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Twice epi-differentiability of extended-real-valued functions with applications in composite optimization

The paper is devoted to the study of the twice epi-differentiablity of extended-real-valued functions, with an emphasis on functions satisfying a certain composite representation. This will be conducted under the parabolic regularity, a second-order regularity condition that was recently utilized in [13] for second-order variational analysis of constraint systems. Besides justifying the twice epi-differentiablity of composite functions, we obtain precise formulas for their second subderivatives under the metric subregularity constraint qualification. The latter allows us to derive second-order optimality conditions for a large class of composite optimization problems.


Introduction
This paper aims to provide a systematic study of the twice epi-differentiability of extend-realvalued functions in finite dimensional spaces. In particular, we pay special attention to the composite optimization problem minimize ϕ(x) + g(F (x)) over all x ∈ X, (1.1) restrictive assumption on the second subderivative, which does not hold for constrained optimization problems. Nor does this condition hold for other major composite functions related to eigenvalue optimization problems; see [24,Theorem 1.2] for more detail. Levy in [11] obtained upper and lower estimates for the second subderivative of the composite function from (1.1), but fell short of establishing the twice epi-differentiability for this framework. The authors and Mordukhovich observed recently in [14] that a second-order regularity, called parabolic regularity (see Definition 3.1), can play a major role toward the establishment of the twice epi-differentiability for constraint systems, namely when the outer function g in (1.1) is the indicator function of a closed convex set. This vastly alleviated the difficulty that was often appeared in the justification of the twice epi-differentiability for the latter framework and opened the door for crucial applications of this concept in theoretical and numerical aspects of optimization. Among these applications, we can list the following: • the calculation of proto-derivatives of subgradient mappings via the connection between the second subderivative of a function and the proto-derivative of its subgradient mapping (see equation (3.21)); • the calculation of the second subderivative of the augmented Lagrangian function associated with the composite problem (1.1), which allows us to characterize the second-order growth condition for the augmented Lagrangian problem (cf. [14,Theorems 8.3 & 8.4]); • the validity of the derivative-coderivative inclusion (cf. [21,Theorem 13.57]), which has important consequences in parametric optimization; see [15,Theorem 5.6] for a recent application in the convergence analysis of the sequential quadratic programming (SQP) method for constrained optimization problems. In this paper, we continue the path, initiated in [14] for constraint systems, and show that the twice epi-differentiability of the objective function in (1.1) can be guaranteed under parabolic regularity. To achieve this goal, we demand that the outer function g from (1.1) be locally Lipschitz continuous relative to its domain; see the next section for the precise definition of this concept. Shapiro in [22] used a similar condition but in addition assumed that this function is finite-valued. The latter does bring certain restrictions for (1.1) by excluding constrained problems as well as piecewise linear-quadratic composite problems. As shown in Example 4.7, major classes of constrained and composite optimization problems satisfy this Lipschitzian condition. However, some composite problems such as the spectral abcissa minimization (cf. [4]), namely the problem of minimizing the largest real parts of eigenvalues, can not be covered by (1.1).
The rest of the paper is organized as follows. Section 2 recalls important notions of variational analysis that are used throughout this paper. Section 3 begins with the definition of parabolic regularity of extended-real-valued functions. Then we justify that parabolic regularity amounts to a certain duality relationship between the second subderivative and parabolic subderivative. Employing this, we show that the twice epi-differentiability of extended-real-valued functions can be guaranteed if they are parabolically regular and parabolic epi-differentiable. Section 4 is devoted to important second-order variational properties of parabolic subderivatives. In particular, we establish a chain rule for parabolic subderivatives of composite functions in (1.1) under the metric subregularity constraint qualification. In Section 5, we establish chain rules for the parabolic regularity and for the second subderivative of composite functions, and consequently establish their twice epi-differentiability. Section 6 deals with important applications of our results in second-order optimality conditions for the composite optimization problem (1.1). We close the paper by achieving a characterization of the strong metric subregularity of the subgradient mapping of the objective function in (1.1) via the second-order sufficient condition for this problem.
In what follows, X and Y are finite-dimensional Hilbert spaces equipped with a scalar product ·, · and its induced norm · . By B we denote the closed unit ball in the space in question and by B r (x) := x + rB the closed ball centered at x with radius r > 0. For any set C in X, its indicator function is defined by δ C (x) = 0 for x ∈ C and δ C (x) = ∞ otherwise. We denote by d(x, C) the distance between x ∈ X and a set C. For v ∈ X, the subspace {w ∈ X| w, v = 0} is denoted by {v} ⊥ . We write x(t) = o(t) with x(t) ∈ X and t > 0 to mean that x(t) /t goes to 0 as t ↓ 0. Finally, we denote by IR + (respectively, IR − ) the set of non-negative (respectively, non-positive) real numbers.

Preliminary Definitions in Variational Analysis
In this section we first briefly review basic constructions of variational analysis and generalized differentiation employed in the paper; see [12,21] for more detail. A family of sets C t in X for t > 0 converges to a set C ⊂ X if C is closed and lim t↓0 d(w, C t ) = d(w, C) for all w ∈ X.
Given a nonempty set C ⊂ X withx ∈ C, the tangent cone T C (x) to C atx is defined by We say a tangent vector w ∈ T C (x) is derivable if there exist a constant ε > 0 and an arc ξ : [0, ε] → C such that ξ(0) =x and ξ ′ + (0) = w, where ξ ′ + signifies the right derivative of ξ at 0, defined by The set C is called geometrically derivable atx if every tangent vector w to C atx is derivable. The geometric derivability of C atx can be equivalently described by the sets [C −x]/t converging to T C (x) as t ↓ 0. Convex sets are important examples of geometrically derivable sets. The second-order tangent set to C atx for a tangent vector w ∈ T C (x) is given by A set C is said to be parabolically derivable atx for w if T 2 C (x, w) is nonempty and for each u ∈ T 2 C (x, w) there are ε > 0 and an are ξ : [0, ε] → C with ξ(0) =x, ξ ′ + (0) = w, and ξ ′′ It is well known that if C ⊂ X is convex and parabolically derivable atx for w, then the secondorder tangent set T 2 C (x, w) is a nonempty convex set in X (cf. [2, page 163]). Given the function f : X → IR := (−∞, ∞], its domain and epigraph are defined, respectively, by The regular subdifferential of f atx ∈ dom f is defined by The subdifferential of f atx is given by where x k f →x stands for x k →x and f (x k ) → f (x). We say that v ∈ X is a proximal subgradient of f atx if there exists r ∈ IR + and a neighborhood U ofx such that for all x ∈ U we have The set of all such v is called the proximal subdifferential of f atx and is denoted by ∂ p f (x). By definitions, it is not hard to obtain the inclusions ∂ p f (x) ⊂ ∂f (x) ⊂ ∂f (x). Given a nonempty set C ⊂ X, the proximal and regular normal cones to C atx ∈ C are defined, respectively, by Similarly, we define the (limiting/Mordukhovich) normal cone of C atx by N C (x) := ∂δ C (x). Consider a set-valued mapping S : X ⇒ Y with its domain and graph defined, respectively, by The graphical derivative of S at (x,ȳ) ∈ gph S is defined by Recall that a set-valued mapping S : X ⇒ Y is metrically regular around (x,ȳ) ∈ gph S if there are constants κ ∈ IR + and ε > 0 such that the distance estimate holds. When y =ȳ in the above estimate, the mapping S is called metrically subregular at (x,ȳ). The set-valued mapping S is called strongly metrically subregular at (x,ȳ) if there are a constant κ ∈ IR + and a neighborhood U ofx such that the estimate x −x ≤ κ d(ȳ, S(x)) for all x ∈ U holds. It is known (cf. [8,Theorem 4E.1]) that the set-valued mapping S is strongly metrically subregular at (x,ȳ) if and only if we have Given a function f : X → IR and a pointx with f (x) finite, the subderivative function df (x) : Define the parametric family of second-order difference quotients for f atx forv ∈ X by If f (x) is finite, then the second subderivative of f atx forv is given by Below, we collect some important properties of the second subderivative that are used throughout this paper. Parts (i) and (ii) were taken from [21,Proposition 13.5] and part (iii) was recently observed in [13, Theorem 4.1(i)]. Proposition 2.1 (properties of second subderivative). Let f : X → IR and (x,v) ∈ X × X with f (x) finite. Then the following conditions hold: is a proper function, meaning that d 2 f (x,v)(w) > −∞ for all w ∈ X and its effective domain, defined by is nonempty, then we always have the inclusion Following [21,Definition 13.6], a function f : X → IR is said to be twice epi-differentiable atx forv ∈ X, with f (x) finite, if the sets epi ∆ 2 t f (x,v) converge to epi d 2 f (x,v) as t ↓ 0. The latter means by [21,Proposition 7.2] that for every sequence t k ↓ 0 and every w ∈ X, there exists a sequence w k → w such that We say that a function f : X → IR is Lipschitz continuous aroundx relative to C ⊂ dom f with constant ℓ ∈ IR + ifx ∈ C and there exists a neighborhood U ofx such that Such a function is called locally Lipschitz continuous relative to C if for everyx ∈ C, it is Lipschitz continuous aroundx relative to C. Piecewise linear-quadratic functions (not necessarily convex) and an indicator function of a nonempty set are important examples of functions that are locally Lipschitz continuous relative to their domains.
Proposition 2.2 (domain of subderivatives). Let f : X → IR be Lipschitz continuous around x relative to its domain with constant ℓ ∈ IR + . Then we have dom df (x) = T dom f (x). In particular, for every w ∈ T dom f (x), the subderivative df (x)(w) is finite.
Proof. The inclusion dom df (x) ⊂ T dom f (x) results directly from the definition. To prove the opposite inclusion, pick w ∈ T dom f (x). This gives us some sequences t k ↓ 0 and w k → w such thatx + t k w k ∈ dom f for all k ∈ IN. Using this and the Lipschitz continuity of f aroundx relative to its domain implies that for all k sufficiently large we have This clearly yields |df (x)(w)| ≤ ℓ w . Thus df (x)(w) is finite and so w ∈ dom df (x). This gives us the inclusion T dom f (x) ⊂ dom df (x) and hence completes the proof.

Twice Epi-Differetiability of Parabolically Regular Functions
This section aims to delineate conditions under which the twice epi-differenibility of extendreal-valued functions can be established. To this end, we appeal to an important second-order regularity condition, called parabolic regularity, which was recently exploited in [14] to study a similar property for constraint systems. We begin with the definition of this regularity condition.
Definition 3.1 (parabolic regularity). A function f : X → IR is parabolically regular atx for v ∈ X if f (x) is finite and if for any w such that d 2 f (x,v)(w) < ∞, there exist, among the sequences t k ↓ 0 and w k → w with A nonempty set C ⊂ X is said to be parabolically regular atx forv if the indicator function δ C is parabolically regular atx forv.
Although the notion of parabolic regularity was introduced first in [21, Definition 13.65], its origin goes back to [5,Theorem 4.4], where Chaney observed a duality relationship between his second-order generalized derivative and the parabolic subderivative, defined in [1] by Ben-Tal and Zowe. This duality relationship was derived later by Rockafellar [18,Proposition 3.5] for convex piecewise linear-quadratic functions. As shown in Proposition 3.6 below, the latter duality relationship is equivalent to the concept of parabolic regularity from Definition 3.1 provided that v, appearing in Definition 3.1, is a proximal subgradient. A different second-order regularity was introduced by Bonnans, Comminetti, and Shapiro [3, Definition 3] for sets, which was later extended in [2, Definition 3.93] for functions. It is not difficult to see that parabolic regularity is implied by the second-order regularity in the sense of [3]; see [2,Proposition 3.103] for a proof of this result. Moreover, the example from [2, page 215] shows that the converse implication may not hold in general.
We showed in [14] that important sets appearing in constrained optimization problems, including polyhedral convex sets, the second-order cone, and the cone of positive semidefinite symmetric matrices, are parabolically regular. Below, we add two important classes of functions for which this property automatically fulfill. We begin first by convex piecewise-linear quadratic functions and then consider eigenvalues functions. While the former was justified in [21,Theorem 13.67], we provide below a different and simpler proof.
where A i is an n × n symmetric matrix, a i ∈ IR n , and α i ∈ IR for i = 1, · · · , s. It was proven in [21,Propsoition 13.9] that the second subderivative of f atx forv ∈ ∂f (x) can be calculated by To prove the parabolic regularity of f atx forv, pick a vector w ∈ IR n with d 2 f (x,v)(w) < ∞. This implies that there is an i with 1 ≤ i ≤ s such that Since C i is a polyhedral convex set, we conclude from [21, Exercise 6.47] that there exists an ε > 0 such thatx + tw ∈ C i for all t ∈ [0, ε]. Pick a sequence t k ↓ 0 such that t k ∈ [0, ε] and let w k := w for all k ∈ IN. Thus a simple calculation tells us that is clearly holds, f is parabolic regular atx forv. . Let X = S n be the space of n × n symmetric real matrices, which is conveniently treated via the inner product A, B := tr AB with tr AB standing for the sum of the diagonal entries of AB. For a matrix A ∈ S n , we denote by A † the Moore-Penrose pseudo-inverse of A and by eig A = (λ 1 (A), . . . , λ n (A)) the vector of eigenvalues of A in decreasing order with eigenvalues repeated according to their multiplicity. Given i ∈ {1, . . . , n}, denote by ℓ i (A) the number of eigenvalues that are equal to λ i (A) but are ranked before i including λ i (A). This integer allows us to locate λ i (A) in the group of the eigenvalues of A as follows: The eigenvalue λ i−ℓ i (A)+1 (A), ranking first in the group of eigenvalues equal to λ i (A), is called the leading eigenvalue. For any i ∈ {1, . . . , n}, define now the function α i : S n → IR by (3.7) It was proven in [24, Theorem 2.1] that ∂α i (A) = ∂α i (A) and that the second subderivative of α i at A for any V ∈ ∂α i (A) is calculated for every W ∈ S n by where I n stands for the n × n identity matrix. Moreover, for any W ∈ S n with d 2 α i (A, H)(W ) < ∞ and any sequence t k ↓ 0, the proof of [24, Theorem 2.1] confirms that This readily verifies (3.5) and thus the functions α i , i ∈ {1, . . . , n}, are parabolically regular at A for any V ∈ ∂α i (A). In particular, for i = 1, the function α i from (3.7) boils down to the maximum eigenvalue function of a matrix, namely So the maximum eigenvalue function λ max is parabolically regular at A for any V ∈ ∂λ max (A). This can be said for any leading eigenvalue for every matrix B ∈ S n sufficiently close to A. Another important function related to the eigenvalues of a matrix A ∈ S n is the sum of the first i components of eig A with i ∈ {1, . . . , n}, namely It is well-known that the functions σ i are convex (cf. [21, Exercise 2.54]). Moreover, we have This together with the parabolic regularity of α i ensures that σ i are parabolically regular at A for any V ∈ ∂σ i (A).
To proceed further in this section, we require the concept of the parabolic subderivative, introduced by Ben-Tal and Zowe in [1]. Let f : X → IR and letx ∈ dom f and w ∈ X with df (x)(w) finite. The parabolic subderivative of f atx for w with respect to z is defined by Recall from [21,Definition 13.59] that f is called parabolically epi-differentiable atx for w if and for every z ∈ X and every sequence t k ↓ 0 there exists a sequences z k → z such that The main interest in parabolic subderivatives in this paper lies in its nontrivial connection with second subderivatives. Indeed, it was shown in [21,Proposition 13.64] that if the function f : X → IR is finite atx, then for any pair (v, w) ∈ X × X with df (x)(w) = w,v we always have As observed below, equality in this estimate amounts to the parabolic regularity of f atx forv.
To proceed, let f : When f is the indicator function of a set, this definition boils down to the classical definition of the critical cone for sets; see [8, page 109]. It is not difficult to see that the set K f (x,v) is a cone in X. Taking into account Proposition 2.1(ii), we conclude that the domain of the second is a proper function. The following result provides conditions under which the domain of the second subderivative is the entire critical cone.

14)
where r ∈ IR + is a constant satisfying (2.1). In particular, we have dom Proof. The lower estimate of d 2 f (x,v) in (3.14) results from Proposition 2.1(iii), which readily implies that d 2 f (x,v)(0) = 0. This tells us that the second subderivative 14) directly comes from (3.12). By assumptions, for any w ∈ This guarantees that the infimum term in (3.14) is finite. Pick w ∈ K f (x,v) and observe from (3.14) that , which completes the proof.
The following example, taken from [21, page 636], shows the domain of the second subderivative can be the entire set K f (x,v) even if the assumption on the domain of the parabolic subderivative in Proposition 3.4 fails. As shown in the next section, however, this condition is automatically satisfied for composite functions appearing in (1.1).
As argued in [21, page 636], the subderivative and subdifferential of f atx = (0, 0), respectively, are . Moreover, the second subderivative of f atx forv has a representation of the form Using the above calculation tells us that which confirms that the assumption related to the domain of the parabolic subderivative in Proposition 3.4 fails.
We proceed next by providing an important characterization of the parabolic regularity that plays a key role in our developments in this paper. Proposition 3.6 (characterization of parabolic regularity). Assume that f : X → IR is finite at Then the function f is parabolically regular atx forv if and only if we have Proof. It follows fromv ∈ ∂ p f (x) and Proposition 2.1(ii)-(iii) that the second subderivative Assume now that f is parabolically regular atx forv. If there exists a w ∈ K f (x,v) \ dom d 2 f (x,v), then (3.15) clearly holds due to (3.12). Suppose now that w ∈ dom d 2 f (x,v). By Definition 3.1, there are sequences t k ↓ 0 and w k → w for which we have Since the sequence z k := 2[w k − w]/t k is bounded, we can assume by passing to a subsequence if necessary that z k →z as k → ∞ for somez ∈ X. Thus we have w k = w + 1 2 t k z k and Combining this and (3.12) implies that (3.15) and (3.16) hold for all w ∈ dom d 2 f (x,v). To obtain the opposite implication, assume that (3.15) holds for all w ∈ K f (x,v). To prove the parabolic regularity of f atx forv, let Employing now [21,Proposition 13.64] results in The last equality clearly justifies (3.5), and thus f is parabolically regular atx forv. This completes the proof.
We next show that the indicator function of the cone of n×n positive semidefinite symmetric matrices, denoted by S n + , is parabolic regular. This can be achieved via [14, Theorem 6.2] using the theory of C 2 -cone reducible sets but below we give an independent proof via Proposition 3.6.
Example 3.7 (parabolic regularity of S n + ). Let S n − stand for the cone of n × n negative semidefinite symmetric matrices. For any A ∈ S n − , we are going to show that f := δ S n − is parabolic regular at A for any V ∈ N S n − (A). Since we have S n + = −S n − , this clearly yields the same property for S n + . Using the notation in Example 3.3, we can equivalently write and conclude from (3.18) and the chain rule from convex analysis that N S n − (A) = IR + ∂λ 1 (A), which implies that V = rB for some r ∈ IR + and B ∈ ∂λ 1 (A). If r = 0, we get V = 0 and parabolic regularity of δ S n − at A for V follows directly from the definition. Assume now r > 0 and pick W ∈ K f (A, V ). The latter amounts to . Employing now [2, Proposition 2.61] tells us that dλ 1 (A)(W ) ≤ 0. Since B ∈ ∂λ 1 (A) and B, W = 0, we arrive at dλ 1 (A)(W ) = 0. We know from [21, Example 10.28] that Using direct calculations, we conclude for any t > 0 and W ′ ∈ S n that which in turn results in Using this together with (3.8) brings us to On the other hand, we conclude from (3.14) that where the last equality comes from [2, This together with Proposition 3.6 tells us that S n − is parabolic regular at A for V .
We are now in a position to establish the main result of this section, which states that parabolically regular functions are always twice epi-differentiable.
Theorem 3.8 (twice epi-differenitability of parabolically regular functions). Let f : X → IR be finite atx andv ∈ ∂ p f (x) and let f be parabolically epi-differentiable atx for every w ∈ K f (x,v). If f is parabolically regular atx forv, then it is properly twice epi-differentiable atx forv with Proof. It follows from the parabolic epi-differentiability of f atx for every w ∈ K f (x,v) and . This together with (3.15) and (3.16) justifies the second subderivative formula (3.19). To establish the twice epi-differentiability of f atx forv, we are going to show that (2.3) holds for all w ∈ X. Pick w ∈ K f (x,v) and an arbitrary sequence t k ↓ 0. Since f is parabolically regular atx forv, by Proposition 3.6, we find az ∈ X such that By the parabolic epi-differentiability of f atx for w, we find a sequence z k →z for which we have Define w k := w + 1 2 t k z k for all k ∈ IN. Using this and w ∈ K f (x,v), we obtain This together with (3.20) results in which justifies (2.3) for every w ∈ K f (x,v). Finally, we are going to show the validity of (2.3) for every w / ∈ K f (x,v). For any such a w, we conclude from (3.19) that d 2 f (x,v)(w) = ∞. Pick an arbitrary sequence t k ↓ 0 and set w k := w for all k ∈ IN. Thus we have which again proves (2.3) for all w / ∈ K f (x,v). This completes the proof of the Theorem.
The above theorem provides a very important generalization of a similar result obtained recently by the authors and Mordukhovich in [14,Theorem 3.6] in which the twice epi-differentiability of set indicator functions was established. It is not hard to see that the assumptions of Theorem 3.8 boils down to those in [14,Theorem 3.6]. To the best of our knowledge, the only results related to the twice epi-differentiability of functions, beyond set indicator functions, are [21, Theorem 13.14] and [24,Theorem 3.1] in which this property was proven for the fully amenable and eigenvalue functions, respectively. We will derive these results in Section 5 as an immediate consequence of our chain rule for the second subderivative.
We proceed with an important consequence of Theorem 3.8 in which the proto-differentiability of subgradient mappings is established under parabolic regularity. Recall that a set-valued mapping S : X ⇒ Y is said to be proto-differentiable atx forȳ with (x,ȳ) ∈ gph S if the set gph S is geometrically derivable at (x,ȳ). When this condition holds for the set-valued mapping S atx forȳ, we refer to DS(x,ȳ) as the proto-derivative of S atx forȳ. The connection between the twice epi-differentiablity of a function and the proto-differentiability of its subgradient mapping was observed first by Rockafellar in [20] for convex functions and was extended later in [16] for prox-regular functions. Recall that a function f : X → IR is called prox-regular atx forv if f is finite atx and is locally l.s.c. aroundx withv ∈ ∂f (x) and there are constant ε > 0 and r ≥ 0 such that for all Moreover, we say that f is subdifferentially continuous atx forv if ( Corollary 3.9 (proto-differentiability under parabolic regularity). Let f : X → IR be proxregular and subdifferentially continuous atx forv and let f be parabolically epi-differentiable at x for every w ∈ K f (x,v). If f is parabolically regular atx forv, then the following equivalent conditions hold: (i) the function f is twice epi-differentiable atx forv; (ii) the subgradient mapping ∂f is proto-differentiable atx forv. Furthermore, the proto-derivative of the subgradient mapping ∂f atx forv can be calculated by

Variational Properties of Parabolic Subderivatives
This section is devoted to second-order analysis of parabolic subderivatives of extended-realvalued functions that are locally Lipschitz continuous relative to their domains. We pay special attention to functions that are expressed as a composition of a convex function and a twice differentiable function. We begin with the following result that gives us sufficient conditions for finding the domain of the parabolic subderivative.
Proposition 4.1 (properties of parabolic subderivatives). Let f : X → IR be finite atx and let f be Lipschitz continuous aroundx relative to its domain with constant ℓ ∈ IR + . Assume that w ∈ T dom f (x) and that f is parabolic epi-differentiable atx for w. Then the following conditions hold: Proof. Since w ∈ T dom f (x), we conclude from Proposition 2.2 that df (x)(w) is finite. To prove (i), observe first that by definition, we always have the inclusion To obtain the opposite inclusion, take z ∈ T 2 dom f (x, w). This tells us that there exist sequences t k ↓ 0 and z k → z so thatx+t k w + 1 Moreover, corresponding to the sequence t k , we find another sequence z ′ k → z w such that Since d 2 f (x)(w z w ) < ∞, we can assume without loss of generality thatx+t k w + 1 2 t 2 k z ′ k ∈ dom f for all k ∈ IN. Using these together with the Lipschitz continuity of f aroundx relative to its domain, we have for all k sufficiently large that Passing to the limit results in the inequality which in turn yields d 2 f (x)(w z) < ∞, i.e., z ∈ dom d 2 f (x)(w ·). This justifies the opposite inclusion in (4.1) and hence proves (i). Turning now to (ii), we conclude from (4.1) and the parabolic epi-differentiability of f atx for w that the second-order tangent set T 2 dom f (x, w) is nonempty. Moreover, it follows from [21, Example 13.62 (b)] that the parabolic epi-differentiability of f atx for w yields the parabolic derivability of epi f at (x, f (x)) for (w, df (x)(w)). The latter clearly enforces the same property for dom f atx for w and hence completes the proof.
It is important to notice the parabolic epi-differentiability of f in Proposition 4.1 is essential to ensure that condition (i) therein, namely the characterization of the domain of the parabolic subderivative, is satisfied. Indeed, as mentioned in the proof of this proposition, inclusion (4.1) always holds. If the latter condition fails, this inclusion can be strict. For example, the function f from Example 3.5 is not parabolic epi-differentiable atx = (0, 0) for any vector w = (w 1 , w 2 ) ∈ K f (x,v) with w 1 = 0 since dom d 2 f (x)(w ·) = ∅. On the other hand, we have dom f = IR 2 and thus T 2 dom f (x, w) = IR 2 for any such a vector w ∈ K f (x,v), and so condition (i) in Proposition 4.1 fails.
Given a function f : X → IR finite atx, in the rest of this paper, we mainly focus on the case when this function has a representation of the form where O is a neighborhood ofx and where the functions F and g are satisfying the following conditions: • F : X → Y is twice differentiable atx; • g : Y → IR is proper, l.s.c., convex, and Lipschitz continuous around F (x) relative to its domain with constant ℓ ∈ IR + . It is not hard to see that the imposed assumptions on g from representation (4.3) implies that dom g is locally closed around F (x), namely for some ε > 0 the set (dom g) ∩ B ε (F (x)) is closed. Taking the neighborhood O from (4.3), we obtain It has been well understood that the second-order variational analysis of the composite form (4.3) requires a certain qualification condition. The following definition provides the one we utilize in this paper.

Definition 4.2 (metric subregularity constraint qualification).
Assume that the function f : X → IR has representation (4.3) aroundx ∈ dom f . We say that the metric subregularity constraint qualification holds for the constraint set (4.4) atx if there exist a constants κ ∈ IR + and a neighborhood U ofx such that By definition, the metric subregularity constraint qualification for the constraint set (4.4) at x amounts to the metric subregularity of the mapping x → F (x) − dom g at (x, 0). The more traditional and well known constraint qualification for (4.3) is where ∂ ∞ ϕ(x) stands for the singular subdifferential of ϕ : IR n → IR atx ∈ dom ϕ defined by By the coderivative criterion from [12, Theorem 3.3], the latter constraint qualification amounts to the metric regularity of the mapping (x, α) → (F (x), α) − epi g around (x, g(F (x))), (0, 0) . Since g is convex, by [12, Proposition 1.25], the aforementioned constraint qualification can be equivalently written as which by the coderivative criterion from [12, Theorem 3.3] is equivalent to the metric regularity of the mapping x → F (x) − dom g around (x, 0). Thus, for the composite form (4.3), the metric regularity of (x, α) → (F (x), α) − epi g around (x, g(F (x))), (0, 0) amounts to that of x → F (x) − dom g around (x, 0). This may not be true if the metric regularity is replaced with the metric subregularity; see [13, Proposition 3.1] and [13, Example 3.3] for more details and discussions about these conditions. So, it is worth reiterating that instead of using the metric subregularity of the epigraphical mapping, our second-order variational analysis will be carried out under a weaker (and simpler) metric subregularity of the domain mapping.
As observed recently in [13], (4.5) suffices to conduct first-and second-order variational analysis of (4.3) when the convex function g therein is merely piecewise linear-quadratic. In what follows, we will show using a different approach that such results can be achieved for (4.3) as well. We continue our analysis by recalling the following first-and second-order chain rules, obtained recently in [13,14]. atx ∈ dom f andv ∈ ∂f (x) and let the metric subregularity constraint qualification hold for the constraint set (4.4) atx. Then the following hold: (i) for any w ∈ X, the following subderivative chain rule for f atx holds: df (x)(w) = dg(F (x))(∇F (x)w); (ii) we have the chain rules If, in addition, w ∈ T dom f (x) and the function g from (4.3) is parabolically epi-differentiable at F (x) for ∇F (x)w, then we have Moreover, dom f is parabolically derivable atx for w.
Proof. The subderivative chain rule in (i) was established recently in [13,Theorem 3.4]. The subdifferential chain rule in (ii) was taken from [13,Theorem 3.6]. As mentioned in Section 2, the inclusion ∂ p f (x) ⊂ ∂f (x) always holds. The opposite inclusion can be justified using the aforementioned subdifferential chain rule and the convexity of g; see [13,Theorem 4.4] for a similar result. The chain rule for the tangent cone to dom f atx results from Proposition 2.2 and the subderivative chain rule for f atx in (i). If in addition w ∈ T dom f (x) and g is parabolically epi-differentiable at F (x) for ∇F (x)w, then it follows from Proposition 4.1 that dom g is parabolically derivable at F (x) for ∇F (x)w. Appealing now to [14,Theorem 4.5] implies that dom f is parabolically derivable atx for w. Finally, the chain rule (4.7) was taken from [14,Theorem 4.5]. This completes the proof.
We continue by establishing a chain rule for the parabolic subderivative, which is important for our developments in the next section. (i) for any z ∈ X we have (ii) the domain of the parabolic subderivative of f atx for w is given by (iii) f is parabolically epi-differentiable atx for w.
Proof. Pick z ∈ X and set u := ∇F (x)z + ∇ 2 F (x)(w, w). We prove (i)-(iii) in a parallel way. Assume that z / ∈ T 2 dom f (x, w). As mentioned in the proof of Proposition 4.1, inclusion (4.1) always holds. This implies that d 2 f (x)(w z) = ∞. On the other hand, by (4.7) we get u / ∈ T 2 dom g F (x), ∇F (x)w . Employing Proposition 4.1(i) for the function g and ∇F (x)w ∈ T dom g (F (x)) gives us Combining these tells us that d 2 g(F (x))(∇F (x)w u) = ∞, which in turn justifies (4.8) for every z / ∈ T 2 dom f (x, w). Consider an arbitrary sequence t k ↓ 0 and set z k := z for all k ∈ IN. Then we have which in turn justifies (3.11) for all z / ∈ T 2 dom f (x, w). Since g is parabolically epi-differentiable at F (x) for ∇F (x)w, Proposition 4.1(ii) tells us that dom g is parabolically derivable at F (x) for ∇F (x)w. We conclude from Proposition 4.3 that dom f is parabolically derivable atx for w. In particular, we have Pick now z ∈ T 2 dom f (x, w) and then consider an arbitrary sequence t k ↓ 0. Thus, by definition, for the aforementioned sequence t k , we find a sequence z k → z as k → ∞ such that Moreover, since g is parabolically epi-differentiable at F (x) for ∇F (x)w, we find a sequence u k → u such that (4.12) It follows from (4.7) that u ∈ T 2 dom g F (x), ∇F (x)w . Combining this with (4.9) tells us that d 2 g(F (x))(∇F (x)w u) < ∞. This implies that y k := F (x) + t k ∇F (x)w + 1 2 t 2 k u k ∈ dom g for all k sufficiently large. Remember that g is Lipschitz continuous around F (x) relative to its domain with constant ℓ. Using this together with Proposition 4.3(i), (4.11), and (4.12), we obtain (4.13) On the other hand, it is not hard to see that for any z ∈ X, we always have Combining this and (4.13) implies that and that These prove (4.8) and (3.11) for any z ∈ T 2 dom f (x, w), respectively, and hence we finish the proof of (i).
Next, we are going to verify (ii). We already know that inclusion (4.1) always holds. To derive the opposite inclusion, pick z ∈ T 2 dom f (x, w), which amounts to u ∈ T 2 dom g F (x), ∇F (x)w due to (4.7). By (i) and (4.9), we obtain This tells us that z ∈ dom d 2 f (x)(w ·) and hence completes the proof of (ii).
Finally, to justify (iii), we require to prove the fulfillment of (3.11) for all z ∈ X and to show that dom d 2 f (x)(w ·) = ∅. The former was proven above and so we proceed with the proof of the latter. This, indeed, follows from (4.10) and the characterization of dom d 2 f (x)(w ·), achieved in (ii), and thus completes the proof.
It is worth mentioning that a chain rule for parabolic subderivatives for the composite form (4.3) was achieved in [21,Exercise 13.63] and [2, Proposition 3.42] when g is merely a proper l.s.c. function and the basic constraint qualification (4.6) is satisfied. Replacing the latter condition with the significantly weaker condition (4.5), we can achieve a similar result if we assume further that g is convex and locally Lipschitz continuous relative to its domain. Another important difference between Theorem 4.4 and those mentioned above is that the chain rule (4.8) obtained in [2,21] does not require the parabolic epi-differentiability of g. Indeed, the usage of the basic constraint qualification (4.6) in [2,21] allows to achieve (4.8) via a chain rule for the epigraphs of f and g similar to the one in (4.7), which is not conceivable under (4.5). These extra assumptions on g automatically fulfill in many important composite and constrained optimization problems and so do not seem to be restrictive in our developments.
We continue by establishing two important properties for parabolic subderivatives that play crucial roles in our developments in the next section. One notable difference between the following results and those obtained in Proposition 4.1 and Theorem 4.4 is that we require the parabolic subderivative be proper. This can be achieved if the parabolic subderivative is bounded below. In general, we may not be able to guarantee this. It turns out, however, that if the vector w in the pervious results is taken from the critical cone to the function in question, which is a subset of the tangent cone to the domain of that function, this can be accomplished via (3.14). Since we only conduct our analysis in the next section over the critical cone, this will provide no harm. Below, we first show that the parabolic subderivative of an extended-real-valued function, which is locally Lipschitz continuous relative to its domain, is Lipschitz continuous relative to its domain. Proposition 4.5 (Lipschitz continuity of of parabolic subderivatives). Let ψ : X → IR be finite atx andv ∈ ∂ p ψ(x), and let ψ be Lipschitz continuous aroundx relative to its domain with constant ℓ ∈ IR + . Assume that w ∈ K ψ (x,v) and that ψ is parabolically epi-differentiable atx for w. Then the parabolic subderivative d 2 ψ(x)(w ·) is proper, l.s.c., and Lipschitz continuous relative to its domain with constant ℓ.
We end this section by obtaining an exact formula for the conjugate function of the parabolic subderivative of a convex function. Proposition 4.6 (conjugate of parabolic subderivatives). Let ψ : X → IR be an l.s.c. convex function with ψ(x) finite,v ∈ ∂ψ(x), and w ∈ K ψ (x,v). Define the function ϕ by ϕ(z) := d 2 ψ(x)(w z) for any z ∈ X. If ψ is parabolically epi-differentiable atx for w and parabolically regular atx for every v ∈ ∂ψ(x), then ϕ is a proper, l.s.c., and convex function and its conjugate function is given by where A(x, w) := {v ∈ ∂ψ(x)| dψ(x)(w) = v, w }.
Proof. It follows from [21,Proposition 13.64] that ϕ is l.s.c. Using similar arguments as the beginning of the proof of Proposition 4.5 together with (4.14) tells us that ϕ is proper. Also we deduce from [21,Example 13.62] that epi ϕ = T 2 epi ψ (x, ψ(x)), (w, dψ(x)(w)) , and thus the parabolic epi-differentiability of ψ atx for w amounts to the parabolic derivability of epi ψ at (x, ψ(x)) for (w, dψ(x)(w)). The latter combined with the convexity of ψ tells us that epi ϕ is a convex set in X × IR and so ϕ is convex.
To verify (4.15), pick v ∈ A(x, w). This yields v ∈ ∂ψ(x) = ∂ p ψ(x) and w ∈ K ψ (x, v), namely the critical cone of ψ at (x, v). Using Proposition 3.6 and parabolic regularity of ψ atx for v implies that which clearly proves (4.15) in this case. Assume now that v / ∈ A(x, w). This means either v / ∈ ∂ψ(x) or dψ(x)(w) = v, w . Define the parabolic difference quotients for ψ atx for w by It is not hard to see that ϑ t are proper, convex, and Remember that by [21,Definition 13.59] the parabolic epi-differentiability of ψ atx for w amounts to the sets epi ϑ t converging to epi ϕ as t ↓ 0 and that the functions ϑ t and ϕ are proper, l.s.c. and convex. Appealing to [21,Theorem 11.34] tells us that the former is equivalent to the sets epi ϑ * t converging to epi ϕ * as t ↓ 0. This, in particular, means that for any v / ∈ A(x, w) and any sequence t k ↓ 0, there exists a sequence v k → v such that If v / ∈ ∂ψ(x), then we have Since ψ * is l.s.c., we get which in turn confirms that If v ∈ ∂ψ(x) but dψ(x)(w) = v, w , we obtain v, w < dψ(x)(w). Since we always have we arrive at which justifies (4.15) when v / ∈ A(x, w) and hence finishes the proof. Proposition 4.6 was first established using a different method in [18,Proposition 3.5] for convex piecewise linear-quadratic functions. It was extended in [7, Theorem 3.1] for any convex functions under a restrictive assumption. Indeed, this result demands that the second subderivative be the same as the second-order directional derivative. Although this condition holds for convex piecewise linear-quadratic functions, it fails for many important functions occurring in constrained and composite optimization problems including the set indicator functions and eigenvalue functions. As discussed below, however, our assumptions are satisfied for all these examples.
Example 4.7. Suppose that g : Y → IR is an l.s.c. convex function andz ∈ Y.
(a) If Y = IR m , g is convex piecewise linear-quadratic (Example 3.2), andz ∈ dom g, then it follows from Example 3.2 and [21, Exercise 13.61] that g is parabolically regular atz for every y ∈ ∂g(z) and parabolically epi-differentiable atz for every w ∈ dom dg(z), respectively, and thus all the assumptions of Proposition 4.6 are satisfied for this function. (b) If Y = S m , g is either the maximum eigenvalue function λ max from (3.9) or the function σ i from (3.10), and A ∈ S n , then by Example 3.3 g is parabolically regular at A for every V ∈ ∂g(A). Moreover, we deduce from [23, Proposition 2.2] that g is parabolically epidifferentiable at A for every W ∈ S n and thus all the assumptions of Proposition 4.6 are satisfied for these functions. (c) If g = δ C andz ∈ C, where C is a closed convex set in Y that is parabolically derivable at z for every w ∈ T C (z) and parabolically regular atz for every v ∈ N C (z), then g satisfies the assumptions imposed in Proposition 4.6. This example of g was recently explored in detail in [14] and encompasses important sets appearing in constrained optimization problems such as polyhedral convex sets, the second-order cone, and the cone of positive semidefinite symmetric matrices. (d) Assume that g is differentiable atz and that there exists a continuous function h : Y → IR, which is positively homogeneous of degree 2, such that Such a function g is called twice semidifferentiable (cf. [21,Example 13.7]) and often appears in the augmented Lagrangian function associated with (1.1); see [14,Section 8] for more detail. This second-order expansion clearly justifies the parabolic epi-differentiability of g atz for every w ∈ Y. Moreover, one has which in turn shows that g is parabolically regular atz for ∇g(z) due to Proposition 3.6.
It is important to mention that the restrictive assumption on the second subderivative, used in [7, Theorem 3.1], does not hold for cases (b)- (d) in Example 4.7.

A Chain Rule for Parabolically Regular Functions
Our main objective in this section is to derive a chain rule for the parabolic regularity of the composite representation (4.3). This opens the door to obtain a chain rule for the second subderivative, and, more importantly, allows us to establish the twice epi-differentiability of the latter composite form.
Taking into account representation (4.3) and picking a subgradientv ∈ ∂f (x), we define the set of Lagrangian multipliers associated with (x,v) by In what follows, we say that a function f : X → IR with (x,v) ∈ gph ∂f and having the composite representation (4.3) atx satisfies the basic assumptions at (x,v) if in addition the following conditions fulfill: (H1) the metric subregularity constraint qualification holds for the constraint set (4.4) atx; (H2) for any y ∈ Λ(x,v), the function g from (4.3) is parabolically epi-differentiable at F (x) for every u ∈ K g (F (x), y); (H3) for any y ∈ Λ(x,v), the function g is parabolically regular at F (x) for y.
We begin with the following result in which we collect lower and upper estimates for the second subderivative of f taken from (4.3).
Moreover, for every w ∈ X we have the lower estimate while for every w ∈ K f (x,v) we obtain the upper estimate Proof. By Proposition 4.3(ii), we have ∂ p f (x) = ∂f (x). Appealing now to Propositions 2.1(iii) and 3.4 confirms, respectively, that d 2 f (x,v) is a proper l.s.c. function and that (5.1) holds. The lower estimate (5.2) can be justified as [21,Theorem 13.14] in which this estimate was derived under condition (4.6). To obtain (5.3), observe first that the basic assumption (H1) yields for every y ∈ Λ(x,v). Pick w ∈ K f (x,v). Since g is parabolically epi-differentiable at F (x) for ∇F (x)w due to (H2), Theorem 4.4(iii) implies that f is parabolically epi-differentiable atx for w, and so dom d 2 f (x)(w ·) = ∅. This combined with (3.14) and (4.8) results in (5.3) and hence completes the proof.
While looking simple, the above result carries important information by which we can achieve a chain rule for the second subderivative. To do so, we should look for conditions under which the lower and upper estimates (5.2) and (5.3), respectively, coincide. This motivates us to consider the unconstrained optimization problem min z∈X − z,v + d 2 g(F (x))(∇F (x)w ∇F (x)z + ∇ 2 F (x)(w, w)). (5.5) When the basic assumptions (H1)-(H3) are satisfied, (5.5) is a convex optimization problem for any w ∈ K f (x,v). Using Proposition 4.6 allows us to obtain the dual problem of (5.5) and then examine whether their optimal values coincide. We pursue this goal in the following result. atx ∈ dom f ,v ∈ ∂f (x), and let the basic assumptions (H1)-(H3) hold for f at (x,v). Then for each w ∈ K f (x,v), the following assertions are satisfied: (i) the dual problem of (5.5) is given by max y∈Y y, ∇ 2 F (x)(w, w) + d 2 g(F (x), y)(∇F (x)w) subject to y ∈ Λ(x,v); (5.6) (ii) the optimal values of the primal and dual problems (5.5) and (5.6), respectively, are finite and coincide; moreover, we have Λ(x,v, w) ∩ (τ B) = ∅, where Λ(x,v, w) stands for the set of optimal solutions to the dual problem (5.6) and where with ℓ and κ taken from (4.3) and (4.5), respectively.
Proof. Pick w ∈ K f (x,v) and observe from (5.4) that ∇F (x)w ∈ K g (F (x), y) for all y ∈ Λ(x,v). This together with Proposition 4.6 ensures that the parabolic subderivative of g at F (x) for ∇F (x)w is a proper, l.s.c., and convex function. Using this combined with [21,Example 11.41] and (4.15) tells us that the dual problem of (5.5) is Combining these confirms that the dual problem of (5.5) is equivalent to (5.6) and thus finishes the proof of (i). To prove (ii), consider the optimal value function ϑ : Y → [−∞, ∞], defined by We proceed with the following claim: Claim.We have ∂ϑ(0) = ∅.
To justify the claim, we first need to show ϑ(0) ∈ IR. To do so, observe thatv ∈ ∂f (x) = ∂ p f (x) due to Proposition 4.3(ii). Thus, it follows from Proposition 2.1(iii) and (5.3) that there is a constant r ∈ IR + such that for any w ∈ K f (x,v) we have which in turn implies that ϑ(0) ∈ IR. Next, we are going to show that where τ is taken from (5.7). To this end, take (p, z) ∈ Y × X such that By (4.9), we get u p ∈ T 2 dom g (F (x), ∇F (x)w). Define now the set-valued mapping S w : Y ⇒ X by S w (p) := z ∈ X| ∇F (x)z + ∇ 2 F (x)(w, w) + p ∈ T 2 dom g (F (x), ∇F (x)w) , p ∈ Y. So, we get z ∈ S w (p). It was recently observed in [14,Theorem 4.3] that the mapping S w enjoys the uniform outer Lipschitzian property at 0 with constant κ taken from (4.5), namely for every This combined with z ∈ S w (p) results in the existence of z 0 ∈ S w (0) and b ∈ B such that z = z 0 + κ p b. It follows from (4.9) and z 0 ∈ S w (0) that Since we have and since the parabolic subderivative d 2 g(F (x))(∇F (x)w ·) is Lipschitz continuous relative to its domain due to Proposition 4.5, we get the relationships which together with (5.7) justify (5.9). Remember that the parabolic subderivative of g at F (x) for ∇F (x)w is a proper and convex function. This implies that the function which completes the proof of the claim. Employing now (5.10) and [2, Theorem 2.142] confirms that the optimal values of the primal and dual problems (5.5) and (5.6), respectively, coincide and that Λ(x,v, w) = ∂ϑ(0). This together with (5.10) justifies (ii) and hence completes the proof.
The above theorem extends the recent results obtained in [14,Propositions 5.4 & 5.5] for constraint sets, namely when the function g in (4.3) is the indicator function of a closed convex set. We should add here that for constraint sets, the dual problem (5.6) can be obtained via elementary arguments. However, for the composite form (4.3) a similar result requires using rather advanced theory of epi-convergence. Remark 5.3 (duality relationship under metric regularity). In the framework of Theorem 5.2, we want to show that replacing assumption (H1) with the strictly stronger constraint qualification (4.6) allows us not only to drop the imposed Lipschitz continuity of g from (4.3) but also to simplify the proof of Theorem 5.2. To this end, let w ∈ K f (x,v) and define the function ψ(u) := d 2 g(F (x))(∇F (x)w u), u ∈ X.
This together with (4.6) and the relationship N dom ψ (u) = ∂ ∞ ψ(u) stemming from the convexity of ψ confirms the validity of (5.12). Appealing now to [2, Theorem 2.165] gives another proof of Theorem 5.2 when assumption (H1) therein is replaced with the strictly stronger constraint qualification (4.6).
The established duality relationships in Theorem 5.2 open the door to derive a chain rule for parabolically regular functions and to find an exact chain rule for the second subderivative of the composite function (4.3) under our basic assumptions.

Second-Order Optimality Conditions for Composite Problems
In this section, we focus mainly on obtaining second-order optimality conditions for the composite problem (1.1), where ϕ : X → IR and F : X → Y are twice differentiable and the function g : Y → IR is an l.s.c. convex function that is locally Lipschitz continuous relative to its domain. The latter means that for any y ∈ dom g, the function g is Lipschitz continuous around y relative to its domain. Important examples of constrained and composite optimization problems can be achieved when g is one of the functions considered in Example 4.7. For any pair (x, y) ∈ X × Y, the Lagrangian associated with the composite problem (1.1) is defined by L(x, y) = ϕ(x) + F (x), y − g * (y), where g * is the Fenchel conjugate of the convex function g. We begin with the following result in which we collect second-order optimality conditions for (1.1) when our basic assumptions are satisfied. Recall that a pointx ∈ X is called a feasible solution to the composite problem (1.1) if we have F (x) ∈ dom g. Theorem 6.1 (second-order optimality conditions). Letx be a feasible solution to problem (1.1) and let f := g • F andv := −∇ϕ(x) ∈ ∂f (x) with ϕ, g, and F taken from (1.1). Assume that the basic assumptions (H1)-(H3) hold for f at (x,v). Then the following second-order optimality conditions for the composite problem (1.1) are satisfied: (i) ifx is a local minimum of (1.1), then the second-order necessary condition max y∈Λ(x,v) ∇ 2 xx L(x, y)w, w + d 2 g(F (x), y)(∇F (x)w) ≥ 0 holds for all w ∈ K f (x,v); (ii) the validity of the second-order condition max y∈Λ(x,v) ∇ 2 xx L(x, y)w, w + d 2 g(F (x), y)(∇F (x)w) > 0 for all w ∈ K f (x,v) \ {0} (6.1) amounts to the existence of constants ℓ > 0 and ε > 0 such that the second-order growth condition ψ(x) ≥ ψ(x) + ℓ 2 x −x 2 for all x ∈ B ε (x) (6.2) holds, where ψ := ϕ + g • F .
The above result was first observed in [8, Theorem 4G.1] for a subclass of nonlinear programming problems and was extended in [6, Theorem 4.6] for C 2 -cone reducible constrained optimization problems and in [14,Theorem 9.2] for parabolically regular constrained optimization problems. The theory of the twice epi-differentiability, obtained in this paper, provides an easy path to achieve a similar result for the composite problem (1.1).
It is worth mentioning that similar characterizations as [14,Theorem 4.2] can be achieved for the KKT system of (1.1). Furthermore, Corollary 3.9 provides a systematic method to calculate proto-derivatives of subgradient mappings of functions enjoying the composite form (4.3), a path we will pursue in our future research.