form of High-Order Subdifferential

By Lemma 1, we have the following Taylor-type formula for a Ck,1 vector-valued function from Rn to Rm which will be useful in the sequel.

Theorem 2.

Let x, u, and f be as in Lemma 1. Then, there exists A∈clco∂k+1f(x,x+u) such that
(8)f(x+u)-f(x)=∑i=1k1i!Dif(x)(ui)+1(k+1)!A(uk+1),
where ui denotes (u,…,u)∈(Rn)i and
(9)clco∂k+1f(x,x+u):=clco(⋃t∈(0,1)∂k+1f(x+tu)).

Proof.

Let α∈Rm be a vector satisfying
(10)f(x+u)-f(x)=∑i=1k1i!Dif(x)(ui)+1(k+1)!α.
We only need to show that there exists A∈clco∂k+1f(x,x+u) such that
(11)α=A(uk+1).
Let g be as Lemma 1. Set φ(t):=(f∘g)(t) and
(12)h(t)≔φ(1)-φ(t)-∑i=1k1i!Diφ(t)(1-t)i-1(k+1)!(1-t)k+1α.
Let y∈Rm be arbitrarily given. Since the function 〈y,h(·)〉 is locally Lipschitz and h(0)=h(1), applying Lebourg mean value theorem [18, Theorem 2.3.7 and Theorem 2.3.9], there exists t0∈(0,1) such that
(13)0∈∂〈y,h(t0)〉⊂〈y,∂h(t0)〉.
Noting that φ(·) and each Diφ(·)(1-·)i(1≤i≤k-1) have derivatives which are continuous, it follows that they are strictly H-differentiable. We have
(14)∂h(t0)=-∂φ(t0)-∑i=1k-11i!∂(Diφ(·)(1-·)i)(t0)-1k!∂(Dkφ(·)(1-·)k)(t0)-1(k+1)!∂((1-·)k+1α)(t0)=-φ′(t0)-∑i=1k-11i!(Di+1φ(t0)(1-t0)i-iDiφ(t0)(1-t0)i-1)-1k!∂(Dkφ(·)(1-·)k)(t0)-1k!(1-t0)kα=-1(k-1)!Dkφ(t0)(1-t0)k-1-1k!∂(Dkφ(·)(1-·)k)(t0)-1k!(1-t0)kα.
Here, the first equation holds by Propositions 7.4.3(b), and 7.3.5 in [19] and the second holds by Proposition 7.3.9 in [19]. By the chain rule [19, Theorem 7.4.5(a)], we also have
(15)∂(Dkφ(·)(1-·)k)(t0)⊂∂k+1φ(t0)(1-t0)k-kDkφ(t0)(1-t0)k.
Hence, we have
(16)∂h(t0)⊂-1k!∂k+1φ(t0)(1-t0)k+1k!(1-t0)kα.
From (13) and (16), we have
(17)0∈〈y,-1k!∂k+1φ(t0)(1-t0)k+1k!(1-t0)kα〉.
Together with Lemma 1, it follows that
(18)0∈〈y,-1k!(1-t0)k∂k+1f(x+t0u)(uk+1)+1k!(1-t0)kα〉;
that is,
(19)〈y,α〉∈〈y,∂k+1f(x+t0u)(uk+1)〉⊂〈y,clco∂k+1f(x,x+u)(uk+1)〉.
Since y is arbitrary in Rn and clco∂k+1f(x,x+u)(uk+1) is convex and compact, by the separation theorem, we can easily show that α∈clco∂k+1f(x,x+u)(uk+1). Hence, we can take A∈clco∂k+1f(x,x+u) such that α=A(uk+1). The proof is completed.

Corollary 3.

Let f be as in Theorem 2 and a∈Rn. Then, for every x∈Rn, there exist Ax∈∂k+1f(a) and a (k+1)-linear mapping r(x) from (Rn)k+1 to Rm such that
(20)limx→a∥r(x)∥=0,f(x)=f(a)+∑i=1k1i!Dif(a)(x-a)if(a)++1(k+1)!Ax(x-a)k+1+r(x)(x-a)k+1.

Proof.

By Theorem 2, for a given x∈Rn, there exists Bx∈clco∂k+1f(a,a+x) such that
(21)f(x)=f(a)+∑i=1k1i!Dif(a)(x-a)i+1(k+1)!Bx(x-a)k+1.
Let Ax∈∂k+1f(a) be an element minimizing the distance from Bx to the convex and compact set ∂k+1f(a). Set
(22)r(x):=Bx-Ax(k+1)!.
Then, from (21), we obtain the formula of the corollary. Moreover, since the mapping ∂k+1f is upper continuous, nonempty, convex, and compact valued (see [18]), for any ε>0, there exists δ>0 such that, for all y∈a+δBRn (where BRn denotes the closed unit ball of Rn),
(23)∂k+1f(y)⊂∂k+1f(a)+(k+1)!εBL((Rn)k+1,Rm),
where BL((Rn)k+1,Rm) denotes the closed unit ball of the space L((Rn)k+1,Rm) of all bounded linear operators from (Rn)k+1 to Rm. If x∈a+δBRn, then
(24)clco∂k+1f(a,a+x)⊂∂k+1f(a)+(k+1)!εBL((Rn)k+1,Rm).
With this we obtain ∥r(x)∥≤ε. The proof is completed.

4. The Positive Definiteness of High-Order Subdifferential and Optimality Conditions

Recall [17] that n-linear symmetric mapping Φ:Xn→Y is said to be positively definite (resp., positively semidefinite) with respect to the ordering cone C if
(25)0<CΦ(xn)(resp.,0≤CΦ(xn)),∀x∈X∖{0},
where xn denotes (x,…,x). If n is odd and the ordering cone C is pointed (i.e., C∩(-C)={0}), then Φ is positively semidefinite if and only if Φ=0; see [17].

By the separation theorem, it is easy to verify that a n-linear symmetric mapping Φ is positively semidefinite with respect to the ordering cone C if and only if the composite c*∘Φ is positively semidefinite for any c*∈C+. Recall that a mapping f:X→Y is C-convex if
(26)f(tx1+(1-t)x2)≤Ctf(x1)+(1-t)f(x2),(1-t)x2∀x1,x2∈X,∀t∈[0,1].
Noting that f is C-convex if and only if c*∘f is convex for all c*∈C+, one can see that a twice differentiable function f is C-convex if and only if f′′(x) is positively semidefinite for all x∈X.

Inspired by the notion of positive definiteness, we introduce positive definiteness of the (k+1)th order subdifferential for Ck,1 functions.

Definition 4.

Let f be a Ck,1 function from Rn to Rm and C a closed convex cone of Rm. We say that the (k+1)th order subdifferential mapping ∂k+1f is positively definite at x-∈Rn with respect to the ordering cone C if each A∈∂k+1f(x-) is positively definite with respect to C.

Proposition 5.

Let f be a Ck,1 function from Rn to Rm, and let C be a closed convex cone of Rm. Suppose that the subdifferential mapping ∂k+1f is positively definite at x-∈Rn with respect to C. Then, there exists η>0 such that
(27)A(xk+1)+ηBRm⊂C,∀A∈∂k+1f(x-),x∈SRn,
where SRn:={x∈Rn:∥x∥=1}.

Proof.

From [17, Proposition 3.4], for any A∈∂k+1f(x-), there exists ηA>0 such that
(28)A(xk+1)+ηABRm⊂C,∀x∈SRn.
If the conclusion is not true, then, for every natural number i, there exist Ai∈∂k+1f(x-), xi∈SRn and bi∈BRn such that
(29)Ai(xik+1)+1ibi∉C.
Since ∂k+1f(x-) and SRn are compact, we can assume that Ai→A0∈∂k+1f(x-), xi→x0∈SRn (passing to a subsequence if necessary). Then,
(30)A0(x0k+1)+(Ai(xik+1)-A0(x0k+1))+1ibi∉C
for all i. But from (28), for large enough i, we have
(31)A0(x0k+1)+(Ai(xik+1)-A0(x0k+1))+1ibi∈A0(x0k+1)+ηA0BRm⊂C,
which is a contradiction with (29). The proof is completed.

Under the positive definiteness assumption, we will provide a (k+1)th order sufficient condition for x- to be a sharp local Pareto solution of (4) for a Ck,1 function f.

Theorem 6.

Let f be a Ck,1 function from Rn to Rm, C a closed convex cone of Rm, and x-∈Rn. Suppose that there exists c*∈C+ with ∥c*∥=1 such that ∑i=1k(1/i!)c*∘Dif(x-)=0, and that ∂(k+1)f is positively definite at x- with respect to the ordering cone C. Then, x- is a local Pareto solution of (4), and there exist η,δ∈(0,+∞) such that
(32)η∥x-x-∥≤[f(x)-f(x-)]+1/(k+1),∀x∈B(x-,δ).

Proof.

Since ∂(k+1)f(x-) is positively definite with respect to C, by Proposition 5, there exists η>0 such that
(33)1(k+1)!∂(k+1)f(x-)(hk+1)+2ηk+1BRm⊂C,∀h∈SRn.
Noting that c*∈C+ and ∥c*∥=1, we have that
(34)1(k+1)!〈c*,Ahk+1〉≥2ηk+1∥h∥k+1,∀A∈∂(k+1)f(x-),h∈Rn.
Let ϕ(x):=〈c*,f(x)〉 for all x∈X. Since f is a Ck,1 function, so is ϕ. Noting that D(i)ϕ(x-)=c*∘D(i)f(x-) with Corollary 3, there exist Ax∈∂k+1f(x-) and (k+1)-linear mapping r(x) with limx→x-∥r(x)∥=0 such that
(35)ϕ(x)=ϕ(x-)+∑i=1k1i!D(i)ϕ(x-)((x-x-)i)+1(k+1)!×〈c*,Ax(x-x-)k+1〉+〈c*,r(x)(x-x-)k+1〉.
It follows that there exists δ>0 such that
(36)ϕ(x)-ϕ(x-)-∑i=1k1i!D(i)ϕ(x-)((x-x-)i)-1(k+1)!〈c*,Ax(x-x-)k+1〉≥-ηk+1∥x-x-∥k+1,
for all x∈B(x-,δ). Since ∑i=1k(1/i!)c*∘Dif(x-)=0, it follows from (34) and (36) that
(37)ηk+1∥x-x-∥k+1≤ϕ(x)-ϕ(x-),∀x∈B(x-,δ).
On the other hand, for any c∈C, one has
(38)ϕ(x)-ϕ(x-)=〈c*,f(x)-f(x-)〉≤〈c*,f(x)-f(x-)+c〉≤∥f(x)-f(x-)+c∥.
This implies that (32) holds. It remains to show that x- is a local Pareto solution of (4). Let x∈B(x-,δ) such that f(x)≤Cf(x-). Then, ∥f(x)-f(x-)∥+=0. It follows from (32) that x=x-, and hence f(x)=f(x-). This shows that x- is a local Pareto solution of (4).

In Theorem 6, if f is a C-convex Ck,1 function, then x- is a global Pareto solution of (4).

Theorem 7.

Let f be a C-convex Ck,1 function from Rn to Rm, C a closed convex cone of Rm, and x-∈Rn. Suppose that there exists c*∈C+ with ∥c*∥=1 such that ∑i=1k(1/i!)c*∘Dif(x-)=0 and that ∂k+1f(x-) is positively definite. Then, x- is a global Pareto solution of (4), and there exists η0∈(0,+∞) such that
(39)η0∥x-x-∥≤max{[f(x)-f(x-)]+1/(k+1),η0∥x-x-∥≤max[f(x)-f(x-)]+{[f(x)-f(x-)]+1/((k+1))}},∀x∈Rn.

Proof.

Similar to the proof of Theorem 6, one can show that (39) implies that x- is a global Pareto solution of (4). It remains to show that (39) holds. By Theorem 6, there exist η,δ∈(0,+∞) such that (32) holds. Since f is C-convex, it is easy to verify that x↦[f(x)-f(x-)]+ is a convex function. Let x∈Rn∖B(x-,δ). Then,
(40)ηk+1δk+1≤[f(x-+δx-x-∥x-x-∥)-f(x-)]+≤(1-δ∥x-x-∥)[f(x-)-f(x-)]++δ∥x-x-∥[f(x)-f(x-)]+=δ∥x-x-∥[f(x)-f(x-)]+.
Hence, ηk+1δk∥x-x-∥≤[f(x)-f(x-)]+. Letting η0:=min{η,ηk+1δk}, it follows from (32) that (39) holds. The proof is completed.

With ∑i=1k(1/i!)c*∘Dif(x-) = 0 in Theorem 7 replaced by a stronger assumption, we have the following sufficient condition for sharp ideal solutions of (4).

Theorem 8.

Let f be a Ck,1 function from Rn to Rm, C a closed convex cone of Rm, and x-∈Rn. Suppose that ∑i=1k(1/i!)Dif(x-)=0 and that ∂k+1f is positively definite at x- with respect to the ordering cone C. Then, there exist η,δ∈(0,+∞) such that
(41)f(x-)≤Cf(x),∀x∈B(x-,δ),(42)η∥x-x-∥≤[f(x)-f(x-)]+1/(k+1),∀x∈B(x-,δ).

Proof.

By Theorem 6, we need only to show that there exists δ>0 such that (41) holds. Since ∂k+1f(x-) is positively definite, there exists η>0 such that
(43)1(k+1)!∂k+1f(x-)(hk+1)+2ηBRm⊂C,∀h∈SRm.
It follows that
(44)1(k+1)!∂k+1f(x-)(xk+1)+η∥x∥k+1BRm⊂C,∀x∈Rm.
On the other hand, since ∑i=1k(1/i!)Dif(x-)=0, with Corollary 3, we can assume that for any x∈Rn close to x-, there exists n-linear symmetric and continuous mapping r(x) from (Rn)k+1 to Rm such that limx→x-r(x)=0 and
(45)f(x)-f(x-)∈1(k+1)!∂(k+1)f(x-)((x-x-)k+1)+r(x)(x-x-)k+1.
Hence, there exists δ>0 such that
(46)f(x)-f(x-)∈1(k+1)!∂k+1f(x-)((x-x-)k+1)+η∥x-x-∥k+1BRm,∀x∈B(x-,δ).
This and (44) imply that (41) holds. The proof is completed.

5. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M391"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">+</mml:mo><mml:mn>1</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>th Order Necessary Conditions for Strongly <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M392"><mml:mrow><mml:mi>C</mml:mi></mml:mrow></mml:math></inline-formula>-Quasiconvex Functions

We recall that a function f from Rn to R is quasiconvex if, for every x,y∈Rn and for every λ∈(0,1), one has f(λx+(1-λ)y)≤max{f(x),f(y)}. Inspired by this, we introduce the notion of strong C-quasiconvexity for functions from Rn to Rm. A function f from Rn to Rm is said to be strongly C-quasiconvex if, for every x,y∈Rn and for every λ∈(0,1), one has
(47)f(λx+(1-λ)y)∈{f(x),f(y)}-C.
Using the generalized Hessian (see [20]), Luc [1] gave a second-order criterion for quasiconvex functions. We will give a (k+1)th order necessary codition for a function to be strongly C-quasiconvex.

Theorem 9.

Let f be a strongly C-quasiconvex function from Rn to Rm, k an odd number and C⊂Rm the closed pointed ordering cone. Then, for any x,u∈Rn with Dif(x)(ui)=0(i=1,…,k), there exists A0∈∂k+1f(x) such that A0(uk+1)∈C.

Proof.

Suppose that the conclusion is not true. Then, there exist some x,u∈Rn, with Dif(x)(ui)=0(i=1,…,k) such that ∂k+1f(x)(uk+1)⊂Rm∖C. Since Rm∖C is open and ∂k+1f(x)(uk+1) is compact, there exists ε>0 such that
(48)∂k+1f(x)(uk+1)+εBL((Rn)k+1,Rm)(uk+1)⊂Rm∖C.
Since ∂k+1f(·) is upper continuous, for the previous ε, there exists t0>0 such that
(49)∂k+1f(x+tu)⊂∂k+1f(x)+εBL((Rn)k+1,Rm),
for any t∈[-t0,t0]. Noting that ∂k+1f(x) is closed convex, from (48) and (49), we have
(50)clco∂k+1f(x-tu,x+tu)(uk+1)⊂∂k+1f(x)(uk+1)+εBL((Rn)k+1,Rm)(uk+1)⊂Rm∖C.
From Theorem 2, for any t∈[-t0,t0], we can take At∈clco∂k+1f(x,x+tu) such that
(51)f(x+tu)-f(x)=∑i=1k1i!tiDif(x)(ui)+1(k+1)!tk+1At(uk+1)=1(k+1)!tk+1At(uk+1)∈Rm∖C.
Noting that k+1 is even, we have f(x+tu)-f(x)∈Rm∖C and f(x-tu)-f(x)∈Rm∖C, for all t∈[-t0,t0].

On the other hand, since f is C-quasiconvex and f(x)=f((1/2)(x+tu)+(1/2)(x-tu)), one has f(x+tu)-f(x)∈C or f(x-tu)-f(x)∈C. This is a contradiction.

If m=1 and C=R+, then C+=R+. We have the following.

Corollary 10 (see [<xref ref-type="bibr" rid="B8">1</xref>]).

Let f be a quasiconvex function from Rn to R and k an odd number. Then, for any x,u∈Rn with Dif(x)(ui)=0(i=1,…,k), one has D+k+1f(x;u)≥0, where D+k+1f(x;u):=max{A(u,…,u):A∈∂k+1f(x)}.

Acknowledgments

This research was supported by the National Natural Science Foundations, China (Grant no. 11061039, 11061038, and 11261067), an internal Grant of Hong Kong Polytechnic University (G-YF17), and IRTSTYN.

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