Differential stability of convex optimization problems under inclusion constraints

Motivated by the recent work of Mordukhovich et al. [Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B. 2009;116:369–396] on the optimal value function in parametric programming under inclusion constraints, this paper presents some new results on differential stability of convex optimization problems under inclusion constraints and functional constraints in Hausdorff locally convex topological vector spaces. By using the Moreau–Rockafellar theorem and appropriate regularity conditions, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function. By virtue of the convexity, several assumptions used in the above paper by Mordukhovich et al., like the nonemptyness of the Fréchet upper subdiffential of the objective function, the existence of a local upper Lipschitzian selection of the solution map, as well as the -inner semicontinuity and the -inner semicompactness of the solution map, are no longer needed. Relationships between our results and the corresponding ones in Aubin’s book [Optima and equilibria. An introduction to nonlinear analysis. 2nd ed. New York (NY): Springer; 1998] are discussed.


Introduction
If a mathematical programming problem depends on a parameter, that is, the objective function and the constraints depend on a certain parameter, then the optimal value is a function of the parameter, and the solution map is a set-valued map on the parameter of the problem. In general, the optimal value function is a fairly complicated function of the parameter; it is often nondifferentiable on the parameter, even if the problem in question is a mathematical program with smooth functions on all the variables and on the parameter. This is the reason of the great interest in having formulas for computing generalized directional derivatives (Dini directional derivative, Dini-Hadarmard directional derivative, and the Clarke generalized directional derivative) and formulas for evaluating subdifferentials (subdifferential in the sense of convex analysis, Clarke subdifferential, Fréchet subdifferential, limiting subdifferential -also called the Mordukhovich subdifferential) of the optimal value function.
Motivated by the recent work of Mordukhovich et al. [8] on the optimal value function in parametric programming under inclusion constraints, this paper presents some new results about differential stability of convex optimization problems under inclusion constraints and functional constraints in Hausdorff locally convex topological vector spaces. By using the Moreau-Rockafellar theorem (see e.g. [11, p.48]) and appropriate regularity conditions, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function. By virtue of the convexity, several assumptions used in the above paper by Mordukhovich et al., like the nonemptyness of the Fréchet upper subdiffential of the objective function, the existence of a local upper Lipschitzian selection of the solution map, as well as the μ-inner semicontinuity and the μ-inner semicompactness of the solution map, are no longer needed. In addition, we can use the Hausdorff locally convex topological vector spaces framework instead of the Banach spaces setting in [8].
Thus, on one hand, our results have the origin in the study of Mordukhovich et al. [8]. On the other hand, they are the results of deepening that study for the case of convex programming problems.
Interestingly, in order to obtain differential stability properties in parametric convex programming, one can use [1] the Fenchel-Moreau theorem [11, p.175]. More precisely, by using that theorem and a series of advanced auxiliary results, Aubin [1, Problem 35 -Subdifferentials of Marginal Functions, p.335] has obtained a formula for computing the subdifferential of the optimal value function under a regularity assumption. This approach requires that the objective function of the problem in question must be convex, lower semicontinuous, and the constraint set mapping must be convex and have closed graph. The above approach of using the Moreau-Rockafellar theorem does not require the last two additional assumptions on the lower semicontinuity of the objective function and on the closedness of the graph of the constraint set mapping. Therefore, although requiring regularity assumptions which are somewhat stronger than that of Aubin, our results are established for a larger class of convex programming problems, and do not coincide with Aubin's result if one considers the special case where the spaces are Hilbert and the objective function does not depend on the parameter.
Applied to parametric optimal control problems, with convex objective functions and linear dynamical systems, either discrete or continuous, our results can lead to some rules for the exact computing of the subdifferential and the singular subdifferential of the optimal value function via the data of the given problem.
The organization of the paper is as follows. Section 2 recalls some definitions from variational analysis [12]. Three motivational results from [8] are described in Section 3. Differential stability under convexity, our focus point, is studied in Sections 4 and 5. The final section compares our results with the results recalled in Section 2 and the above-mentioned result of Aubin.

Preliminaries 2.1. Normal cones
Let X be a Banach space with the dual denoted by X * . For any set-valued map F : X ⇒ X * , by one denotes the sequential Painlevé-Kuratowski upper limit of F as x tends tox with respect to the norm topology of X and the weak * topology of X * . Here x * k w * − → x * means that the sequence {x * k } ⊂ X * weakly * converges to x * ∈ X * . If ⊂ X is a given subset, the notation x − →x means that x →x and x ∈ . (i) For any x ∈ and ε ≥ 0, the set of ε-normals of at x is defined by is called the Mordukhovich normal cone or the limiting normal cone of atx. We put N (x; ) = ∅ ifx ∈ .
for all ε ≥ 0 andx ∈ . Especially, the set N (x; ) coincides with the convex cone of atx in the sense of convex analysis, that is, (2.1) The notions in Definition 2.1 have a local character, since they just depend on the structure of in an arbitrarily small neighborhood of the point in question. Thus, one can formulate the results in Proposition 2.1 for locally convex sets, as follows.
where N (x) denotes the family of the neighborhoods ofx, such that ∩ U is convex, then

Subdifferentials
Consider a function f : X → R having values in the extended real line R = [−∞, +∞].
One says that f is proper if f (x) > −∞ for all x ∈ X , and the domain is nonempty. The epigraph and the hypograph of f are given, respectively, by In the sequel, the notation (i) The set is said to be the Mordukhovich subdifferential or the limiting subdifferential of f atx. (iv) The set is said to be the singular subdifferential of f atx.
, and ∂ ∞ f (x) to be empty sets.
i.e. the Fréchet subdifferential and the Mordukhovich subdifferential of f atx coincide with the subdifferential of f atx in the sense of convex analysis [11, p.196]. In particular, f is lower regular atx. Given a subset ⊂ X , one defines the indicator function δ(·; ) : X → R of by setting

Coderivatives
Let F : X ⇒ Y be a set-valued map between Banach spaces. The graph and the domain of F are given, respectively, by the formulas Equipping the product space X × Y with the norm (x, y) := x + y , by the above notions of normal cones, one can define the concepts of Fréchet coderivative and Mordukhovich coderivative (also called the limiting coderivative) of set-valued maps as follows.

Optimal value function
Consider a set-valued map G : X ⇒ Y between Banach spaces, a function ϕ : X ×Y → R. The optimal value function of the parametric optimization problem under an inclusion constraint, defined by G and ϕ, is the function μ : By the convention inf ∅ = +∞, we have μ(x) = +∞ for any x / ∈ dom G. The set-valued map G (resp., the function ϕ) is called the map describing the constraint set (resp., the objective function) of the problem on the right-hand side of (2.4).
Corresponding to each data pair {G, ϕ} we have one optimization problem depending on a parameter x: Formulas for computing exactly or estimating the Fréchet subdifferential and the Mordukhovich subdifferential of the optimal value function μ(x), to be considered in forthcoming sections, are connected tightly with the solution map M : of the problem (P x ).
and if epi ϕ is a convex set in X × Y × R (that is, ϕ is a convex function), then (2.5) is said to be a parametric convex optimization problem.
It is a simple matter to show that if (2.5) is a convex optimization problem, then μ is a convex function. Hence, under that convexity assumption, the Fréchet subdifferential and the Mordukhovich subdifferential of μ atx ∈ dom μ, with μ(x) = −∞, coincide with the subdifferential of μ atx in the sense of convex analysis, and these sets can be computed by formula (2.2) with f being replaced by μ.

Fréchet subdifferential of µ(.)
The following theorem gives us an upper estimate for the Fréchet subdifferential of the optimal value function in formula (2.4) at a given parameterx. This estimate is established via the Fréchet coderivative of the map G describing the constraint set and the Fréchet upper subdifferentials of the objective function ϕ.
The estimate (3.1) is valid in the form of an inclusion. It is natural to ask when that inclusion holds as an equality. Additional assumptions based on the following definitions will be made to get the equality.
The next theorem gives a sufficient condition for the inclusion (3.1) to hold as equality.
The reader is referred to [8] for illustrative examples for Theorems 3.1 and 3.2.
The assumption ∂ + ϕ(x,ȳ) = ∅ in Theorem 3.1 is rather strict. It excludes from our consideration convex, Lipschitzian functions of the type ϕ( The above remark can be strengthened as follows: Theorem 3.1 cannot be used for any problem of the form (2.5) with ϕ being proper, convex, continuous, and nondifferentiable at a given point (x,ȳ) ∈ gph M. Indeed, since ϕ is convex, the Fréchet subdifferential ∂ϕ(x,ȳ) coincides with the subdifferential in the sense of [11, Subsection 4.2.1]. As ϕ is continuous at (x,ȳ), the latter set is nonempty by [11,Prop. 3,p.199]. Hence, if ∂ + ϕ(x,ȳ) = ∅ then ϕ is Fréchet differentiable at (x,ȳ) by [12,Prop. 1.87]. This contradicts the condition saying that ϕ is nondifferentiable at (x,ȳ). In the two subsequent sections, we will obtain some results for parametric convex problems of the form (2.5) which allow us to avoid not only the assumption ∂ + ϕ(x,ȳ) = ∅ in Theorem 3.1, but also the requirement that the solution map admits a local upper Lipschitzian selection in Theorem 3.2.

Mordukhovich subdifferential of µ(.)
In order to recall a main result from [8] on computing the Mordukhovich subdifferential of μ(.), we have to consider the following definitions.  The only difference is that the condition x k →x in [12] is now replaced by the weaker condition x k μ − →x.
Definition 3.6 A set-valued map F : X ⇒ Y is said to be sequentially normally compact (SNC) at (x,ȳ) ∈ gph F if its graph possesses this property. (i) For a given vectorȳ ∈ M(x), suppose that M is μ-inner semicontinuous at (x,ȳ) ∈ gph M, that either ϕ is SNEC at (x,ȳ) or G is SNC at (x,ȳ), and the regularity condition N ((x,ȳ); is satisfied (these assumptions are automatically satisfied if ϕ is Lipschitz continuous around (x,ȳ)). Then one has the inclusions (ii) Assume that M is μ-inner semicompact atx and that the other assumptions of (i) are satisfied at any (x,ȳ) ∈ gph M. Then one has the inclusions We are going to show that if the problem in question is convex then several assumptions in Theorems 3.1-3.3 are no longer needed and, surprisingly, all the upper estimates become equalities.

Differential stability under convexity
We now present some new results on differential stability of convex optimization problems under inclusion constraints. By using the Moreau-Rockafellar theorem and appropriate regularity conditions, we will obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value functions.
From now on, if not otherwise stated, we assume that X, Y are Hausdorff locally convex topological vector spaces with the topological duals denoted, respectively, by X * and Y * .
For a convex set ⊂ X , the normal cone of atx ∈ is given by As noted in Section 2, formula (4.1) fully agrees with formula (2.1), which was given in a Banach space setting.
For a convex function f : X → R, the subdifferential of f at with f (x) = −∞, is given by This is a generalization of formula (2.2) for the case of functions defined on Hausdorff locally convex topological vector spaces. The set If (x,ȳ) / ∈ gph F, then we put D * F(x,ȳ)(y * ) = ∅ for any y * ∈ Y * . The following theorem from convex analysis is the main tool in our subsequent proofs. Then for all x ∈ X . If, at a point x 0 ∈ dom f 1 ∩ · · · ∩ dom f m , all the functions f 1 , . . . , f m , except, possibly, one are continuous, then for all x ∈ X.
We denote the interior of a subset of X by int . Using indicator functions of convex sets, one can easily derive from Theorem 4.1 the next intersection formula.

. , A m be convex subsets of X . Set
Proof Indeed, since epi f is a convex set, by (4.2) we have Let us turn back our attention to the parametric optimization problem (2.5). The next theorem provides us with formulas for computing the subdifferential and the singular subdifferential of μ in the case G and ϕ are assumed to be convex.
Theorem 4.2 Let G : X ⇒ Y be a convex set-valued mapping and ϕ : X × Y → R be a proper convex function. If at least one of the following regularity conditions is satisfied: is continuous at a point (x 0 , y 0 ) ∈ gph G, then for anyx ∈ dom μ, with μ(x) = −∞, and for anyȳ ∈ M(x) we have and
We are going to obtain (4.4) with the aid of some arguments suggested by the anonymous referee of this paper. This new proof is much shorter than our original proof. Observe that x ∈ dom μ if and only if

Convex programming problem under functional constraints
We now apply the above general results to convex optimization problems under geometrical and functional constraints.As in the preceding section, X and Y are Hausdorff locally convex topological vector spaces. Consider the problem min ϕ(x, y) | (x, y) ∈ C, g i (x, y) ≤ 0, i ∈ I, h j (x, y) = 0, j ∈ J , (5.1) in which ϕ : X × Y → R is a convex function, C ⊂ X × Y is a convex set, I = {1, . . . , m}, J = {1, . . . , k}, g i : X × Y → R (i ∈ I ) are continuous convex functions, and h j : X × Y → R ( j ∈ J ) are continuous affine functions. For each x ∈ X , we put It clear that the set-valued map G(·) given by (5.2) is convex and The following statement is a Farkas lemma for infinite dimensional vector spaces.  . . . , m). Then, the inequality γ (x) ≤ 0 is a consequence of the inequalities system if and only if there exist nonnegative real numbers λ 1 , λ 2 , . . . , λ m ≥ 0 such that γ = λ 1 α 1 + · · · + λ m α m .
The following lemma describes the normal cone of the intersection of finitely many affine hyperplanes.
We omit the proof of this lemma, because it can be done quite easily by applying Lemma 5.1.
The next lemma will play an important role in the subsequent application of Theorem 4.2 for problem (5.1). [11, p.206]) Let f be a proper convex function on X , continuous at a point x 0 ∈ X . Assume that the inequality f (x 1 ) < f (x 0 ) = α 0 holds for some x 1 ∈ X. Then,

Lemma 5.3 (See
is the cone generated by the subdifferential of f at x 0 . Let us go back to considering the parametric convex programming problem (5.1). Our first result in this section can be formulated as follows.
Proof Recall that G : X ⇒ Y , with G(x) being defined by (5.2), is a convex set-valued mapping, and the objective function ϕ(x, y) of (5.1) is convex. If (a1) is satisfied then it is clear that (u 0 , v 0 ) ∈ int(gph G), hence the condition (a) in Theorem 4.2 is fulfilled. If (b1) is satisfied then ϕ is continuous at the point (x 0 , y 0 ) which belongs to gph G, so the condition (b) in Theorem 4.2 is fulfilled. Therefore, our assumptions guarantee that (4.3) and (4.4) hold.
We now consider the case where the affine constraints h j (x, y) = 0 ( j ∈ J ) are available in (5.1). The second result of this section reads as follows.

Comparisons with the results of Aubin [1]
Recall that if X is a Hausdorff locally convex topological vector space, f : X → R is a function having values in the extended real line, then the function f * : X * → R given by is said to be the conjugate function of f . The conjugate function of f * , denoted by f * * , is a function defined on X and having values in R: Clearly, the function f * * is convex and closed (in the sense that epi f * * is closed in the weak topology of X × R or, the same, f * * is lower semicontinuous w.r.t. the weak topology of X ). According to the Fenchel-Moreau theorem (see [11,Theorem 1,p.175]), if f is a function on X everywhere greater than −∞, then f = f * * if and only if f is closed and convex. By a different approach, Aubin [1, Problem 35 -Subdifferentials of Marginal Functions, p.335] has studied a problem similar to that one considered in the preceding two sections. Namely, in our notation, Aubin has studied the parametric problem: where X, Y are Hilbert spaces, ϕ : Y → R ∪ {+∞} is a proper, convex, lower semicontinuous function, G : X ⇒ Y is convex, of closed graph. The optimal value function of that problem is given by Using the notion of conjugate function, the above Fenchel-Moreau theorem, and some auxiliary results related to continuous linear mappings, convex functions, and convex sets on Hilbert spaces, Aubin has proved the following theorem.  N ((x,ȳ); gph G).
The proof of Aubin is long and rather complicated. The requirements that ϕ is lower semicontinuous and gph G is closed are really needed in Aubin's proof.
The next two claims clarify the connections between the regularity conditions in Theorem 4.2 and the regularity condition in Theorem 6.1. Indeed, if (a) is fulfilled, then there exist (x 0 , y 0 ) ∈ gph G with y 0 ∈ dom ϕ and U ∈ N (0 X ), V ∈ N (0 Y ), U and V are open sets, such that that is for any x ∈ x 0 +U and y ∈ y 0 + V, y ∈ G(x). Hence x ∈ G −1 (y), for all x ∈ x 0 +U and y ∈ y 0 + V . In particular, y 0 + V ⊂ dom G −1 . As y 0 ∈ dom ϕ, it follows that 0 ∈ −V = y 0 − (y 0 + V ) ⊂ dom ϕ − dom G −1 . Indeed, suppose that ϕ is continuous at such a point y 0 that there is x 0 ∈ X with (x 0 , y 0 ) ∈ gph G. Then, for every ε > 0, there exists V ∈ N (0) such that |ϕ(y 0 + v) − ϕ(y 0 )| < ε, ∀v ∈ V.

Comparisons with the results of Mordukhovich et al. [8]
The assertions of Theorem 4.2 are similar to those of the three theorems of Mordukhovich et al. [8] which have been recalled in Section 3. By imposing the strong convexity requirement on (2.5), we need not to rely on the assumption ∂ + ϕ(x,ȳ) = ∅ in Theorem 3.1, the condition saying that the solution map M : dom G ⇒ Y has a local upper Lipschitzian selection at (x,ȳ) in Theorem 3.2, as well as the μ-inner semicontinuity and the μ-inner semicompactness conditions on the solution map M(·), in Theorem 3.3.