Generalized Envelope Theorems: Applications to Dynamic Programming

Abstract

We show in this paper that the class of Lipschitz functions provides a suitable framework for the generalization of classical envelope theorems for a broad class of constrained programs relevant to economic models, in which nonconvexities play a key role, and where the primitives may not be continuously differentiable. We give sufficient conditions for the value function of a Lipschitz program to inherit the Lipschitz property and obtain bounds for its upper and lower directional Dini derivatives. With strengthened assumptions we derive sufficient conditions for the directional differentiability, Clarke regularity, and differentiability of the value function, thus obtaining a collection of generalized envelope theorems encompassing many existing results in the literature. Some of our findings are then applied to decision models with discrete choices, to dynamic programming with and without concavity, to the problem of existence and characterization of Markov equilibrium in dynamic economies with nonconvexities, and to show the existence of monotone controls in constrained lattice programming problems.

Appendices

Appendices

Appendix A discusses some of the features of Lipschitz functions, Appendix B presents some properties of correspondences, and Appendix C discusses some features of lattices and lattice programming.

Appendix A. Properties of Lipschitz Functions

Lipschitz Property. Given an open set $\Omega \subset \mathbb {R}^{n},$ a function $f:\Omega \rightarrow \mathbb {R} ^{m}$ is said to be Lipschitz continuous (or simply “Lipschitz”) at $x\in \Omega$ of modulus $k\ge 0$ if $\exists$ $\delta >0$ such that:

$$\begin{aligned} \forall x^{\prime },x^{\prime \prime }\in \delta B(x),\quad \left| f(x^{\prime \prime })-f(x^{\prime })\right| \le k\left| x^{\prime \prime }-x^{\prime }\right| , \end{aligned}$$

where B(x) is the open ball of radius 1 centered on x. If the modulus k can be chosen independently of x on an entire subset of $\Omega$, f is said to be globally Lipschitz on that subset. Note, for instance, that the function $x\longmapsto Log(x)$ is globally Lipschitz on any [a, b] for $0<a<b$, but only Lipschitz at every $x\in \Omega =]0,+\infty [$.

Dini Derivatives. Intuitively, to be Lipschitz at x means that the rate of change of f around x, no matter how it is calculated, cannot exceed the modulus k. In particular, it implies that the Dini derivatives, upper and lower bounds for the rate of growth of f at x in the direction d, respectively, defined as the functions:

$$\begin{aligned} d\longmapsto & {} D^{+}f(x;d)=\lim \sup _{t\downarrow 0}\frac{f(x+td)-f(x)}{t} \text { and} \\ d\longmapsto & {} D_{+}f(x;d)=\lim \inf _{t\downarrow 0}\frac{f(x+td)-f(x)}{t}, \end{aligned}$$

always exist.

In the event the two Dini bounds coincide, f is said to be Gâ teaux (or directionally) differentiable at x, with Gâteaux derivative given by the common bound:

$$\begin{aligned} f^{\prime }(x;d)=\lim _{t\downarrow 0}\frac{f(x+td)-f(x)}{t}. \end{aligned}$$

If this Gâteaux derivative is a linear function of d, i.e., if $f^{\prime }(x;d)=\nabla f(x)\cdot d$, then f is said to be differentiable at x. Note, for instance, that the Lipschitz function $x\longmapsto \left| x\right|$ is directionally (i.e., Gâteaux) differentiable, but not differentiable at $x=0$. Failure to be differentiable “rarely happens,” since Rademacher’s theorem guarantees that if f is Lipschitz at all point of an open set $\Theta \subset \Omega$, then it is almost everywhere differentiable on $\Theta$. Finally, if the function $x\longrightarrow \nabla f(.)$ is continuous at x, then f is said to be continuously differentiable at x.

Clarke Derivatives and Clarke Gradients. Being Lipschitz at x is a local condition, as it requires that the rate of change around x be bounded by k. Consequently, the upper and lower Clarke derivatives, respectively, defined as:

$$\begin{aligned} d\longmapsto & {} f^{o}(x;d)=\lim \sup _{\begin{array}{c} y\rightarrow x \\ t\downarrow 0 \end{array}}\frac{f(y+td)-f(y)}{t}\,\,\text { and:} \\ d\longmapsto & {} f^{-o}(x;d)=\lim \inf _{_{\begin{array}{c} y\rightarrow x \\ t\downarrow 0 \end{array}}}\frac{f(y+td)-f(y)}{t} \end{aligned}$$

also always exist when f is Lipschitz at x.

The upper Clarke derivative is upper semicontinuous in x (hence the lower Clarke derivative is lower semicontinuous) as established by Clarke [7] (Proposition 2.1.1), and the Clarke derivatives define wider bounds than the Dini derivatives since:

$$\begin{aligned} f^{-o}(x;d)\le Df_{+}(x;d)\le Df^{+}(x;d)\le f^{o}(x;d)=-f^{-o}(x;-d). \end{aligned}$$

The Clarke gradient of a Lipschitz function f at x is the nonempty compact convex set:

$$\begin{aligned} \partial f(x)=cl \quad conv\left\{ \lim \nabla f(x_{i}):x_{i}\rightarrow x,x_{i}\notin \Theta ,x_{i}\notin \Omega _{f}\right\} , \end{aligned}$$

where cl conv denotes the closure of the convex hull, $\Theta$ is any set of Lebesgue measure zero in the domain, and $\Omega _{f}$ is a set of points at which f fails to be differentiable. Clarke [7] (Proposition 2.1.5) shows that the correspondence $x\rightrightarrows \partial f(x)$ is upper hemicontinuous, and Clarke [7] (Proposition 2.1.2) shows that:

$$\begin{aligned} f^{o}(x;d)=\max _{\zeta \in \partial f(x)}\{\zeta .d\}; \end{aligned}$$

hence, $f^{o}(x;d)$ is a convex function of d.

Clarke Regular Functions. An important class of Lipschitz functions is the upper Clarke regular (“upper Clarke regular”) functions, Lipschitz functions that are Gâteaux differentiable and for which the Gâteaux derivative coincides with the upper Clarke derivative, that is, $f^{o}(x;d)=f^{\prime }(x;d)$. Lower Clarke regular functions are similarly defined, i.e., as Gâteaux differentiable Lipschitz functions such that $f^{-o}(x;d)=f^{\prime }(x;d)$.

Several features combine to make the set of upper Clarke regular functions a natural extension of convex functions well suited to the study of economies with nonconvexities. First, except for some pathological cases, convex functions are upper Clarke regular. (Concave functions are lower Clarke regular, since f is upper Clarke regular iff $-f$ is lower Clarke regular.) Second, the theory, calculus, and properties of Clarke gradients have its precise counterpart in smooth and convex analysis. In particular, the Clarke gradient of a convex function coincides with the subgradient of convex analysis, i.e., the set of p $\in M_{m\times n}$ satisfying $\forall d,p\cdot d\le f(x_{0}+d)-f(x_{0})$. Third, upper Clarke regularity grants more power to a Gâteaux derivative at a specific point and in a particular direction, since it then becomes an approximation for the maximum rate of growth of f in a whole neighborhood of that point in that direction. Such local behavior is one step short of continuous differentiability, as shown in the following result.

Lemma A. 1

If $f:\Omega \rightarrow \mathbb {R}^{m}$ is upper Clarke regular and differentiable at x, then f is continuously differentiable at x.

Proof

Differentiability and upper Clarke regular together imply:

$$\begin{aligned} f^{o}(x;d)=f^{\prime }(x,d)=\nabla f(x).d; \end{aligned}$$

hence,

$$\begin{aligned} f^{-o}(x;d)= & {} -f^{o}(x;-d)=-\nabla f(x).(-d) \\= & {} \nabla f(x).d=f^{o}(x;d). \end{aligned}$$

Function $x\rightarrow \nabla f(x)$ is thus both upper and lower semicontinuous at x; hence, f is continuously differentiable at x.$\square$

Similarly, lower Clarke regular differentiable functions are continuously differentiable. Consequently, we note that if f is both upper and lower Clarke regular at x then f is continuously differentiable at x since:

$$\begin{aligned} f^{o}(x;d)=f^{\prime }(x;d)=f^{-o}(x;d) \end{aligned}$$

implies that $f^{\prime }(x;d)$ is both convex and concave in d,  hence linear in d. Thus, f is differentiable at x and therefore continuously differentiable by the lemma above.

Appendix B. Properties of Correspondences

A significant advantage to working in metric spaces is that the topological properties of correspondences can be stated exclusively in terms of sequences.

Definition B. 1

Given $A\subset \mathbb {R }^{n}$ and $S\subset \mathbb {R}^{m}$, a nonempty-valued correspondence $D:S\twoheadrightarrow A$ is:

• (i) lower hemicontinuous at s if for every $a\in D(s)$ and every sequence $s_{n}\rightarrow s$ there exists a sequence $\{a_{n}\}$ such that $a_{n}\rightarrow a$ and $a_{n}\in D(s_{n})$.

• (ii) upper hemicontinuous at s if for every sequence $s_{n}\rightarrow s$ and every sequence $\{a_{n}\}$ such that $a_{n}\in D(s_{n})$ there exists a convergent subsequence of $\{a_{n}\}$ whose limit point a is in D(s).

• (iii) closed at s if $s_{n}\rightarrow s$, $a_{n}\in D(s_{n})$ and $a_{n}\rightarrow a$ implies that $a\in D(s)$. (In particular, this implies that D(s) is a closed set.)

• (iv) open at s if for any sequence $s_{n}\rightarrow s$ and any $a\in D(s)$, there exists a sequence $\{a_{n}\}$ and a number N such that $a_{n}\rightarrow a$ and $a_{n}\in D(s_{n})$ for all $n\ge N$.

Note that $D(s)=\{a\in A,$ $g_{i}(a,s)\le 0,\ i=1,\dots ,p\}$, where the $g_{i}$ are Lipschitz (thus continuous), is necessarily closed at s. The same property holds true in the presence of Lipschitz equality constraints.

Another property of correspondences which is critical in our analysis is that of uniform compactness.

Definition B. 2

A nonempty-valued correspondence D is said to be uniformly compact near s if there exists a neighborhood $S^{\prime }$ of s such that $cl\left[ \cup _{s^{\prime }\in S^{\prime }}D(s)\right]$ is compact.

We note the result in Hogan [44] that if D is uniformly compact near s, then D is closed at s if and only if D(s) is a compact set and D is upper hemicontinuous at s. When D is defined by a system of continuous equality and inequality constraints, uniform compactness near s thus implies compactness and upper hemicontinuity at s. In fact, for any $s^{\prime }$ sufficiently close to s, since $D(s^{\prime })$ is a closed subset of $cl\left[ \cup _{s^{\prime }\in S^{\prime }}D(s)\right]$ it is therefore compact.

Finally, we will need the following property of hemicontinuous correspondences (and thus of Clarke gradients).

Proposition B. 1

If D is an upper hemicontinuous correspondence, then for every compact neighborhood K of x, the set:

$$\begin{aligned} \mathop {\textstyle \bigcup }\limits _{z\in K}D(z) \end{aligned}$$

is compact.

Proof

Consider a sequence $\{y_{n}\}$ in $\mathop {\textstyle \bigcup }\nolimits _{z\in K}D(z)$ so that $y_{n}\in D(z_{n})$ for some $z_{n}$ in K. The sequence $\{z_{n}\}$ is the compact K, so there exists a subsequence of $\{z_{\varphi (n)}\}$ of $\{z_{n}\}$ converging to some $z^{\prime }\in K$. By upper hemicontinuity of D at $z^{\prime }$, there exists a subsequence of $\{y_{\varphi (n)}\}$ converging to some $y\in D(z^{\prime })$. This proves that the initial sequence $\{y_{n}\}$ has a convergent subsequence and therefore that the set $\mathop {\textstyle \bigcup }\nolimits _{x\in K}D(x)$ is compact.$\square$

Appendix C. Posets, Lattices, Supermodularity, and Lattice Programming

A partially ordered set (or poset) is a set X ordered with a reflexive, transitive, and antisymmetric relation. If any two elements of X are comparable, X is referred to as a complete partially ordered set, or chain. An upper (resp., lower) bound of $B\subset X$ is an element $x^{u}$ (resp., $x^{l}$) in B such that $\forall x\in B$, $x\le x^{u}$ (resp., $x^{l}\le x$). A lattice is a set X ordered with a reflexive, transitive, and antisymmetric relation $\ge$ such that any two elements x and $x^{\prime }$ in X have a least upper bound in X, denoted $x\wedge x^{\prime }$, and a greatest lower bound in X, denoted $x\vee x^{\prime }$. The product of an arbitrary collection of lattices equipped with the product (coordinatewise) order is a lattice. $B\subset X$ is a sublattice of X if it contains the sup and the inf (with respect to X) of any pair of points in B.

Let $(X,\ge _{X})$ and $(Y,\ge _{Y})$ be posets. A mapping $f:X\rightarrow Y$ is isotone (or increasing) on X if $f(x^{\prime })\ge _{Y}f(x),$ when $x^{\prime }\ge _{X}x$, for $x,x^{\prime }\in X$. A correspondence (or multifunction) $F:X\rightarrow 2^{Y}$ is ascending in the set relation on $2^{Y}$ denoted by $\ge _{S}$ if $F(x^{\prime })\ge _{S}F(x),$ when $x^{\prime }\ge _{X}x$. A particular set relation of interest is Veinott’s strong set order (see Veinott [35], Chapter 4). Let $L(Y)=\{A|A\subset Y,$ A a nonempty sublattice} be ordered with the strong set order $\ge _{a}$: if $A_{1},A_{2}\in L(Y),$ we say $A_{1}\ge _{a}A_{2}$ if $\forall (a,b)\in A_{1}\times A_{2}$, $a\wedge b\in A_{2}$ and $a\vee b\in A_{1}.$

Let X be a lattice. A function $f:X\rightarrow R$ is supermodular (resp., strictly supermodular) in x if $\forall (x,y)\in X^{2}$, $f(x\vee y)+$ $f(x\wedge y)\ge$(resp., >) f(x) $+$ f(y). The class of supermodular functions is closed under pointwise limits (see Topkis [22], Lemma 2.6.1). Consider a partially ordered set $\Psi =X_{1}\times P$ (with order $\ge$), and $B\subset X_{1}\times P$. The function $f:B\longrightarrow R$ has increasing differences in $(x_{1},p)$ if for all $p_{1},p_{2}\in P$, $p_{1}\le p_{2}$ $\Longrightarrow$ $f(x,p_{2})-f(x,p_{1})$ is nondecreasing in $x\in B_{p_{1}},$ where $B_{p}$ is the p section of B. If this difference is strictly increasing in x then f has strictly increasing differences on B.