Positive Definiteness of High-Order Subdifferential and High-Order Optimality Conditions in Vector Optimization Problems

Abstract

We obtain a new Taylor's formula in terms of the 𝑘+1 order subdifferential of a 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚. 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 𝑅^𝑛 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 𝐶^𝑘,1 vector-valued function from 𝑅^𝑛 to 𝑅^𝑚. We will first prove a new Taylor's formula in the terms of 𝑘+1 order subdifferential for 𝐶^𝑘,1 functions from 𝑅^𝑛 to 𝑅^𝑚, 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 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚. 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 𝐶^𝑘,1 functions from 𝑅^𝑛 to 𝑅 to that for 𝐶^𝑘,1 vector-valued functions from 𝑅^𝑛 to 𝑅^𝑚 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 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚. 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, 𝐶⁺=𝑦^∗∈𝑌^∗∶0≤𝑦^∗,𝑐∀𝑐∈𝐶. (1) 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 ()⟺((𝐶()⟺(){𝑎},𝐼()()().𝑎∈WE𝐴,𝐶𝑎−int))∩𝐴=∅,𝑎∈𝐸𝐴,𝐶𝑎−𝐶∩𝐴=𝐴,𝐶⊂𝐸𝐴,𝐶⊂WE𝐴,𝐶 (2)

Let 𝑋^𝑛∶={(𝑥₁,…,𝑥_𝑛)∶𝑥_𝑖∈𝑋,𝑖=1,…,𝑛} be equipped with the norm ‖(𝑥₁,…,𝑥_𝑛)‖=∑^𝑛_𝑖=1‖𝑥_𝑖‖.

Let Φ∶𝑋^𝑛→𝑌 be 𝑛-linear and symmetric mapping [17]; that is, for any 𝑠,𝑡∈ℝ and 𝑥₁,𝑧₁,𝑥₂,…,𝑥_𝑛∈𝑋, Φ𝑠𝑥₁+𝑡𝑧₁,𝑥₂,…,𝑥_𝑛=𝑠Φ𝑥₁,𝑥₂,…,𝑥_𝑛+𝑡Φ𝑧₁,𝑥₂,…,𝑥_𝑛,Φ𝑥₁,…,𝑥_𝑛=Φ𝑥_𝑖1,…,𝑥_𝑖𝑛, (3) where (𝑖₁,…,𝑖_𝑛) 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 𝑅^𝑛 to 𝑅^𝑚 and 𝐶⊂𝑅^𝑚 be a closed convex cone. Consider the following vector optimization problem 𝐶−min_{𝑥∈𝑅𝑛}𝑓(𝑥). (4) 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 𝜂‖𝑥−𝑥‖≤𝑓(𝑥)−𝑓(𝑥)^𝑟₊,∀𝑥∈𝐵(𝑥,𝛿), (5) where [𝑓(𝑥)−𝑓(𝑥)]₊∶=𝑑(𝑓(𝑥)−𝑓(𝑥),−𝐶).

We denote by 𝐶^𝑘,1,𝑘>0, the class of 𝑘-time differentiable mappings from 𝑅^𝑛 to 𝑅^𝑚 whose 𝑘th order derivatives are locally Lipschitz mappings and by 𝐶^0,1 the class of locally Lipschitz functions from 𝑅^𝑛 to 𝑅^𝑚. By Rademacher's theorem (see [18]), for any 𝑓∈𝐶^𝑘,1, 𝑓(𝑥)=(𝑓₁(𝑥),…,𝑓_𝑚(𝑥)), its 𝑘th order derivative 𝐷^𝑘𝑓(𝑥) is a function differentiable almost everywhere. The (𝑘+1)th order subdifferential of 𝑓 at 𝑥∈𝑅^𝑛 is defined as “generalized Jacobian” of 𝐷^𝑘+1𝑓 at 𝑥 in Clarke's sense [18] as follows: 𝜕^𝑘+1𝑓(𝑥)∶=colim𝐷^𝑘+1𝑓𝑥_𝑖∶𝑥_𝑖𝐷→𝑥,_𝑘+1𝑓𝑥_𝑖existsat𝑥_𝑖. (6) It is worth mentioning that each element in 𝜕^𝑘+1𝑓(𝑥) is a 𝑘+1 linear and symmetric mapping from (𝑅^𝑛)^𝑘+1 to 𝑅^𝑚. 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 𝑅^𝑛, 𝑔 be a function from 𝑅 to 𝑅^𝑛 defined by 𝑔(𝑡)=𝑥+𝑡𝑢 for every 𝑡∈𝑅, and let 𝑓 be a 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚. Then, 𝜕^𝑘+1𝑓∘𝑔(𝑡)⊆𝜕^𝑘+1𝑓(𝑥+𝑡𝑢)(𝑢,…,𝑢). (7)

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

By Lemma 1, we have the following Taylor-type formula for a 𝐶^𝑘,1 vector-valued function from 𝑅^𝑛 to 𝑅^𝑚 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 𝑓(𝑥+𝑢)−𝑓(𝑥)=^𝑘_𝑖=11𝐷𝑖!^𝑖𝑓(𝑥)𝑢^𝑖+1()!𝐴𝑢𝑘+1^𝑘+1, (8) where 𝑢^𝑖 denotes (𝑢,…,𝑢)∈(𝑅^𝑛)^𝑖 and clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢)∶=clco_()𝑡∈0,1𝜕^𝑘+1𝑓(𝑥+𝑡𝑢). (9)

Proof. Let 𝛼∈𝑅^𝑚 be a vector satisfying 𝑓(𝑥+𝑢)−𝑓(𝑥)=^𝑘_𝑖=11𝐷𝑖!^𝑖𝑓(𝑥)𝑢^𝑖+1(𝑘+1)!𝛼. (10) We only need to show that there exists 𝐴∈clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢) such that 𝛼=𝐴𝑢^𝑘+1. (11) Let 𝑔 be as Lemma 1. Set 𝜑(𝑡)∶=(𝑓∘𝑔)(𝑡) and ℎ(𝑡)(1)(𝑡)−∶=𝜑−𝜑_𝑘𝑖=11𝐷𝑖!^𝑖𝜑(𝑡)(1−𝑡)^𝑖−1()!()𝑘+11−𝑡^𝑘+1𝛼. (12) Let 𝑦∈𝑅^𝑚 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 0∈𝜕𝑦,ℎ𝑡₀⊂𝑦,𝜕ℎ𝑡₀. (13) Noting that 𝜑(⋅) and each 𝐷^𝑖𝜑(⋅)(1−⋅)^𝑖  (1≤𝑖≤𝑘−1) have derivatives which are continuous, it follows that they are strictly 𝐻-differentiable. We have 𝜕ℎ𝑡₀=−𝜕𝜑𝑡₀−^𝑘−1_𝑖=11𝜕𝐷𝑖!^𝑖𝜑(⋅)(1−⋅)^𝑖𝑡₀−1𝜕𝐷𝑘!^𝑘𝜑(⋅)(1−⋅)^𝑘𝑡₀−1()!𝜕()𝑘+11−⋅^𝑘+1𝛼𝑡₀=−𝜑^′𝑡₀−^𝑘−1_𝑖=11𝐷𝑖!^𝑖+1𝜑𝑡₀1−𝑡₀^𝑖−𝑖𝐷^𝑖𝜑𝑡₀1−𝑡₀^𝑖−1−1𝜕𝐷𝑘!^𝑘𝜑(⋅)(1−⋅)^𝑘𝑡₀−1𝑘!1−𝑡₀^𝑘𝛼1=−()!𝐷𝑘−1^𝑘𝜑𝑡₀1−𝑡₀^𝑘−1−1𝜕𝐷𝑘!^𝑘𝜑(⋅)(1−⋅)^𝑘𝑡₀−1𝑘!1−𝑡₀^𝑘𝛼. (14) 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 𝜕𝐷^𝑘𝜑(⋅)(1−⋅)^𝑘𝑡₀⊂𝜕^𝑘+1𝜑𝑡₀1−𝑡₀^𝑘−𝑘𝐷^𝑘𝜑𝑡₀1−𝑡₀^𝑘. (15) Hence, we have 𝜕ℎ𝑡₀1⊂−𝜕𝑘!^𝑘+1𝜑𝑡₀1−𝑡₀^𝑘+1𝑘!1−𝑡₀^𝑘𝛼. (16) From (13) and (16), we have 10∈𝑦,−𝜕𝑘!^𝑘+1𝜑𝑡₀1−𝑡₀^𝑘+1𝑘!1−𝑡₀^𝑘𝛼. (17) Together with Lemma 1, it follows that 10∈𝑦,−𝑘!1−𝑡₀^𝑘𝜕^𝑘+1𝑓𝑥+𝑡₀𝑢𝑢^𝑘+1+1𝑘!1−𝑡₀^𝑘𝛼; (18) that is, 𝑦,𝛼∈𝑦,𝜕^𝑘+1𝑓𝑥+𝑡₀𝑢𝑢^𝑘+1⊂𝑦,clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢)𝑢^𝑘+1. (19) Since 𝑦 is arbitrary in 𝑅^𝑛 and clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢)(𝑢^𝑘+1) is convex and compact, by the separation theorem, we can easily show that 𝛼∈clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢)(𝑢^𝑘+1). Hence, we can take 𝐴∈clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑢) such that 𝛼=𝐴(𝑢^𝑘+1). The proof is completed.

Corollary 3. Let 𝑓 be as in Theorem 2 and 𝑎∈𝑅^𝑛. Then, for every 𝑥∈𝑅^𝑛, there exist 𝐴_𝑥∈𝜕^𝑘+1𝑓(𝑎) and a (𝑘+1)-linear mapping 𝑟(𝑥) from (𝑅^𝑛)^𝑘+1 to 𝑅^𝑚 such that lim_𝑥→𝑎‖𝑟(𝑥𝑓(𝑥)(𝑎)+)‖=0,=𝑓^𝑘_𝑖=11𝐷𝑖!^𝑖𝑓(𝑎)(𝑥−𝑎)^𝑖+1()!𝐴𝑘+1^𝑥(𝑥−𝑎)^𝑘+1+𝑟(𝑥)(𝑥−𝑎)^𝑘+1. (20)

Proof. By Theorem 2, for a given 𝑥∈𝑅^𝑛, there exists 𝐵_𝑥∈clco𝜕^𝑘+1𝑓(𝑎,𝑎+𝑥) such that 𝑓(𝑥)=𝑓(𝑎)+^𝑘_𝑖=11𝐷𝑖!^𝑖𝑓(𝑎)(𝑥−𝑎)^𝑖+1()!𝐵𝑘+1^𝑥(𝑥−𝑎)^𝑘+1. (21) Let 𝐴_𝑥∈𝜕^𝑘+1𝑓(𝑎) be an element minimizing the distance from 𝐵_𝑥 to the convex and compact set 𝜕^𝑘+1𝑓(𝑎). Set 𝑟(𝑥)𝐵∶=^𝑥−𝐴_𝑥()!.𝑘+1 (22) 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+𝛿𝐵_𝑅^𝑛 (where 𝐵_𝑅^𝑛 denotes the closed unit ball of 𝑅^𝑛), 𝜕^𝑘+1𝑓𝑦⊂𝜕^𝑘+1𝑓(𝑎)+(𝑘+1)!𝜀𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚), (23) where 𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚) denotes the closed unit ball of the space 𝐿((𝑅^𝑛)^𝑘+1,𝑅^𝑚) of all bounded linear operators from (𝑅^𝑛)^𝑘+1 to 𝑅^𝑚. If 𝑥∈𝑎+𝛿𝐵_𝑅^𝑛, then clco𝜕^𝑘+1𝑓(𝑎,𝑎+𝑥)⊂𝜕^𝑘+1𝑓(𝑎)+(𝑘+1)!𝜀𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚). (24) 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∶𝑋^𝑛→𝑌 is said to be positively definite (resp., positively semidefinite) with respect to the ordering cone 𝐶 if 0<_𝐶Φ𝑥^𝑛resp.,0≤_𝐶Φ𝑥^𝑛,∀𝑥∈𝑋⧵{0}, (25) where 𝑥^𝑛 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 𝑓𝑡𝑥₁+(1−𝑡)𝑥₂≤_𝐶𝑡𝑓𝑥₁+(1−𝑡)𝑓𝑥₂,∀𝑥₁,𝑥₂∈𝑋,∀𝑡∈[0,1]. (26) 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 𝐶^𝑘,1 functions.

Definition 4. Let 𝑓 be a 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚 and 𝐶 a closed convex cone of 𝑅^𝑚. We say that the (𝑘+1)th order subdifferential mapping 𝜕^𝑘+1𝑓 is positively definite at 𝑥∈𝑅^𝑛 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 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚, and let 𝐶 be a closed convex cone of 𝑅^𝑚. Suppose that the subdifferential mapping 𝜕^𝑘+1𝑓 is positively definite at 𝑥∈𝑅^𝑛 with respect to 𝐶. Then, there exists 𝜂>0 such that 𝐴𝑥^𝑘+1+𝜂𝐵_𝑅^𝑚⊂𝐶,∀𝐴∈𝜕^𝑘+1𝑓(𝑥),𝑥∈𝑆_𝑅^𝑛, (27) where 𝑆_𝑅^𝑛∶={𝑥∈𝑅^𝑛∶‖𝑥‖=1}.

Proof. From [17, Proposition 3.4], for any 𝐴∈𝜕^𝑘+1𝑓(𝑥), there exists 𝜂_𝐴>0 such that 𝐴𝑥^𝑘+1+𝜂_𝐴𝐵_𝑅^𝑚⊂𝐶,∀𝑥∈𝑆_𝑅^𝑛. (28) If the conclusion is not true, then, for every natural number 𝑖, there exist 𝐴_𝑖∈𝜕^𝑘+1𝑓(𝑥), 𝑥_𝑖∈𝑆_𝑅^𝑛 and 𝑏_𝑖∈𝐵_𝑅^𝑛 such that 𝐴_𝑖𝑥_𝑖^𝑘+1+1𝑖𝑏_𝑖∉𝐶. (29) Since 𝜕^𝑘+1𝑓(𝑥) and 𝑆_𝑅^𝑛 are compact, we can assume that 𝐴_𝑖→𝐴₀∈𝜕^𝑘+1𝑓(𝑥), 𝑥_𝑖→𝑥₀∈𝑆_𝑅^𝑛 (passing to a subsequence if necessary). Then, 𝐴₀𝑥₀^𝑘+1+𝐴_𝑖𝑥_𝑖^𝑘+1−𝐴₀𝑥₀^𝑘+1+1𝑖𝑏_𝑖∉𝐶 (30) for all 𝑖. But from (28), for large enough 𝑖, we have 𝐴₀𝑥₀^𝑘+1+𝐴_𝑖𝑥_𝑖^𝑘+1−𝐴₀𝑥₀^𝑘+1+1𝑖𝑏_𝑖∈𝐴₀𝑥₀^𝑘+1+𝜂_𝐴0𝐵_𝑅^𝑚⊂𝐶, (31) 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 𝐶^𝑘,1 function 𝑓.

Theorem 6. Let 𝑓 be a 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚, 𝐶 a closed convex cone of 𝑅^𝑚, and 𝑥∈𝑅^𝑛. Suppose that there exists 𝑐^∗∈𝐶⁺ with ‖𝑐^∗‖=1 such that ∑^𝑘_𝑖=1(1/𝑖!)𝑐^∗∘𝐷^𝑖𝑓(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 𝜂‖𝑥−𝑥‖≤𝑓(𝑥)−𝑓(𝑥)₊^1/(𝑘+1),∀𝑥∈𝐵(𝑥,𝛿). (32)

Proof. Since 𝜕^(𝑘+1)𝑓(𝑥) is positively definite with respect to 𝐶, by Proposition 5, there exists 𝜂>0 such that 1()!𝜕𝑘+1^(𝑘+1)𝑓(𝑥)ℎ^𝑘+1+2𝜂^𝑘+1𝐵_𝑅^𝑚⊂𝐶,∀ℎ∈𝑆_𝑅^𝑛. (33) Noting that 𝑐^∗∈𝐶⁺ and ‖𝑐^∗‖=1, we have that 1()!𝑐𝑘+1^∗,𝐴ℎ^𝑘+1≥2𝜂^𝑘+1‖ℎ‖^𝑘+1,∀𝐴∈𝜕^(𝑘+1)𝑓(𝑥),ℎ∈𝑅^𝑛. (34) Let 𝜙(𝑥)∶=⟨𝑐^∗,𝑓(𝑥)⟩ for all 𝑥∈𝑋. Since 𝑓 is a 𝐶^𝑘,1 function, so is 𝜙. Noting that 𝐷^(𝑖)𝜙(𝑥)=𝑐^∗∘𝐷^(𝑖)𝑓(𝑥) with Corollary 3, there exist 𝐴_𝑥∈𝜕^𝑘+1𝑓(𝑥) and (𝑘+1)-linear mapping 𝑟(𝑥) with lim_𝑥→𝑥‖𝑟(𝑥)‖=0 such that 𝜙(𝑥)=𝜙(𝑥)+^𝑘_𝑖=11𝐷𝑖!^(𝑖)𝜙(𝑥)(𝑥−𝑥)^𝑖+1()!×𝑐𝑘+1_∗,𝐴_𝑥(𝑥−𝑥)^𝑘+1+𝑐^∗,𝑟(𝑥)(𝑥−𝑥)^𝑘+1. (35) It follows that there exists 𝛿>0 such that 𝜙(𝑥)−𝜙(𝑥)−^𝑘_𝑖=11𝐷𝑖!^(𝑖)𝜙(𝑥)(𝑥−𝑥)^𝑖−1()!𝑐𝑘+1^∗,𝐴_𝑥(𝑥−𝑥)^𝑘+1≥−𝜂^𝑘+1‖𝑥−𝑥‖^𝑘+1, (36) for all 𝑥∈𝐵(𝑥,𝛿). Since ∑^𝑘_𝑖=1(1/𝑖!)𝑐^∗∘𝐷^𝑖𝑓(𝑥)=0, it follows from (34) and (36) that 𝜂^𝑘+1‖𝑥−𝑥‖^𝑘+1≤𝜙(𝑥)−𝜙(𝑥),∀𝑥∈𝐵(𝑥,𝛿). (37) On the other hand, for any 𝑐∈𝐶, one has 𝜙(𝑥)−𝜙(𝑥)=𝑐^∗,𝑓(𝑥)−𝑓(𝑥≤𝑐_∗),𝑓(𝑥)−𝑓(𝑥)+𝑐≤𝑓(𝑥)−𝑓(𝑥)+𝑐. (38) 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 𝐶^𝑘,1 function, then 𝑥 is a global Pareto solution of (4).

Theorem 7. Let 𝑓 be a 𝐶-convex 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚, 𝐶 a closed convex cone of 𝑅^𝑚, and 𝑥∈𝑅^𝑛. Suppose that there exists 𝑐^∗∈𝐶⁺ with ‖𝑐^∗‖=1 such that ∑^𝑘_𝑖=1(1/𝑖!)𝑐^∗∘𝐷^𝑖𝑓(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 𝜂₀‖𝑥−𝑥‖≤max𝑓(𝑥)−𝑓(𝑥)₊^1/(𝑘+1),𝑓(𝑥)−𝑓(𝑥)₊,∀𝑥∈𝑅^𝑛. (39)

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∈𝑅^𝑛⧵𝐵(𝑥,𝛿). Then, 𝜂^𝑘+1𝛿^𝑘+1≤𝑓𝑥+𝛿𝑥−𝑥‖𝑥−(𝑥‖−𝑓𝑥)₊≤𝛿1−‖𝑥−𝑥‖𝑓(𝑥)−𝑓(𝑥)₊+𝛿‖𝑥−𝑥‖𝑓(𝑥)−𝑓(𝑥)₊=𝛿‖𝑥−𝑥‖𝑓(𝑥)−𝑓(𝑥)₊. (40) Hence, 𝜂^𝑘+1𝛿^𝑘‖𝑥−𝑥‖≤[𝑓(𝑥)−𝑓(𝑥)]₊. Letting 𝜂₀∶=min{𝜂,𝜂^𝑘+1𝛿^𝑘}, it follows from (32) that (39) holds. The proof is completed.

With ∑^𝑘_𝑖=1(1/𝑖!)𝑐^∗∘𝐷^𝑖𝑓(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 𝐶^𝑘,1 function from 𝑅^𝑛 to 𝑅^𝑚, 𝐶 a closed convex cone of 𝑅^𝑚, and 𝑥∈𝑅^𝑛. Suppose that ∑^𝑘_𝑖=1(1/𝑖!)𝐷^𝑖𝑓(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 𝑓(𝑥)≤_𝐶𝑓(𝑥),∀𝑥∈𝐵(),𝜂‖𝑥,𝛿𝑥−𝑥‖≤𝑓(𝑥)−𝑓(𝑥)₊^1/(𝑘+1),∀𝑥∈𝐵(𝑥,𝛿). (41)(42)

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 1()!𝜕𝑘+1^𝑘+1𝑓(𝑥)ℎ^𝑘+1+2𝜂𝐵_𝑅^𝑚⊂𝐶,∀ℎ∈𝑆_𝑅^𝑚. (43) It follows that 1()!𝜕𝑘+1^𝑘+1𝑓(𝑥)𝑥^𝑘+1+𝜂‖𝑥‖^𝑘+1𝐵_𝑅^𝑚⊂𝐶,∀𝑥∈𝑅^𝑚. (44) On the other hand, since ∑^𝑘_𝑖=1(1/𝑖!)𝐷^𝑖𝑓(𝑥)=0, with Corollary 3, we can assume that for any 𝑥∈𝑅^𝑛 close to 𝑥, there exists 𝑛-linear symmetric and continuous mapping 𝑟(𝑥) from (𝑅^𝑛)^𝑘+1 to 𝑅^𝑚 such that lim_𝑥→𝑥𝑟(𝑥)=0 and 𝑓(𝑥)−𝑓(𝑥)∈1()!𝜕𝑘+1^(𝑘+1)𝑓(𝑥)(𝑥−𝑥)^𝑘+1+𝑟(𝑥)(𝑥−𝑥)^𝑘+1. (45) Hence, there exists 𝛿>0 such that 𝑓(𝑥)−𝑓(𝑥)∈1()!𝜕𝑘+1^𝑘+1𝑓(𝑥)(𝑥−𝑥)^𝑘+1+𝜂‖𝑥−𝑥‖^𝑘+1𝐵_𝑅^𝑚,∀𝑥∈𝐵(𝑥,𝛿). (46) This and (44) imply that (41) holds. The proof is completed.

5. (𝑘+1)th Order Necessary Conditions for Strongly 𝐶-Quasiconvex Functions

We recall that a function 𝑓 from 𝑅^𝑛 to 𝑅 is quasiconvex if, for every 𝑥,𝑦∈𝑅^𝑛 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 𝑅^𝑛 to 𝑅^𝑚. A function 𝑓 from 𝑅^𝑛 to 𝑅^𝑚 is said to be strongly 𝐶-quasiconvex if, for every 𝑥,𝑦∈𝑅^𝑛 and for every 𝜆∈(0,1), one has 𝑓𝜆𝑥+(1−𝜆)𝑦∈𝑓(𝑥),𝑓𝑦−𝐶. (47) 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 𝑅^𝑛 to 𝑅^𝑚, 𝑘 an odd number and 𝐶⊂𝑅^𝑚 the closed pointed ordering cone. Then, for any 𝑥,𝑢∈𝑅^𝑛 with 𝐷^𝑖𝑓(𝑥)(𝑢^𝑖)=0(𝑖=1,…,𝑘), there exists 𝐴₀∈𝜕^𝑘+1𝑓(𝑥) such that 𝐴₀(𝑢^𝑘+1)∈𝐶.

Proof. Suppose that the conclusion is not true. Then, there exist some 𝑥,𝑢∈𝑅^𝑛, with 𝐷^𝑖𝑓(𝑥)(𝑢^𝑖)=0(𝑖=1,…,𝑘) such that 𝜕^𝑘+1𝑓(𝑥)(𝑢^𝑘+1)⊂𝑅^𝑚⧵𝐶. Since 𝑅^𝑚⧵𝐶 is open and 𝜕^𝑘+1𝑓(𝑥)(𝑢^𝑘+1) is compact, there exists 𝜀>0 such that 𝜕^𝑘+1𝑓(𝑥)𝑢^𝑘+1+𝜀𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚)𝑢^𝑘+1⊂𝑅^𝑚⧵𝐶. (48) Since 𝜕^𝑘+1𝑓(⋅) is upper continuous, for the previous 𝜀, there exists 𝑡₀>0 such that 𝜕^𝑘+1𝑓(𝑥+𝑡𝑢)⊂𝜕^𝑘+1𝑓(𝑥)+𝜀𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚), (49) for any 𝑡∈[−𝑡₀,𝑡₀]. Noting that 𝜕^𝑘+1𝑓(𝑥) is closed convex, from (48) and (49), we have clco𝜕^𝑘+1𝑓(𝑥−𝑡𝑢,𝑥+𝑡𝑢)𝑢^𝑘+1⊂𝜕^𝑘+1𝑓(𝑥)𝑢^𝑘+1+𝜀𝐵_{𝐿((𝑅𝑛)}^𝑘+1_,𝑅𝑚)𝑢^𝑘+1⊂𝑅^𝑚⧵𝐶. (50) From Theorem 2, for any 𝑡∈[−𝑡₀,𝑡₀], we can take 𝐴_𝑡∈clco𝜕^𝑘+1𝑓(𝑥,𝑥+𝑡𝑢) such that 𝑓(𝑥+𝑡𝑢)−𝑓(𝑥)=^𝑘_𝑖=11𝑡𝑖!^𝑖𝐷^𝑖𝑓(𝑥)𝑢^𝑖+1()!𝑡𝑘+1^𝑘+1𝐴_𝑡𝑢^𝑘+1=1()!𝑡𝑘+1^𝑘+1𝐴_𝑡𝑢^𝑘+1∈𝑅^𝑚⧵𝐶. (51) Noting that 𝑘+1 is even, we have 𝑓(𝑥+𝑡𝑢)−𝑓(𝑥)∈𝑅^𝑚⧵𝐶 and 𝑓(𝑥−𝑡𝑢)−𝑓(𝑥)∈𝑅^𝑚⧵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 [1]). Let 𝑓 be a quasiconvex function from 𝑅^𝑛 to 𝑅 and 𝑘 an odd number. Then, for any 𝑥,𝑢∈𝑅^𝑛 with 𝐷^𝑖𝑓(𝑥)(𝑢^𝑖)=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.