# Second-order optimality conditions for nonlinear programs and mathematical programs

## Abstract

It is well known that second-order information is a basic tool notably in optimality conditions and numerical algorithms. In this work, we present a generalization of optimality conditions to strongly convex functions of order γ with the help of first- and second-order approximations derived from (Optimization 40(3):229-246, 2011) and we study their characterization. Further, we give an example of such a function that arises quite naturally in nonlinear analysis and optimization. An extension of Newton’s method is also given and proved to solve Euler equation with second-order approximation data.

## 1 Introduction

The concept of approximations of mappings was introduced by Thibault [2]. Sweetser [3] considered approximations by subsets of the space of continuous linear maps $$L(X,Y)$$, where X and Y are Banach spaces, and Ioffe [4] by the so-called fans. This approach was revised by Jourani and Thibault [5]. Another approach belongs to Allali and Amahroq [1]. Following the same ideas, Amahroq and Gadhi [6, 7] have established optimality conditions to some optimization problems under set-valued mapping constraints.

In this work, we explore the notion of strongly convex functions of order γ; see, for instance, [815] and references therein. Let f be a mapping from a Banach space X into $$\mathbb{R}$$, and let $$C\subset X$$ be a closed convex set. It is well known that the notion of strong convexity plays a central role. On the one hand, it ensures the existence and uniqueness of the optimal solution for the problem

$$(\mathcal{P})\quad \min_{x\in C} f(x).$$

On the other hand, if f is twice differentiable, then the strong convexity of f implies that its Hessian matrix is nonsingular, which is an important tool in numerical algorithms. Here we adopt the definition of a second-order approximation [1] to detect some equivalent properties of strongly convex functions of order γ and to characterize the latter. Furthermore, for a $$C^{1,1}$$ function f on a finite-dimensional setting, we show some simple facts. We also provide an extension of Newton’s method to solve an Euler equation with second-order approximation data.

The rest of the paper is written as follows. Section 2 contains basic definitions and preliminary results. Section 3 is devoted to mains results. In Section 4, we point out an extension of Newton’s method and prove its local convergence.

## 2 Preliminaries

Let X and Y be two Banach spaces. We denote by $$\mathcal{L}(X,Y)$$ the set of all continuous linear mappings from X into Y, by $$\mathcal{B}(X\times X,Y)$$ the set of all continuous bilinear mappings from $$X\times X$$ into Y, and by $$\mathbb{B}_{Y}$$ the closed unit ball of Y centered at the origin.

Throughout this paper, $$X^{*}$$ and $$Y^{*}$$ denote the continuous duals of X and Y, respectively, and we write $$\langle\cdot,\cdot\rangle$$ for the canonical bilinear forms with respect to the dualities $$\langle X^{*},X\rangle$$ and $$\langle Y^{*},Y\rangle$$.

### Definition 1

[1]

Let f be a mapping from X into Y, $$\bar{x}\in X$$. A set of mappings $$\mathcal{A}_{f}(\bar{x})\subset\mathcal{L}(X,Y)$$ is said to be a first-order approximation of f at if there exist $$\delta >0$$ and a function $$r: X\to\mathbb{R}$$ satisfying $$\lim _{x\to \bar {x} }r(x)=0$$ such that

$$f(x)-f(\bar{x})\in\mathcal{A}_{f}(\bar{x}) (x-\bar{x})+ \Vert x-\bar{x} \Vert r(x)\mathbb{B}_{Y}$$
(1)

for all $$x\in\bar{x} +\delta\mathbb{B}_{X}$$.

It is easy to check that Definition 1 is equivalent to the following: for all $$\varepsilon>0$$, there exists $$\delta>0$$ such that

$$f(x)-f(\bar{x})\in\mathcal{A}_{f}(\bar{x}) (x-\bar{x})+ \varepsilon \Vert x-\bar{x} \Vert \mathbb{B}_{Y}$$
(2)

for all $$x\in\bar{x} +\delta\mathbb{B}_{X}$$.

### Remark 1

If $$\mathcal{A}_{f}(\bar{x})$$ is a first-order approximation of f at , then (2) means that for any $$x\in\bar{x} +\delta\mathbb{B}_{X}$$, there exist $$A(x)\in\mathcal {A}_{f}(\bar{x})$$ and $$b\in\mathbb{B}_{Y}$$ such that

$$f(x)-f(\bar{x})=A(x) (x-\bar{x})+\varepsilon \Vert x-\bar{x} \Vert b.$$

Hence, for any $$x\in\mathbb{B}(\bar{x},\delta)$$ and $$A(x)\in \mathcal{A}_{f}(\bar{x})$$,

$$\bigl\Vert f(x)-f(\bar{x})-A(x) (x-\bar{x}) \bigr\Vert \leq \varepsilon \Vert x-\bar{x} \Vert .$$
(3)

If $$\mathcal{A}_{f}(\bar{x})$$ is norm-bounded (resp. compact), then it is called a bounded (resp. compact) first-order approximation. Recall that $$\mathcal{A}_{f}(\bar{x})$$ is a singleton if and only if f is Fréchet differentiable at .

The following proposition proved by Allali and Amahroq [1] plays an important role in the sequel in a finite-dimensional setting.

### Proposition 1

[1]

Let $$f: \mathbb{R}^{p} \to\mathbb{R}$$ be a locally Lipschitz function at . Then the Clarke subdifferential of f at ,

$$\partial_{c}f (\bar{x}):=\operatorname{co} \bigl\{ \lim\nabla f(x_{n}): x_{n}\in \operatorname{dom} \nabla f\textit{ and }x_{n}\to\bar{x} \bigr\} ,$$
(4)

is a first-order approximation of f at .

In [6], it is also shown that when f is a continuous function, it admits as an approximation the symmetric subdifferential defined and studied in [16].

The next proposition shows that Proposition 1 holds also when f is a vector-valued function. Let us first recall the definition of the generalized Jacobian for a vector-valued function (see [17, 18] for more details) and the definition of upper semicontinuity.

### Definition 2

The generalized Jacobian of a function $$g: \mathbb{R}^{p} \to\mathbb {R}^{q}$$ at , denoted $$\partial_{c} g(\bar{x})$$, is the convex hull of all matrices M of the form

$$M=\underset{n \to+\infty}{\lim} Jg(x_{n}),$$

where $$x_{n}\to\bar{x}$$, g is differentiable at $$x_{n}$$ for all n, and Jg denotes the $$q\times p$$ usual Jacobian matrix of partial derivatives.

### Definition 3

A set-valued mapping $$F: \mathbb{R}^{p} \rightrightarrows\mathbb {R}^{q}$$ is said to be upper semicontinuous at a point $$\bar{x}\in \mathbb {R}^{p}$$ if, for every $$\varepsilon>0$$, there exists $$\delta>0$$ such that

$$F(x)\subset F(\bar{x}) +\varepsilon\mathbb{B}$$

for every $$x\in\mathbb{R}^{p}$$ such that $$\Vert x-\bar{x} \Vert <\delta$$.

### Proposition 2

Let $$g: \mathbb{R}^{p} \to\mathbb{R}^{q}$$ be a locally Lipschitz function at . Then the generalized Jacobian $$\partial_{c} g(\bar {x})$$ of g at is a first-order approximation of g at .

### Proof

Since the set-valued mapping $$\partial_{c} g(\cdot)$$ is upper semicontinuous, for all $$\varepsilon>0$$, there exists $$r_{0}>0$$ such that

$$\partial_{c} g(x)\subset\partial_{c} g(\bar{x})+ \varepsilon\mathbb{B}_{\mathcal{L}(\mathbb{R}^{p},\mathbb{R}^{q})}\quad \mbox{for all } x\in\bar{x} +r_{0} \mathbb{B}_{\mathbb{R}^{p}}.$$

We may assume that g is Lipschitzian in $$\bar{x} +r_{0}\mathbb {B}_{\mathbb{R}^{p}}$$. Let $$x\in\bar{x} +r_{0}\mathbb{B}_{\mathbb{R}^{p}}$$. We apply [17], Prop. 2.6.5, to derive that there exits $$c\in\mathopen{]}x,\bar{x}[$$ such that

$$g(x)-g(\bar{x}) \in\partial_{c} g(c) (x-\bar{x})\subset \partial_{c} g(\bar{x}) (x-\bar{x})+ \varepsilon\mathbb{B}_{\mathcal {L}(\mathbb {R}^{p},\mathbb{R}^{q})}(x- \bar{x}).$$

Since

$$\mathbb{B}_{\mathcal{L}(\mathbb{R}^{p},\mathbb{R}^{q})}(x-\bar {x})\subset \Vert x-\bar{x} \Vert \mathbb{B}_{\mathbb{R}^{q}},$$

we have

$$g(x)-g(\bar{x}) \in\partial_{c} g(\bar{x}) (x-\bar{x})+ \varepsilon \Vert x-\bar{x} \Vert \mathbb{B}_{\mathbb{R}^{q}},$$

which means that $$\partial_{c} g(\bar{x})$$ is a first-order approximation of g at . □

Recall that a mapping $$f: X \to Y$$ is said to be $$C^{1,1}$$ at if it is Fréchet differentiable in neighborhood of and if its Fréchet derivative $$\nabla f(\cdot)$$ is Lipschitz at .

Let $$\bar{x}\in\mathbb{R}^{p}$$, and let $$f: \mathbb{R}^{p} \rightarrow \mathbb{R}$$ be a $$C^{1,1}$$ function at . The generalized Hessian matrix of f at was introduced and studied by Hiriart-Urruty et al. [19] is the compact nonempty convex set

$$\partial^{2}_{H} f(\bar{x}):=\operatorname{co} \bigl\{ \lim \nabla^{2} f(x_{n}): (x_{n}) \in \operatorname{dom} \nabla^{2} f \textit{ and } x_{n} \to\bar{x} \bigr\} ,$$
(5)

where $$\operatorname{dom} \nabla^{2} f$$ is the effective domain of $$\nabla^{2} f(\cdot)$$.

### Corollary 1

Let $$\bar{x}\in\mathbb{R}^{p}$$, and $$f: \mathbb{R}^{p} \rightarrow \mathbb{R}$$ be a $$C^{1,1}$$ function at . Then, f admits $$\partial^{2}_{H} f(\bar{x})$$ as a first-order approximation at .

### Definition 4

[1]

We say that $$f: X \rightarrow Y$$ admits a second-order approximation at  if there exit two sets $$\mathcal{A}_{f} (\bar{x})\subset \mathcal{L}(X,Y)$$ and $$\mathcal{B}_{f} (\bar{x})\subset\mathcal {B}(X\times X,Y)$$ such that

1. (i)

$$\mathcal{A}_{f} (\bar{x})$$ is a first-order approximation of f at ;

2. (ii)

For all $$\varepsilon>0$$, there exists $$\delta>0$$ such that

$$f(x)-f(\bar{x})\in\mathcal{A}_{f} (\bar{x}) (x-\bar{x})+ \mathcal{B}_{f} (\bar{x}) (x-\bar{x}) (x-\bar{x})+\varepsilon \Vert x- \bar{x} \Vert ^{2}\mathbb{B}_{Y}$$

for all $$x\in\bar{x}+\delta\mathbb{B}_{X}$$.

In this case the pair $$(\mathcal{A}_{f} (\bar{x}),\mathcal{B}_{f} (\bar {x}))$$ is called a second-order approximation of f at . It is called a compact second-order approximation if $$\mathcal{A}_{f} (\bar {x})$$ and $$\mathcal{B}_{f} (\bar{x})$$ are compacts.

Every $$C^{2}$$ mapping $$f: X \to Y$$ at admits $$(\nabla f(\bar {x}), \nabla^{2} f(\bar{x}))$$ as a second-order approximation, where $$\nabla f(\bar{x})$$ and $$\nabla^{2} f(\bar{x})$$ are, respectively, the first- and second-order Fréchet derivatives of f at .

### Proposition 3

[1]

Let $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ be a $$C^{1,1}$$ function at . Then f admits $$(\nabla f(\bar{x}),\frac {1}{2}\partial ^{2}_{H} f(\bar{x}))$$ as a second-order approximation at .

### Proposition 4

Let $$f: X\to Y$$ be a Fréchet-differentiable mapping. If $$(\nabla f(\bar{x}),\mathcal{B}_{f}(\bar{x}))$$ is a bounded second-order approximation of f at . Then $$\nabla f(\cdot)$$ is stable at , that is, there exist $$c, r>0$$ such that

$$\bigl\Vert \nabla f(x)-\nabla f(\bar{x}) \bigr\Vert \leq c \Vert x-\bar{x} \Vert$$
(6)

for all $$x\in\bar{x} +r\mathbb{B}_{X}$$.

To derive some results for γ-strong convex functions, the following notions are needed.

### Definition 5

[8]

Let $$\gamma>0$$. We say that a map $$f: X \to\mathbb{R}\cup\{ +\infty\}$$ is γ-strongly convex if there exist $$c\geq0$$ and $$g: [0,1]\to\mathbb{R}^{+}$$ satisfying

$$g(0)=g(1)=0 \quad\mbox{and}\quad \underset{\theta\to0}{\lim} \frac {g(\theta )}{\theta}=1$$
(7)

and such that

$$f \bigl(\theta x+(1-\theta)y \bigr)\leq\theta f(x)+(1- \theta)f(y)-c g(\theta) \Vert x-y \Vert ^{\gamma}$$
(8)

for all $$\theta\in[0,1]$$ and $$x, y\in X$$.

Of course, when $$c=0$$, f is called a convex function. Otherwise, f is said γ-strongly convex. This class has been introduced by Polyak [11] when $$\gamma=2$$ and $$g(\theta)=\theta(1-\theta)$$ and studied by many authors. Recently, a characterization of γ-strongly convex functions has been shown in [8]. For example, if f is $$C^{1}$$ and $$\gamma\geq1$$, then (8) is equivalent to

$$\bigl\langle \nabla f(x),y-x \bigr\rangle \leq f(y)-f(x)- \frac{c}{\gamma} \Vert y-x \Vert ^{\gamma},\quad \forall x, y\in X.$$
(9)

Let $$f: X \to\mathbb{R}\cup\{+\infty\}$$ and $$\bar{x} \in \operatorname{dom} f:=\{x\in X, f(x)<+\infty\}$$ (the effective domain of f). The Fenchel-subdifferential of f at is the set

$$\partial_{\mathrm{Fen}} f(\bar{x})= \bigl\{ x^{*}\in X^{*}: \bigl\langle x^{*},y-\bar{x} \bigr\rangle \leq f(y)-f(\bar{x}), \forall y\in X \bigr\} .$$
(10)

Let $$\gamma>0$$ and $$c>0$$. The $$(\gamma, c)$$-subdifferential of f at is the set

$$\partial_{(\gamma, c)} f(\bar{x})= \bigl\{ x^{*}\in X^{*}: \bigl\langle x^{*},y-\bar{x} \bigr\rangle \leq f(y)-f(\bar{x}) - c \Vert \bar {x}-y \Vert ^{\gamma }, \forall y\in X \bigr\} .$$
(11)

For more details on $$(\gamma, c)$$-subdifferential, see [8]. Note that if $$x\notin \operatorname{dom} f$$, then $$\partial_{(\gamma,c)} f(\bar {x})=\partial_{\mathrm{Fen}} f(\bar{x})=\emptyset$$. Clearly, we have $$\partial_{(\gamma,c)}f(\bar{x})\subset\partial_{\mathrm{Fen}} f(\bar{x})$$. Note that the Fenchel-subdifferential defined by (10) coincides with the Clarke subdifferential of f at if the function f is convex. We also need to recall the following definitions.

### Definition 6

[20]

We say that a map $$f: X \to\mathbb{R}\cup\{+\infty\}$$ is 2-paraconvex if there exists $$c>0$$ such that

$$f \bigl(\theta x+(1-\theta)y \bigr)\leq\theta f(x)+(1- \theta)f(y)+c \min(\theta,1-\theta) \Vert x-y \Vert ^{2}$$
(12)

for all $$\theta\in[0,1]$$ and $$x, y\in X$$.

It has been proved in [20] that if f is a $$C^{1}$$ mapping, then (12) is equivalent to

$$\bigl\langle \nabla f(x),y-x \bigr\rangle \leq f(y)-f(x)+c \Vert y-x \Vert ^{2}, \quad\forall x, y\in X.$$
(13)

## 3 Main results

In this section, we obtain the main results of the paper related to strongly convex functions of order γ defined by (7)-(8). We begin by showing some interesting facts of functions that admit a first-order approximation.

For any subset A of $$X^{*}$$, we define the support function of A as

$$s(A,x)=\sup \bigl\{ \bigl\langle x^{*},x \bigr\rangle , x^{*}\in A \bigr\} .$$
(14)

It is well known that, for any convex function f: $$X\rightarrow \mathbb{R}\cup\{+\infty\}$$, the ‘right-hand’ directional derivative at x in domf (the domain of f ) exists and, for each $$h\in X$$, is

$$d^{+}f(x) (h)=\underset{t \rightarrow0^{+}}{\lim}\frac{f(x+th)-f(x)}{t}.$$

### Theorem 1

Let $$\bar{x}\in X$$. If $$f:X\to\mathbb{R}\cup\{+\infty\}$$ is convex and continuous at and if $$\mathcal{A}_{f}(\bar{x})\subset X^{*}$$ is a convex $$w(X^{*},X)$$-closed approximation of f at , then

$$\partial_{(\gamma,c)}f(\bar{x})\subset\mathcal{A}_{f}(\bar{x}).$$

### Proof

By the definition of $$\mathcal{A}_{f} (\bar{x})$$, there exist $$\delta >0$$ and $$r:X \to\mathbb{R}$$ with $$\lim_{x\to\bar{x}} r(x)=0$$ such that, for all $$x\in\bar{x}+\delta\mathbb{B}_{X}$$, $$t\in ]0,\delta[$$, and $$h\in X$$, there exist $$A\in\mathcal{A}_{f} (\bar{x})$$ and $$b\in[-1,1]$$ satisfying

$$\frac{f(\bar{x}+th)-f(\bar{x})}{t} - \Vert h \Vert r(\bar {x}+th)b=\langle A,h\rangle\leq s \bigl(\mathcal{A}_{f} (\bar{x});h \bigr).$$

By letting $$t\to0^{+}$$ the directional derivative of f at satisfies

$$d^{+}f(\bar{x}) (h)\leq s \bigl(\mathcal{A}_{f} (\bar{x});h \bigr),\quad \forall h\in X.$$
(15)

Using [21], Prop. 2.24, we get

$$s \bigl(\partial_{\mathrm{Fen}} f (\bar{x});h \bigr)\leq s \bigl( \mathcal{A}_{f} (\bar{x});h \bigr).$$

Since $$\partial_{(\gamma,c)}f(\bar{x})\subset\partial_{\mathrm{Fen}} f(\bar {x})$$, we deduce that

$$s \bigl(\partial_{(\gamma,c)}f(\bar{x});h \bigr)\leq s \bigl( \mathcal{A}_{f} (\bar{x});h \bigr).$$

Hence we conclude that $$\partial_{(\gamma,c)}f(\bar{x})\subset \mathcal {A}_{f} (\bar{x})$$. □

### Proposition 5

Let $$f: X \to\mathbb{R}\cup\{+\infty\}$$ be a γ-strongly convex function. Assume that $$\mathcal{A}_{f}(\bar{x})$$ is a compact approximation at . Then $$\mathcal{A}_{f}(\bar{x})\cap \partial _{(\gamma,c)}f(\bar{x})\neq \emptyset$$.

### Proof

Let $$d\in X$$ be fixed and define $$x_{n}:=\bar{x}+\frac{1}{n}d$$. Using Definition 1, we get, for n large enough, $$A_{n}\in\mathcal {A}_{f}(\bar{x})$$ and $$b_{n}\in[-1,1]$$ such that

$$\frac{1}{n}\langle A_{n},d\rangle=f \biggl(\bar{x}+ \frac{1}{n}d \biggr)-f(\bar{x})-\frac {1}{n} \Vert d \Vert r(x_{n})b_{n}.$$

By γ-strong convexity we obtain

$$\frac{1}{n}\langle A_{n},d\rangle\leq\frac{1}{n} \bigl( f( \bar{x}+d)-f(\bar{x}) \bigr)-c g \biggl(\frac{1}{n} \biggr) \Vert d \Vert ^{\gamma}- \frac {1}{n} \Vert d \Vert r(x_{n})b_{n}.$$

By the compactness of $$\mathcal{A}_{f}(\bar{x})$$, extracting a subsequence if necessary, we may assume that there exists $$A\in \mathcal {A}_{f}(\bar{x})$$ such that $$\langle A_{n},d\rangle \to\langle A,d\rangle$$; and hence we obtain

$$\langle A,d\rangle \leq f(\bar{x}+d)-f(\bar{x}) -c \Vert d \Vert ^{\gamma}.$$
(16)

Assume that $$A\in\mathcal{A}_{f}(\bar{x})\cap\partial_{(\gamma ,c)}f(\bar {x})$$. By the separation theorem there exists $$h\in X$$ with $$\Vert h \Vert =1$$ such that

$$\min_{A\in\mathcal{A}_{f} (\bar{x}) }\langle A,h\rangle > \sup_{x^{*}\in\partial _{(\gamma,c)} f(\bar{x})} \bigl\langle x^{*},h\bigr\rangle .$$

Let $$t >0$$ sufficiently small, so that

$$\min_{A\in\mathcal{A}_{f} (\bar{x}) }\langle A,h\rangle >\frac {f(\bar{x}+th)-f(\bar{x})}{t},$$

in contradiction with relation (16) by taking $$d=th$$. □

Following a result by Rademacher, which states that a locally Lipschitzian function between finite-dimensional spaces is differentiable (Lebesgue) almost everywhere, we can prove the following result.

### Proposition 6

Let $$\gamma\geq1$$, $$\bar{x}\in\mathbb{R}^{p}$$, and let $$f: \mathbb {R}^{p} \to\mathbb{R}$$ be continuous at . Assume that f is a γ-strongly convex function. Then $$\partial_{c} f (\bar{x})= \partial_{(\gamma,c)}f(\bar{x})$$.

### Proof

Obviously, we have $$\partial_{(\gamma,c)}f(\bar{x})\subset\partial_{c} f (\bar{x})$$. Now let $$A\in\partial_{c} f (\bar{x})$$. For all n, there exists $$x_{n}\in \operatorname{dom} \nabla f$$ such that $$x_{n}\to\bar{x}$$ and $$\nabla f(x_{n})\to A$$. Since f is γ-strongly convex and Fréchet differentiable at $$x_{n}$$ for all $$n\in\mathbb{N}$$, it follows by (9) that

$$\bigl\langle \nabla f(x_{n}),y-x_{n}\bigr\rangle \leq f(y)-f(x_{n})-c \Vert y-x_{n} \Vert ^{\gamma}, \quad\forall y\in \mathbb{R}^{p}, \forall n\in\mathbb{N}.$$

Letting $$n\to+\infty$$, we get

$$\langle A,y-\bar{x}\rangle \leq f(y)-f(\bar{x})-c \Vert y-\bar {x} \Vert ^{\gamma},\quad \forall y\in\mathbb{R}^{p},$$

which means that $$\partial_{c} f (\bar{x}) \subset\partial_{(\gamma ,c)}f(\bar{x})$$. □

### Corollary 2

Let $$\gamma\geq1$$, $$\bar{x}\in\mathbb{R}^{p}$$, and let $$f: \mathbb {R}^{p} \to\mathbb{R}$$ be continuous at . Assume that f is a γ-strongly convex function. Then, for all $$\varepsilon>0$$, there exists $$r>0$$ such that

$$f(x)-f(\bar{x})\in\partial_{(\gamma,c)} f(\bar{x}) (x-\bar {x})+\varepsilon \Vert x-\bar{x} \Vert \mathbb{B}_{\mathbb{R}}$$
(17)

for all $$x\in\bar{x}+r\mathbb{B}_{\mathbb{R}^{p}}$$, which means that $$\partial_{(\gamma,c)} f(\bar{x})$$ is a first-order approximation of f at .

### Proof

It is clear that $$\partial_{c} f (\bar{x})$$ is a first-order approximation of at . We end the proof by Propositions 1 and 6. □

The converse of Proposition 5 holds if (16) is valid for any $$A\in\mathcal{A}_{f}(x)$$ and $$x\in X$$.

### Proposition 7

Let $$\gamma\geq1$$ and $$f:X\to\mathbb{R}\cup\{+\infty\}$$. Assume that, for each $$x\in X$$, f admits a first-order approximation $$\mathcal{A}_{f}(x)$$ such that $$\mathcal{A}_{f}(x)\subset\partial _{(\gamma ,c)} f(x)$$. Then f is γ-strongly convex.

### Proof

Define $$x_{\theta}:=\theta u+(1-\theta)v$$ for $$\theta\in[0,1]$$ and $$u, v\in X$$. Let us take $$A\in\mathcal{A}_{f} (x_{\theta})$$. Then

$$\langle A,u-x_{\theta}\rangle \leq f(u)-f(x_{\theta})-c \Vert u-x_{\theta } \Vert ^{\gamma}.$$

Multiplying this inequality by θ, we obtain

$$\bigl(\mathrm{a}' \bigr)\quad \theta(1-\theta)\langle A,u-v\rangle \leq\theta f(u)-\theta f(x_{\theta})-c(1-\theta)^{\gamma} \theta \Vert u-v \Vert ^{\gamma}.$$

In a similar way, since

$$\langle A,v-x_{\theta}\rangle \leq f(v)-f(x_{\theta})-c \Vert v-x_{\theta } \Vert ^{\gamma},$$

we get

$$\bigl(\mathrm{a}'' \bigr)\quad {-}\theta(1-\theta)\langle A,u-v\rangle \leq(1- \theta) f(v)- (1-\theta) f(x_{\theta})-c(1-\theta) \theta^{\gamma} \Vert u-v \Vert ^{\gamma}.$$

We deduce by addition of $$(\mathrm{a}')$$ and $$(\mathrm{a}'')$$ that

$$f(x_{\theta})\leq\theta f(u)+(1-\theta) f(v)-cg(\theta) \Vert u-v \Vert ^{\gamma} \quad\mbox{for all } u, v\in X,$$

where $$g(\theta)=(1-\theta) \theta^{\gamma} +(1-\theta)^{\gamma} \theta$$, so that f is γ-strongly convex. □

The next results are devoted to presenting some useful properties of the generalized Hessian matrix for a $$C^{1,1}$$ function in the finite-dimensional setting and a characterization of γ-strongly convex functions with the help of a second-order approximation.

### Proposition 8

Let $$\bar{x}\in X$$, and let $$f: X \rightarrow\mathbb{R}\cup\{ +\infty\}$$ be convex and Fréchet differentiable at . Suppose that f admits $$(\nabla f(\bar{x}),\mathcal{B}_{f}(\bar{x}))$$ as a second-order approximation at and that $$\mathcal {B}_{f}(\bar {x})$$ is compact. Then there exists $$B\in\mathcal{B}_{f}(\bar{x})$$ such that

$$\sup_{B\in\mathcal{B}_{f}(\bar{x})}\langle Bd,d\rangle \geq0,\quad \forall d \in X.$$
(18)

If f is 2-strongly convex, then we obtain

$$\sup_{B\in\mathcal{B}_{f}(\bar{x})} \langle Bd,d\rangle \geq c \Vert d \Vert ^{2},\quad \forall d\in X,$$
(19)

for some $$c>0$$.

### Proof

We prove only the case where f is convex. In a similar way, we can prove the other case. Let $$d\in X$$ and $$\varepsilon>0$$ be fixed. We get for n large enough $$B_{n}\in\mathcal{B}_{f}(\bar{x})$$ and $$b_{n}\in [-1,1]$$ such that

$$f \biggl(\bar{x}+\frac{1}{n}d \biggr)-f(\bar{x})=\frac {1}{n}\bigl\langle \nabla f(\bar{x}),d\bigr\rangle +\frac{1}{n^{2}}\langle B_{n} d,d\rangle +\varepsilon \frac{1}{n^{2}} \Vert d \Vert ^{2}b_{n}.$$

Since f is convex, we obtain

$$\langle B_{n} d,d\rangle +\varepsilon \Vert d \Vert ^{2}b_{n} \geq0.$$

By the compactness of $$\mathcal{B}_{f}(\bar{x})$$, extracting a subsequence if necessary, we may assume that there exits $$B\in \mathcal{B}_{f}(\bar{x})$$ such that $$B_{n}$$ converges to B; therefore

$$\langle Bd,d\rangle \geq0,$$

and hence

$$\sup_{B\in\mathcal{B}_{f}(\bar{x})}\langle Bd,d\rangle \geq0, \quad \forall d\in X.$$

□

When X is a finite-dimensional space, we get the following essential result.

### Proposition 9

Let $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ be a $$C^{1,1}$$ function at . Assume that f is γ-strongly convex. Then, for any $$B\in\partial^{2}_{H} f(\bar{x})$$, we have the following inequality:

$$\langle Bd,d\rangle \geq c \Vert d \Vert ^{\gamma},\quad \forall d\in \mathbb{R}^{p},$$
(20)

for some $$c>0$$.

### Proof

It is clear that $$(\nabla f(\bar{x}),\frac{1}{2}\partial^{2}_{H} f(\bar {x}))$$ is a second-order approximation of f at . Now let $$B\in\partial^{2}_{H} f(\bar{x})$$, so that there exists a sequence $$(x_{n})\in \operatorname{dom} \nabla^{2} f$$ such that $$x_{n}\to\bar{x}$$ and $$\nabla^{2} f(x_{n})\to B$$. Since f is γ-strongly convex, there exists $$c>0$$ such that

$$\bigl\langle \nabla^{2} f(x_{n}) d,d\bigr\rangle \geq c \Vert d \Vert ^{\gamma}, \quad\forall d\in\mathbb{R}^{p}, \forall n \in\mathbb{N}.$$

Letting $$n\to+\infty$$, we have

$$\langle Bd,d\rangle \geq c \Vert d \Vert ^{\gamma},\quad \forall d\in \mathbb{R}^{p}.$$

□

The preceding result shows that γ-strongly convex functions enjoy a very desirable property for generalized Hessian matrices. In fact, in this case, any matrix $$B\in\partial^{2}_{H} f(\bar{x})$$ is invertible. The next result proves the converse of Proposition 9. Let us first recall the following characterization of l.s.c. γ-strongly convex functions.

### Theorem 2

Amahroq et al. [8]

Let f: $$X\rightarrow \mathbb{R}\cup\{+\infty\}$$ be a proper and l.s.c. function. Then f is γ-strongly convex iff $$\partial_{c} f$$ is γ-strongly monotone, that is, there exists a positive real number c such that, for all $$x, y\in X$$, $$x^{*}\in\partial_{c} f(x)$$, and $$y^{*} \in\partial_{c} f(y)$$, we have

$$\bigl\langle x^{*}-y^{*},x-y\bigr\rangle \geq c \Vert x-y \Vert ^{\gamma}.$$

We are now in position to state our main second result.

### Theorem 3

Let $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ be a $$C^{1,1}$$ function. Assume that $$\partial^{2}_{H} f(\cdot)$$ satisfies relation (20) at any $$x\in\mathbb{R}^{p}$$. Then f is γ-strongly convex.

### Proof

Let $$t\in[0,1]$$ and $$u, v\in\mathbb{R}^{p}$$. Define $$\varphi:\mathbb {R}\to\mathbb{R}$$ as

$$\varphi(t):=f \bigl(u+t(v-u) \bigr),$$

so that $$\varphi'(t):=\langle \nabla f(u+t(v-u)),v-u\rangle$$. By the Lebourg mean value theorem [22] there exists $$t_{0}\in\mathopen{]}0,1[$$ such that

$$\varphi'(1)-\varphi'(0)\in\partial_{c} \varphi'(t_{0}).$$

By using calculus rules it follows that

$$\varphi'(1)-\varphi'(0)\in\partial_{c} \varphi'(t_{0})\subset\partial^{2}_{H} f \bigl(u+t_{0}(v-u) \bigr) (v-u) (v-u).$$

Hence, there exists $$B_{t_{0}} \in\partial^{2}_{H} f(u+t_{0}(v-u))$$ such that $$\langle \nabla f(v)-\nabla f(u),v-u\rangle =\langle B_{t_{0}} (v-u),v-u\rangle$$. The result follows from Theorem 2. □

Hiriart-Urruty et al. [19] have presented many examples of $$C^{1,1}$$ functions. The next proposition shows another example of a $$C^{1,1}$$ function.

### Theorem 4

Let $$f: H \rightarrow\mathbb{R}$$ be continuous on a Hilbert space H. Suppose that f is convex (or 2-strongly convex) and thatf is 2-paraconvex. Then f is Fréchet differentiable on H, and for some $$c>0$$, we have that

$$\bigl\Vert \nabla f(x)-\nabla f(y) \bigr\Vert \leq c \Vert x-y \Vert \quad\textit{for all }x, y\in H.$$
(21)

### Proof

Let $$x_{0}\in X$$. Clearly, f is locally Lipschitzian at $$x_{0}$$. Now let $$x_{1}^{*}$$ and $$x_{2}^{*}$$ be arbitrary elements of $$\partial_{c} f(x_{0})$$ and $$\partial_{c} (-f)(x_{0})$$, respectively. By [20], Thm. 3.4, there exists $$c>0$$ such that $$\partial_{c} (-f)(x_{0})=\partial^{(2,c)} (-f)(x_{0})$$, and for any $$y\in H$$ and positive real θ, we have

$$(\mathrm{a})\quad \theta\bigl\langle x_{2}^{*},y\bigr\rangle \leq-f(x_{0}+ \theta y)+f(x_{0})+c \theta^{2} \Vert y \Vert ^{2}$$

and

$$\bigl(\mathrm{a}' \bigr)\quad \theta\bigl\langle x_{1}^{*},y\bigr\rangle \leq f(x_{0}+\theta y)-f(x_{0}).$$

Adding (a) and (a′), we get

$$\theta\bigl\langle x_{1}^{*}+x_{2}^{*},y\bigr\rangle \leq c \theta^{2} \Vert y \Vert ^{2},$$

and hence

$$\bigl\langle x_{1}^{*}+x_{2}^{*},y\bigr\rangle \leq c \theta \Vert y \Vert ^{2}.$$

Letting $$\theta\to0$$, we have $$\langle x_{1}^{*}+x_{2}^{*},y\rangle \leq 0$$, so that $$x_{1}^{*}=-x_{2}^{*}$$. Since $$x_{1}^{*}$$ and $$x_{2}^{*}$$ are arbitrary in $$\partial_{c} f(x_{0})$$ and $$\partial_{c} (-f)(x_{0})$$, it follows that $$\partial_{c} f(x_{0})$$ is single-valued. Put $$\partial_{c} f(x_{0})=\{p(x_{0})\}$$. Since (a) and (a′) hold for any $$\theta> 0$$ and $$y\in H$$, we deduce that, for $$\theta=1$$,

$$\bigl\langle p(x_{0}),y\bigr\rangle \leq f(x_{0}+y)-f(x_{0})$$

and

$$f(x_{0}+y)-f(x_{0})-\bigl\langle p(x_{0}),y \bigr\rangle \leq c \Vert y \Vert ^{2}.$$

Hence, for all $$y\neq0$$, we obtain

$$\frac{ | f(x_{0}+ y)-f(x_{0})-\langle p(x_{0}),y\rangle |}{ \Vert y \Vert } \leq c \Vert y \Vert .$$
(22)

Letting $$\Vert y \Vert \to0$$ in (22), we conclude that f is Fréchet differentiable at $$x_{0}$$. Now since −f is 2-paraconvex and f is Fréchet differentiable, we may prove that there exists $$c>0$$ such that

$$-\bigl\langle \nabla f(x),y-x\bigr\rangle \leq-f(y)+f(x)+c \Vert x-y \Vert ^{2} \quad\mbox{for all } x, y\in H.$$
(23)

For every $$z\in H$$, we have that

$$-f(z)\geq-f(x)+\bigl\langle \nabla f(x),x\bigr\rangle -\bigl\langle \nabla f(x),z\bigr\rangle -c \Vert x-z \Vert ^{2}.$$

Thus

$$-f(z)\geq f^{*} \bigl(\nabla f(x) \bigr)-\bigl\langle \nabla f(x),z\bigr\rangle -c \Vert x-z \Vert ^{2},$$

so that

\begin{aligned} &f^{*} \bigl(\nabla f(y) \bigr)\geq\bigl\langle \nabla f(y),z\bigr\rangle -f(z), \\ &f^{*} \bigl(\nabla f(y) \bigr)\geq\bigl\langle \nabla f(y),z\bigr\rangle +f^{*} \bigl(\nabla f(x) \bigr)-\bigl\langle \nabla f(x),z\bigr\rangle -c \Vert x-z \Vert ^{2}, \end{aligned}

and hence

\begin{aligned} &f^{*} \bigl(\nabla f(y) \bigr)- f^{*} \bigl(\nabla f(x) \bigr)-\bigl\langle \nabla f(y)-\nabla f(x),x\bigr\rangle \\ &\quad\geq\bigl\langle \nabla f(y)-\nabla f(x),z-x\bigr\rangle -c \Vert x-z \Vert ^{2} \\ &\quad\geq\underset{z\in H}{\sup} \bigl\{ \bigl\langle \nabla f(y)-\nabla f(x),z-x \bigr\rangle -c \Vert x-z \Vert ^{2} \bigr\} . \end{aligned}

This means that, for all $$x, y \in H$$,

$$f^{*} \bigl(\nabla f(y) \bigr)- f^{*} \bigl(\nabla f(x) \bigr)-\bigl\langle \nabla f(y)-\nabla f(x),x\bigr\rangle \geq\frac{1}{2c} \bigl\Vert \nabla f(y)- \nabla f(x) \bigr\Vert ^{2}.$$

Changing the roles of x and y, we obtain

$$f^{*} \bigl(\nabla f(x) \bigr)- f^{*} \bigl(\nabla f(y) \bigr)-\bigl\langle \nabla f(x)-\nabla f(y),y\bigr\rangle \geq\frac{1}{2c} \bigl\Vert \nabla f(x)- \nabla f(y) \bigr\Vert ^{2}.$$

$$\bigl\langle \nabla f(x)-\nabla f(y),x-y\bigr\rangle \geq \frac {1}{c} \bigl\Vert \nabla f(x)-\nabla f(y) \bigr\Vert ^{2}.$$
(24)

Consequently, by the Cauchy-Schwarz inequality we obtain

$$\bigl\Vert \nabla f(x)-\nabla f(y) \bigr\Vert \leq c \Vert x-y \Vert \quad \mbox{for all }x, y\in H.$$

□

## 4 Newton’s method

The aim of this section is to solve the Euler equation

$$\nabla f(x)=0$$
(25)

by Newton’s method. The classic assumption is that $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ a $$C^{2}$$ mapping and the Hessian matrix $$\nabla ^{2} f(x)$$ of f at x is nonsingular. Here we prove the convergence of a natural extension of Newton’s method to solve (25) assuming that $$\nabla f(\cdot)$$ admits $$\beta_{f}(\cdot)$$ as a first-order approximation. Clearly, if $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ is a $$C^{1,1}$$ mapping, then using Corollary 1, we obtain that $$\nabla f(\cdot)$$ admits $$\partial_{H}^{2} f(\cdot)$$ as a first-order approximation.

This algorithm has been proposed by Cominetti et al. [23] with $$C^{1,1}$$ data. Only some ideas were given, but it remains as an open question to state results on rate of convergence and local convergence of that algorithm. In the sequel, $$f: \mathbb{R}^{p} \rightarrow\mathbb {R}$$ is a Fréchet-differentiable mapping such that its Fréchet derivative admits a first-order approximation, and is a solution of (25).

### Theorem 5

Let $$f: \mathbb{R}^{p} \rightarrow\mathbb{R}$$ be a Fréchet-differentiable function, and be a solution of (25). Let $$\varepsilon, r, K >0$$ be such that $$\nabla f(\cdot)$$ admits $$\beta_{f}(\bar{x})$$ as a first-order approximation at such that, for each $$x\in\mathbb{B}_{\mathbb{R}^{p}} (\bar{x},r)$$, there exists an invertible element $$B(x) \in\mathcal{B}_{f}(x)$$ satisfying $$\Vert B(x)^{-1} \Vert \leq K$$ and $$\xi:= \varepsilon K<1$$. Then the sequence $$(x_{k})$$ generated by Algorithm $$(\mathcal {M})$$ is well defined for every $$x_{0} \in\mathbb{B}_{\mathbb{R}^{p}}(\bar {x},r)$$ and converges linearly to with rate ξ.

### Proof

Since $$\nabla f(\bar{x})=0$$, we have

$$x_{k+1}-\bar{x} =B(x_{k})^{-1} \bigl( \nabla f( \bar{x})-\nabla f(x_{k})+B(x_{k}) (x_{k} -\bar{x}) \bigr).$$

We inductively obtain that

$$\Vert x_{k+1}-\bar{x} \Vert \leq K \bigl\Vert \nabla f (\bar {x})- \nabla f(x_{k})+B(x_{k}) (x_{k}-\bar{x}) \bigr\Vert .$$

Thus

$$\Vert x_{k+1}-\bar{x} \Vert \leq\xi \Vert x_{k}- \bar{x} \Vert ,$$

which means that $$x_{k+1} \in\mathbb{B}_{\mathbb{R}^{p}}(\bar{x},r)$$, and we have $$\Vert x_{k+1}-\bar{x} \Vert \leq\xi^{k} \Vert x_{0}-\bar{x} \Vert$$. Therefore the whole sequence $$(x_{k})$$ is well defined and converges to . □

Now let us consider the following algorithm under less assumptions.

### Theorem 6

Let U be an open set of $$\mathbb{R}^{p}$$, $$x_{0}\in U$$, and $$f: \mathbb {R}^{p} \rightarrow\mathbb{R}$$ be a Fréchet-differentiable function on U. Let $$\varepsilon, r, K >0$$ be such that $$\nabla f(\cdot)$$ admits $$\beta_{f}(x_{0})$$ as a strict first-order approximation at $$x_{0}$$ such that, for each $$x\in\mathbb{B}_{\mathbb{R}^{p}} (x_{0},r)$$, there exists a right inverse of $$B(x)\in\beta_{f}(x_{0})$$, denoted by $$\tilde {B}(x)$$, satisfying $$\Vert \tilde{B}(x)(\cdot) \Vert \leq K \Vert \cdot \Vert$$ and $$\xi:= \varepsilon K<1$$.

If $$\Vert \nabla f(x_{0}) \Vert \leq K^{-1}(1-\xi)r$$ and f is continuous, then the sequence $$(x_{k})$$ generated by Algorithm $$(\mathcal {M}')$$ is well defined and converges to a solution of (25). Moreover, we have $$\Vert x_{k}-\bar {x} \Vert \leq r\xi^{k}$$ for all $$k\in\mathbb{N}$$ and $$\Vert \bar {x}-x_{0} \Vert \leq \Vert \nabla f(x_{0}) \Vert K(1-\xi)^{-1}< r$$.

### Proof

We prove by induction that $$x_{k}\in x_{0} +r \mathbb{B}_{ \mathbb{R}^{p}}$$, $$\Vert x_{k+1}-x_{k} \Vert \leq K \xi^{k} \Vert \nabla f(x_{0}) \Vert$$, and $$\Vert \nabla f(x_{k}) \Vert \leq\xi ^{k} \Vert \nabla f(x_{0}) \Vert$$ for all $$k\in\mathbb{N}$$. For $$k=0$$, these relations are obvious. Assuming that they are valid for $$k< n$$, we get

\begin{aligned} \Vert x_{n} -x_{0} \Vert &\leq\underset{k=0}{ \overset{n-1}{\sum}} \Vert x_{k+1}-x_{k} \Vert \leq K \bigl\Vert \nabla f(x_{0}) \bigr\Vert \underset{k=0}{ \overset{\infty}{\sum}} \xi^{k} \\ &\leq K \bigl\Vert \nabla f(x_{0}) \bigr\Vert (1-\xi )^{-1}< r. \end{aligned}

Thus $$x_{n} \in x_{0} +r \mathbb{B}_{ \mathbb{R}^{p}}$$ and since $$\nabla f(x_{n-1})+B(x_{n-1})(x_{n}-x_{n-1})=0$$, from Algorithm $$(\mathcal {M}')$$ we have

\begin{aligned} \bigl\Vert \nabla f(x_{n}) \bigr\Vert &\leq \bigl\Vert \nabla f(x_{n})- \nabla f(x_{n-1})-B(x_{n-1}) (x_{n}-x_{n-1}) \bigr\Vert \leq\varepsilon \Vert x_{n}-x_{n-1} \Vert \\ &\leq\xi^{n} \bigl\Vert \nabla f (x_{0}) \bigr\Vert \end{aligned}

and

$$\Vert x_{n+1}-x_{n} \Vert \leq K \xi^{n} \bigl\Vert \nabla f(x_{0}) \bigr\Vert .$$

Since $$\xi<1$$, the sequence $$(x_{n})$$ is a Cauchy sequence and hence converges to some $$\bar{x}\in\mathbb{R}^{p}$$ with $$\Vert x_{0}- \bar {x} \Vert < r$$. Since f is a continuous function, we get $$\nabla f (\bar{x})=0$$. □

## 5 Conclusions

In this paper, we investigate the concept of first- and second-order approximations to generalize some results such as optimality conditions for a subclass of convex functions called strongly convex functions of order γ. We also present an extension of Newton’s method to solve the Euler equation under weak assumptions.

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## Acknowledgements

The author wishes to express his heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.

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Correspondence to Ikram Daidai.

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Daidai, I. Second-order optimality conditions for nonlinear programs and mathematical programs. J Inequal Appl 2017, 212 (2017). https://doi.org/10.1186/s13660-017-1487-8