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.
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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, [8–15] 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
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 x̄ if there exist \(\delta >0\) and a function \(r: X\to\mathbb{R}\) satisfying \(\lim _{x\to \bar {x} }r(x)=0\) such that
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
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 x̄, 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
Hence, for any \(x\in\mathbb{B}(\bar{x},\delta)\) and \(A(x)\in \mathcal{A}_{f}(\bar{x})\),
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 x̄.
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 x̄. Then the Clarke subdifferential of f at x̄,
is a first-order approximation of f at x̄.
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 x̄, denoted \(\partial_{c} g(\bar{x})\), is the convex hull of all matrices M of the form
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
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 x̄. Then the generalized Jacobian \(\partial_{c} g(\bar {x})\) of g at x̄ is a first-order approximation of g at x̄.
Proof
Since the set-valued mapping \(\partial_{c} g(\cdot)\) is upper semicontinuous, for all \(\varepsilon>0\), there exists \(r_{0}>0\) such that
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
Since
we have
which means that \(\partial_{c} g(\bar{x})\) is a first-order approximation of g at x̄. □
Recall that a mapping \(f: X \to Y\) is said to be \(C^{1,1}\) at x̄ if it is Fréchet differentiable in neighborhood of x̄ and if its Fréchet derivative \(\nabla f(\cdot)\) is Lipschitz at x̄.
Let \(\bar{x}\in\mathbb{R}^{p}\), and let \(f: \mathbb{R}^{p} \rightarrow \mathbb{R}\) be a \(C^{1,1}\) function at x̄. The generalized Hessian matrix of f at x̄ was introduced and studied by Hiriart-Urruty et al. [19] is the compact nonempty convex set
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 x̄. Then, ∇f admits \(\partial^{2}_{H} f(\bar{x})\) as a first-order approximation at x̄.
Definition 4
[1]
We say that \(f: X \rightarrow Y\) admits a second-order approximation at x̄ 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
-
(i)
\(\mathcal{A}_{f} (\bar{x})\) is a first-order approximation of f at x̄;
-
(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 x̄. 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 x̄ 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 x̄.
Proposition 3
[1]
Let \(f: \mathbb{R}^{p} \rightarrow\mathbb{R}\) be a \(C^{1,1}\) function at x̄. Then f admits \((\nabla f(\bar{x}),\frac {1}{2}\partial ^{2}_{H} f(\bar{x}))\) as a second-order approximation at x̄.
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 x̄. Then \(\nabla f(\cdot)\) is stable at x̄, that is, there exist \(c, r>0\) such that
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
and such that
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
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 x̄ is the set
Let \(\gamma>0\) and \(c>0\). The \((\gamma, c)\)-subdifferential of f at x̄ is the set
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 x̄ 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
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
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
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
Theorem 1
Let \(\bar{x}\in X\). If \(f:X\to\mathbb{R}\cup\{+\infty\}\) is convex and continuous at x̄ and if \(\mathcal{A}_{f}(\bar{x})\subset X^{*}\) is a convex \(w(X^{*},X)\)-closed approximation of f at x̄, then
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
By letting \(t\to0^{+}\) the directional derivative of f at x̄ satisfies
Using [21], Prop. 2.24, we get
Since \(\partial_{(\gamma,c)}f(\bar{x})\subset\partial_{\mathrm{Fen}} f(\bar {x})\), we deduce that
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 x̄. 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
By γ-strong convexity we obtain
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
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
Let \(t >0\) sufficiently small, so that
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 x̄. 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
Letting \(n\to+\infty\), we get
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 x̄. Assume that f is a γ-strongly convex function. Then, for all \(\varepsilon>0\), there exists \(r>0\) such that
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 x̄.
Proof
It is clear that \(\partial_{c} f (\bar{x})\) is a first-order approximation of at x̄. 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
Multiplying this inequality by θ, we obtain
In a similar way, since
we get
We deduce by addition of \((\mathrm{a}')\) and \((\mathrm{a}'')\) that
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 x̄. Suppose that f admits \((\nabla f(\bar{x}),\mathcal{B}_{f}(\bar{x}))\) as a second-order approximation at x̄ and that \(\mathcal {B}_{f}(\bar {x})\) is compact. Then there exists \(B\in\mathcal{B}_{f}(\bar{x})\) such that
If f is 2-strongly convex, then we obtain
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
Since f is convex, we obtain
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
and hence
□
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 x̄. Assume that f is γ-strongly convex. Then, for any \(B\in\partial^{2}_{H} f(\bar{x})\), we have the following inequality:
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 x̄. 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
Letting \(n\to+\infty\), we have
□
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
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
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
By using calculus rules it follows that
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 that −f is 2-paraconvex. Then f is Fréchet differentiable on H, and for some \(c>0\), we have that
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
and
Adding (a) and (a′), we get
and hence
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 \(\theta=1\),
and
Hence, for all \(y\neq0\), we obtain
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
For every \(z\in H\), we have that
Thus
so that
and hence
This means that, for all \(x, y \in H\),
Changing the roles of x and y, we obtain
So by addition we get
Consequently, by the Cauchy-Schwarz inequality we obtain
□
4 Newton’s method
The aim of this section is to solve the Euler equation
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 x̄ is a solution of (25).
Theorem 5
Let \(f: \mathbb{R}^{p} \rightarrow\mathbb{R}\) be a Fréchet-differentiable function, and x̄ 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 x̄ 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 x̄ with rate ξ.
Proof
Since \(\nabla f(\bar{x})=0\), we have
We inductively obtain that
Thus
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 x̄. □
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 x̄ 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
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
and
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.
References
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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
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DOI: https://doi.org/10.1186/s13660-017-1487-8