 Research
 Open access
 Published:
Secondorder optimality conditions for nonlinear programs and mathematical programs
Journal of Inequalities and Applications volume 2017, Article number: 212 (2017)
Abstract
It is well known that secondorder 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 secondorder approximations derived from (Optimization 40(3):229246, 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 secondorder 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 socalled 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 setvalued 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 secondorder 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 finitedimensional setting, we show some simple facts. We also provide an extension of Newton’s method to solve an Euler equation with secondorder 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 firstorder 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 firstorder 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 normbounded (resp. compact), then it is called a bounded (resp. compact) firstorder 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 finitedimensional 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 firstorder 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 vectorvalued function. Let us first recall the definition of the generalized Jacobian for a vectorvalued 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 setvalued 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 firstorder approximation of g at x̄.
Proof
Since the setvalued 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 firstorder 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 HiriartUrruty 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 firstorder approximation at x̄.
Definition 4
[1]
We say that \(f: X \rightarrow Y\) admits a secondorder 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 firstorder 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 secondorder approximation of f at x̄. It is called a compact secondorder 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 secondorder approximation, where \(\nabla f(\bar{x})\) and \(\nabla^{2} f(\bar{x})\) are, respectively, the first and secondorder 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 secondorder approximation at x̄.
Proposition 4
Let \(f: X\to Y\) be a Fréchetdifferentiable mapping. If \((\nabla f(\bar{x}),\mathcal{B}_{f}(\bar{x}))\) is a bounded secondorder 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 γstrong 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 γstrongly 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 γstrongly 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 γstrongly 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 Fenchelsubdifferential 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 Fenchelsubdifferential 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 2paraconvex 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 firstorder 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 ‘righthand’ 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 γstrongly 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 γstrong 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 finitedimensional 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 γstrongly 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 γstrongly 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 γstrongly 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 firstorder approximation of f at x̄.
Proof
It is clear that \(\partial_{c} f (\bar{x})\) is a firstorder 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 firstorder approximation \(\mathcal{A}_{f}(x)\) such that \(\mathcal{A}_{f}(x)\subset\partial _{(\gamma ,c)} f(x)\). Then f is γstrongly 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 γstrongly convex. □
The next results are devoted to presenting some useful properties of the generalized Hessian matrix for a \(C^{1,1}\) function in the finitedimensional setting and a characterization of γstrongly convex functions with the help of a secondorder 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 secondorder 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 2strongly 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 finitedimensional 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 γstrongly 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 secondorder 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 γstrongly convex, there exists \(c>0\) such that
Letting \(n\to+\infty\), we have
□
The preceding result shows that γstrongly 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. γstrongly 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 γstrongly convex iff \(\partial_{c} f\) is γstrongly 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 γstrongly convex.
Proof
Let \(t\in[0,1]\) and \(u, v\in\mathbb{R}^{p}\). Define \(\varphi:\mathbb {R}\to\mathbb{R}\) as
so that \(\varphi'(t):=\langle \nabla f(u+t(vu)),vu\rangle \). 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 \(B_{t_{0}} \in\partial^{2}_{H} f(u+t_{0}(vu))\) such that \(\langle \nabla f(v)\nabla f(u),vu\rangle =\langle B_{t_{0}} (vu),vu\rangle \). The result follows from Theorem 2. □
HiriartUrruty 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 2strongly convex) and that −f is 2paraconvex. 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 singlevalued. 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 2paraconvex 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 CauchySchwarz 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 firstorder 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 firstorder 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 Fréchetdifferentiable mapping such that its Fréchet derivative admits a firstorder approximation, and x̄ is a solution of (25).
Theorem 5
Let \(f: \mathbb{R}^{p} \rightarrow\mathbb{R}\) be a Fréchetdifferentiable 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 firstorder 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 Fréchetdifferentiable function on U. Let \(\varepsilon, r, K >0\) be such that \(\nabla f(\cdot)\) admits \(\beta_{f}(x_{0})\) as a strict firstorder 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 \(\nabla f(x_{n1})+B(x_{n1})(x_{n}x_{n1})=0\), 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 secondorder 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
Allali, K, Amahroq, T: Second order approximations and primal and dual necessary optimality conditions. Optimization 40(3), 229246 (1997)
Thibault, L: Subdifferentials of compactly Lipschitzian vectorvalued functions. Ann. Mat. Pura Appl. (4) 125, 157192 (1980)
Sweetser, TH: A minimal setvalued strong derivative for setvalued Lipschitz functions. J. Optim. Theory Appl. 23, 539562 (1977)
Ioffe, AD: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266, 156 (1981)
Jourani, A, Thibault, L: Approximations and metric regularity in mathematical programming in Banach space. Math. Oper. Res. 18(2), 390401 (1993)
Amahroq, T, Gadhi, N: On the regularity condition for vector programming problems. J. Glob. Optim. 21(4), 435443 (2001)
Amahroq, T, Gadhi, N: Second order optimality conditions for the extremal problem under inclusion constraints. J. Math. Anal. Appl. 285(1), 7485 (2003)
Amahroq, T, Daidai, I, Syam, A: γstrongly convex functions, γstrong monotonicity of their presubdifferential and γsubdifferentiability, application to nonlinear PDE. J. Nonlinear Convex Anal. (2017, to appear)
Crouzeix, JP, Ferland, JA, Zalinescu, C: αconvex sets and strong quasiconvexity. SIAM J. Control Optim. 22, 9981022 (1997)
Lin, GH, Fukushima, M: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 118, 6780 (2003)
Polyak, BT: Introduction to Optimization. Optimization Software, New York (1987). Translated from the Russian, with a foreword by Dimitri P. Bertsekas
Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877898 (1976)
Vial, JP: Strong convexity of sets and functions. J. Math. Econ. 9(12), 187205 (1982). doi:10.1016/03044068(82)90026X
Vial, JP: Strong and weak convexity of sets and functions. Math. Oper. Res. 8(2), 231259 (1983)
Zălinescu, C: On uniformly convex functions. J. Math. Anal. Appl. 95(2), 344374 (1983)
Mordukhovich, BS, Shao, YH: On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2(12), 211227 (1995)
Clarke, FH: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Clarke, FH: On the inverse function theorem. Pac. J. Math. 64(1), 97102 (1976)
HiriartUrruty, JB, Strodiot, JJ, Nguyen, VH: Generalized Hessian matrix and secondorder optimality conditions for problems with \(C^{1,1}\) data. Appl. Math. Optim. 11(1), 4356 (1984)
Jourani, A: Subdifferentiability and subdifferential monotonicity of γparaconvex functions. Control Cybern. 25(4), 721737 (1996)
Phelps, RR: Convex Functions, Monotone Operators and Differentiability Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1989)
Lebourg, G: Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris 281, 795797 (1975)
Cominetti, R, Correa, R: A generalized secondorder derivative in nonsmooth optimization. SIAM J. Control Optim. 28(4), 789809 (1990)
Acknowledgements
The author wishes to express his heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The author read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Daidai, I. Secondorder optimality conditions for nonlinear programs and mathematical programs. J Inequal Appl 2017, 212 (2017). https://doi.org/10.1186/s1366001714878
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366001714878