Generalized Envelope Theorems

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Generalized Envelope Theorems

Generalized Envelope Theorems

01 Jan 2013-

TL;DR: In this paper, the value function of a nonconvex and nonsmooth Lipschitz program was shown to be locally locally localizable, and bounds for upper and lower Dini derivatives of this value function were obtained.

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Abstract: We develop new envelope theorems for a broad class of parameterized nonsmooth optimization problems typical of economic applications where nonconvexities play a key role. We provide su¢ cient conditions for the value function of a nonconvex and nonsmooth Lipschitz program to be locally Lipschitz. We obtain bounds for upper and lower Dini derivatives of this value function, as well as su¢ cient conditions for the

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Book Chapter•DOI•

Fixed Point Theory

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Klaus Deimling

01 Jan 1985

TL;DR: The first group of results in fixed point theory were derived from Banach's fixed point theorem as discussed by the authors, which is a nice result since it contains only one simple condition on the map F, since it is easy to prove and since it nevertheless allows a variety of applications.

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Abstract: Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

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994 citations

Journal Article•DOI•

A Nonsmooth Approach to Envelope Theorems

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Olivier F. Morand¹, Kevin Reffett², Suchismita Tarafdar³•Institutions (3)

University of Connecticut¹, Arizona State University², Shiv Nadar University³

01 Jan 2015-Social Science Research Network

TL;DR: In this paper, a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or non-smooth objectives was developed.

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Abstract: We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian-Fromowitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.

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13 citations

Journal Article•DOI•

A nonsmooth approach to envelope theorems

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Olivier F. Morand¹, Kevin Reffett², Suchismita Tarafdar³•Institutions (3)

University of Connecticut¹, Arizona State University², Shiv Nadar University³

01 Dec 2015-Journal of Mathematical Economics

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10 citations

References

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Book•

Optimization and nonsmooth analysis

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Frank H. Clarke

01 Jan 1983

TL;DR: The Calculus of Variations as discussed by the authors is a generalization of the calculus of variations, which is used in many aspects of analysis, such as generalized gradient descent and optimal control.

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Abstract: 1. Introduction and Preview 2. Generalized Gradients 3. Differential Inclusions 4. The Calculus of Variations 5. Optimal Control 6. Mathematical Programming 7. Topics in Analysis.

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9,498 citations

"Generalized Envelope Theorems" refers background or methods in this paper

...As in Clarke [8], we work with a minimization problem, and explain how the results follow for a maximization problem....
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...The Lipschitz property of V and the formula for the generalized gradient follow directly from Clarke [8] Corollary 1 and rely on the combination of Clarke’s Hypothesis and GMFCQ....
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...We judiciously rewrite our maximization problem so as to apply results from Clarke [8], omitting the equality constraints h to simplify notations....
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...We adopt the general hypothesis made in Clarke [8] (Hypothesis 6....
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...1 in Clarke [8], there exists λ ≥ 0, and θ such that λg(y) = 0 and: 0 ∈ ∂yL(y, λ, θ, s)...
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Book•

Introduction to lattices and order

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Brian A. Davey¹, Hilary A. Priestley²•Institutions (2)

La Trobe University¹, University of Oxford²

01 Jan 1990

TL;DR: The Stone Representation Theorem for Boolean algebras and its application to lattices in algebra can be found in this article, where the structure of finite distributive lattices and finite Boolean algebraic structures are discussed.

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Abstract: 1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.

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4,715 citations

Book•

Perturbation Analysis of Optimization Problems

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J. Frédéric Bonnans, Alexander Shapiro

11 May 2000

TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.

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Abstract: Basic notation.- Introduction.- Background material.- Optimality conditions.- Basic perturbation theory.- Second order analysis of the optimal value and optimal solutions.- Optimal Control.- References.

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2,067 citations

Book•

Supermodularity and Complementarity

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Donald M. Topkis

13 Apr 1998

TL;DR: In this article, the authors introduce the concept of lattices, supermodular functions, and optimal decision models for cooperative games and non-cooperative games, and present a review of the literature.

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Abstract: PrefaceCh. 1Introduction3Ch. 2Lattices, Supermodular Functions, and Related Topics7Ch. 3Optimal Decision Models94Ch. 4Noncooperative Games175Ch. 5Cooperative Games207Bibliography263Index269

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1,981 citations

"Generalized Envelope Theorems" refers background in this paper

...First, the term Ev∗(y, h(K, z), z′) in Bellman’s equation above has increasing differences with respect to (y;h) so by Topkis[38] (Theorems 2.8.1 and 2.8.3) the optimal correspondence Y ∗ is strong set order ascending in h and both ∨Y ∗ and ∧Y ∗ are isotone in h....
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...The traditional result of Topkis (e.g. [38], Theorem 2.7.6) does not apply, so our approach is based on the generalized envelope theorems in the previous section....
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...Second, given that h is increasing in its arguments (and given our assumptions on u and F ), the right hand side in Bellman’s equation above has increasing differences with respect to (y, (k,K, z)) which implies again by the same Theorems in Topkis[38] that Y ∗ is strong set order ascending in (k,K, z), so that both both ∨Y ∗ and ∧Y ∗ are isotone in (k,K, z)....
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...First, the term Ev∗(y, h(K, z), z′) in Bellman’s equation above has increasing differences with respect to (y;h) so by Topkis[38] (Theorems 2....
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...By Topkis (Theorems 2.8.1 and 2.8.3), Y ∗n is strong set order ascending, and ∨Y ∗n and ∧Y ∗n are both increasing selections in (k,K, z)....
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Journal Article•DOI•

Envelope Theorems for Arbitrary Choice Sets

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Paul Milgrom¹, Ilya Segal¹•Institutions (1)

Stanford University¹

01 Mar 2002-Econometrica

TL;DR: The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative as mentioned in this paper.

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Abstract: The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative. This paper studies optimization with arbitrary choice sets and shows that the traditional envelope formula holds at any differentiability point of the value function. We also provide conditions for the value function to be, variously, absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to mechanism design, convex programming, continuous optimization problems, saddle-point problems, problems with parameterized constraints, and optimal stopping problems.

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1,183 citations

"Generalized Envelope Theorems" refers background in this paper

...Corollary 8 Under Clarke’s Hypothesis, if the SMFCQ holds at every optimal solution a∗(s) ∈ A∗(s), and the primitive data is C(1)in s, then the value function is directionally differentiable with: V ′(s;x) = max a∗(s)∈A(s) L2(a ∗(s), s, λ, μ) · x An alternative to SMFCQ is to assume enough concavity to "squeeze" the lower and upper bounds to obtain a directional envelope, as done in Milgrom and Segal [26] Corollary 5....
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...Corollary 8 Under Clarke’s Hypothesis, if the SMFCQ holds at every optimal solution a∗(s) ∈ A∗(s), and the primitive data is C1in s, then the value function is directionally differentiable with: V ′(s;x) = max a∗(s)∈A(s) L2(a ∗(s), s, λ, µ) · x An alternative to SMFCQ is to assume enough concavity to "squeeze" the lower and upper bounds to obtain a directional envelope, as done in Milgrom and Segal [26] Corollary 5....
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