Second-order optimality conditions for nonlinear programs and mathematical programs

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.


Introduction
The concept of approximations of mappings was introduced by Thibault []. Sweetser [] considered approximations by subsets of the space of continuous linear maps L(X, Y ), where X and Y are Banach spaces, and Ioffe [] by the so-called fans. This approach was revised by Jourani and Thibault []. Another approach belongs to Allali and Amahroq []. Following the same ideas, Amahroq and Gadhi [, ] 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, [-] and references therein. Let f be a mapping from a Banach space X into R, and let C ⊂ 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 [] to detect some equivalent properties of strongly convex functions of order γ and to characterize the latter. Further-more, for a C , 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 secondorder approximation data.
The rest of the paper is written as follows. Section  contains basic definitions and preliminary results. Section  is devoted to mains results. In Section , we point out an extension of Newton's method and prove its local convergence.

Preliminaries
Let X and Y be two Banach spaces. We denote by L(X, Y ) the set of all continuous linear mappings from X into Y , by B(X × X, Y ) the set of all continuous bilinear mappings from X × X into Y , and by 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 ·, · for the canonical bilinear forms with respect to the dualities X * , X and Y * , Y .
for all x ∈x + δB X .
It is easy to check that Definition  is equivalent to the following: for all ε > , there exists δ >  such that for all x ∈x + δB X .

Remark  If
is a first-order approximation of f atx, then () means that for any Hence, for any x ∈ B(x, δ) and is a first-order approximation of f atx.
In [], it is also shown that when f is a continuous function, it admits as an approximation the symmetric subdifferential defined and studied in []. The next proposition shows that Proposition  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 [, ] for more details) and the definition of upper semicontinuity.
Definition  The generalized Jacobian of a function g : R p → R q atx, denoted ∂ c g(x), is the convex hull of all matrices M of the form where x n →x, g is differentiable at x n for all n, and Jg denotes the q × p usual Jacobian matrix of partial derivatives.
Definition  A set-valued mapping F : R p ⇒ R q is said to be upper semicontinuous at a pointx ∈ R p if, for every ε > , there exists δ >  such that for every x ∈ R p such that x -x < δ.
Proposition  Let g : R p → R q be a locally Lipschitz function atx. Then the generalized Jacobian ∂ c g(x) of g atx is a first-order approximation of g atx.
Proof Since the set-valued mapping ∂ c g(·) is upper semicontinuous, for all ε > , there exists r  >  such that We may assume that g is Lipschitzian inx + r  B R p . Let x ∈x + r  B R p . We apply [], Prop. .., to derive that there exits c ∈ ]x,x[ such that which means that ∂ c g(x) is a first-order approximation of g atx.
Recall that a mapping f : X → Y is said to be C , atx if it is Fréchet differentiable in neighborhood ofx and if its Fréchet derivative ∇f (·) is Lipschitz atx.
Letx ∈ R p , and let f : R p → R be a C , function atx. The generalized Hessian matrix of f atx was introduced and studied by Hiriart-Urruty et al. [] is the compact nonempty convex set where dom ∇  f is the effective domain of ∇  f (·).
In this case the pair (A f (x), B f (x)) is called a second-order approximation of f atx. It is called a compact second-order approximation if A f (x) and B f (x) are compacts.
) as a second-order approximation, where ∇f (x) and ∇  f (x) are, respectively, the first-and second-order Fréchet derivatives of f atx.
) as a second-order approximation atx.
for all x ∈x + rB X .
To derive some results for γ -strong convex functions, the following notions are needed.
and such that for all θ ∈ [, ] and x, y ∈ X.
Of course, when c = , f is called a convex function. Otherwise, f is said γ -strongly convex. This class has been introduced by Polyak [] when γ =  and g(θ ) = θ (θ ) and studied by many authors. Recently, a characterization of γ -strongly convex functions has been shown in []. For example, if f is C  and γ ≥ , then () is equivalent to . Note that the Fenchel-subdifferential defined by () coincides with the Clarke subdifferential of f atx if the function f is convex.
We also need to recall the following definitions.
It has been proved in [] that if f is a C  mapping, then () is equivalent to

Main results
In this section, we obtain the main results of the paper related to strongly convex functions of order γ defined by ()-(). 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 → R∪{+∞}, the 'right-hand' directional derivative at x in dom f (the domain of f ) exists and, for each h ∈ X, is Proof By the definition of A f (x), there exist δ >  and r : By letting t →  + the directional derivative of f atx satisfies .
Proof Let d ∈ X be fixed and define x n :=x +  n d. Using Definition , we get, for n large enough, A n ∈ A f (x) and b n ∈ [-, ] such that By γ -strong convexity we obtain By the compactness of A f (x), extracting a subsequence if necessary, we may assume that there exists A ∈ A f (x) such that A n , d → A, d ; and hence we obtain . By the separation theorem there exists h ∈ X with h =  such that Let t >  sufficiently small, so that in contradiction with relation () 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  Let γ ≥ ,x ∈ R p , and let f : . For all n, there exists x n ∈ dom ∇f such that x n →x and ∇f (x n ) → A. Since f is γ -strongly convex and Fréchet differentiable at x n for all n ∈ N, it follows by () that Letting n → +∞, we get Corollary  Let γ ≥ ,x ∈ R p , and let f : R p → R be continuous atx. Assume that f is a γ -strongly convex function. Then, for all ε > , there exists r >  such that for all x ∈x + rB R p , which means that ∂ (γ ,c) f (x) is a first-order approximation of f atx.
Proof It is clear that ∂ c f (x) is a first-order approximation of atx. We end the proof by Propositions  and .
The converse of Proposition  holds if () is valid for any A ∈ A f (x) and x ∈ X.
Proposition  Let γ ≥  and f : X → R ∪ {+∞}. Assume that, for each x ∈ X, f admits a first-order approximation . Then f is γ -strongly convex.
Proof Define x θ := θ u + (θ )v for θ ∈ [, ] and u, v ∈ X. Let us take A ∈ A f (x θ ). Then Multiplying this inequality by θ , we obtain In a similar way, since We deduce by addition of (a ) and (a ) that The next results are devoted to presenting some useful properties of the generalized Hessian matrix for a C , function in the finite-dimensional setting and a characterization of γ -strongly convex functions with the help of a second-order approximation.
Proposition  Letx ∈ X, and let f : X → R ∪ {+∞} be convex and Fréchet differentiable atx. Suppose that f admits (∇f (x), B f (x)) as a second-order approximation atx and that If f is -strongly convex, then we obtain for some c > .
Proof We prove only the case where f is convex. In a similar way, we can prove the other case. Let d ∈ X and ε >  be fixed. We get for n large enough B n ∈ B f (x) and b n ∈ [-, ] such that Since f is convex, we obtain When X is a finite-dimensional space, we get the following essential result.
Proposition  Let f : R p → R be a C , function atx. Assume that f is γ -strongly convex. Then, for any B ∈ ∂  H f (x), we have the following inequality: Letting n → +∞, 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 ∈ ∂  H f (x) is invertible. The next result proves the converse of Proposition . Let us first recall the following characterization of l.s.c. γ -strongly convex functions. a proper and l.s.c. function. Then f is γ -strongly convex iff ∂ c f is γ -strongly monotone, that is, there exists a positive real number c such that, for all x, y ∈ X, x * ∈ ∂ c f (x), and y * ∈ ∂ c f (y), we have We are now in position to state our main second result.
Theorem  Let f : R p → R be a C , function. Assume that ∂  H f (·) satisfies relation () at any x ∈ R p . Then f is γ -strongly convex.
Proof Let t ∈ [, ] and u, v ∈ R p . Define ϕ : R → R as By using calculus rules it follows that Theorem  Let f : H → R be continuous on a Hilbert space H. Suppose that f is convex (or -strongly convex) and that -f is -paraconvex. Then f is Fréchet differentiable on H, and for some c > , we have that Proof Let x  ∈ X. Clearly, f is locally Lipschitzian at x  . Now let x *  and x *  be arbitrary elements of ∂ c f (x  ) and ∂ c (-f )(x  ), respectively. By [], Thm. ., there exists c >  such that ∂ c (-f )(x  ) = ∂ (,c) (-f )(x  ), and for any y ∈ H and positive real θ , we have Adding (a) and (a ), we get and hence Since (a) and (a ) hold for any θ >  and y ∈ H, we deduce that, for θ = , Hence, for all y = , we obtain Letting y →  in (), we conclude that f is Fréchet differentiable at x  . Now since -f is -paraconvex and f is Fréchet differentiable, we may prove that there exists c >  such that For every z ∈ H, we have that and hence This means that, for all x, y ∈ H, Changing the roles of x and y, we obtain So by addition we get Consequently, by the Cauchy-Schwarz inequality we obtain ∇f (x) -∇f (y) ≤ c xy for all x, y ∈ H.

Newton's method
The aim of this section is to solve the Euler equation by Newton's method. The classic assumption is that f : R p → R a C  mapping and the Hessian matrix ∇  f (x) of f at x is nonsingular. Here we prove the convergence of a natural extension of Newton's method to solve () assuming that ∇f (·) admits β f (·) as a firstorder approximation. Clearly, if f : R p → R is a C , mapping, then using Corollary , we obtain that ∇f (·) admits ∂  H f (·) as a first-order approximation. This algorithm has been proposed by Cominetti et al. [] with C , 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 : R p → R is a Fréchetdifferentiable mapping such that its Fréchet derivative admits a first-order approximation, andx is a solution of ().
Algorithm (M) Starting from an arbitrary point x  ∈ B(x, r), generate the sequences (x k ) and (h k ) as follows: Theorem  Let f : R p → R be a Fréchet-differentiable function, andx be a solution of (). Let ε, r, K >  be such that ∇f (·) admits β f (x) as a first-order approximation atx such that, for each x ∈ B R p (x, r), there exists an invertible element B(x) ∈ B f (x) satisfying B(x) - ≤ K and ξ := εK < . Then the sequence (x k ) generated by Algorithm (M) is well defined for every x  ∈ B R p (x, r) and converges linearly tox with rate ξ .
Proof Since ∇f (x) = , we have We inductively obtain that which means that x k+ ∈ B R p (x, r), and we have x k+ -x ≤ ξ k x  -x . Therefore the whole sequence (x k ) is well defined and converges tox. Now let us consider the following algorithm under less assumptions.
Algorithm (M ) Starting from an arbitrary point x  ∈ R p , generate the sequences (x k ) and (h k ) as follows: (i) h k ∈ R p is a solution of  ∈ ∇f (x k ) + β f (x  )(h k ), and (ii) x k+ = x k + h k .
Theorem  Let U be an open set of R p , x  ∈ U, and f : R p → R be a Fréchet-differentiable function on U. Let ε, r, K >  be such that ∇f (·) admits β f (x  ) as a strict first-order approximation at x  such that, for each x ∈ B R p (x  , r), there exists a right inverse of B(x) ∈ β f (x  ), denoted byB(x), satisfying B (x)(·) ≤ K · and ξ := εK < .
If ∇f (x  ) ≤ K - (ξ )r and ∇f is continuous, then the sequence (x k ) generated by Algorithm (M ) is well defined and converges to a solutionx of (). Moreover, we have x k -x ≤ rξ k for all k ∈ N and xx  ≤ ∇f (x  ) K(ξ ) - < r.
Proof We prove by induction that x k ∈ x  + rB R p , x k+x k ≤ Kξ k ∇f (x  ) , and ∇f (x k ) ≤ ξ k ∇f (x  ) for all k ∈ N. For k = , these relations are obvious. Assuming that they are valid for k < n, we get Thus x n ∈ x  + rB R p and since ∇f (x n- ) + B(x n- )(x nx n- ) = , from Algorithm (M ) we have and x n+x n ≤ Kξ n ∇f (x  ) .
Since ξ < , the sequence (x n ) is a Cauchy sequence and hence converges to somex ∈ R p with x  -x < r. Since ∇f is a continuous function, we get ∇f (x) = .

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.