Optimization in economies with nonconvexities
01 Jan 2010-
TL;DR: Conditions under which the Classical Lagrangian serves as an exact penalization of a nonconvex programming of a constrained optimization problems in economics are given.
Abstract: Nonconvex optimization is becoming the fashion to solve constrained optimization problems in economics. Classical Lagrangian does not necessarily represent a nonconvex optimization problem. In this paper, we give conditions under which the Classical Lagrangian serves as an exact penalization of a nonconvex programming. This has a simple interpretation and is easy to solve. We use this Classical Lagrangian to provide su¢ cient conditions under which value function is Clarke dif"
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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.
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.
13 citations
Cites background from "Optimization in economies with nonc..."
...Alternatively, one can also construct examples in which the SMFCQ fails, the MFCQ holds, and the value function is not C1 (see Tarafdar [24] for such examples, or an earlier draft of this paper....
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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.
10 citations
References
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Posted Content•
01 Jan 1995
TL;DR: In this paper, the authors developed a general framework to study the relationship between the structure of information flows and the process of social learning and showed that in a connected society, local learning ensures that all agents obtain the same utility, in the long run.
Abstract: textWhen payoffs from different actions are unknown, agents use their own past experience as well as the experience of their neighbors to guide their current decision making. This paper develops a general framework to study the relationship between the structure of information flows and the process of social learning.
We show that in a connected society, local learning ensures that all agents obtain the same utility, in the long run. We develop conditions under which this utility is the maximal attainable, i.e. optimal actions are adopted.
This analysis identifies a structural property of information structures -- local independence -- which greatly facilitates social learning. Our analysis also suggests that there exists a negative relationship between the degree of social integration and the likelihood of diversity.
Simulations of the model generate spatial and temporal patterns of adoption that are consistent with empirical work.
233 citations
"Optimization in economies with nonc..." refers background in this paper
...Bala, Venkatesh and Sanjeev Goyal (1998)....
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...Bala and Goyal [1] advance beyond the simple networks by taking into account an arbitrary structure, and investigate the process of social learning from neighbors....
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TL;DR: In this paper, the authors examine a model of lumpy investment where establishments face persistent shocks to common and plant-specific productivity, and nonconvex adjustment costs lead them to pursue generalized (S,s) investment rules.
Abstract: We examine a model of lumpy investment wherein establishments face persistent shocks to common and plant-specific productivity, and nonconvex adjustment costs lead them to pursue generalized (S,s) investment rules. We allow persistent heterogeneity in both capital and total factor productivity alongside low-level investments exempt from adjustment costs to develop the first model consistent with available evidence on establishment-level investment rates. Reassessing the implications of lumpy investment for aggregate dynamics in this setting, we find that they remain substantial when factor supply considerations are ignored, but are quantitatively irrelevant in general equilibrium. The substantial implications of general equilibrium extend beyond the dynamics of aggregate series. While the presence of idiosyncratic shocks makes the time-averaged distribution of plant-level investment rates largely invariant to market-clearing movements in real wages and interest rates, we show that the dynamics of plants’ investments differ sharply in their presence. Thus, model-based estimations of capital adjustment costs involving panel data may be quite sensitive to the assumption about equilibrium. Our analysis also offers new insights about how nonconvex adjustment costs influence investment at the plant. When establishments face large and weakly persistent idiosyncratic productivity shocks consistent with existing estimates, we find that nonconvex costs do not cause lumpy investments, but act to eliminate them.
230 citations
Book•
07 Dec 2010
TL;DR: In this paper, Newton's method for Lipschitz Equations is used to analyze the regularity of further non-nonsmooth maps in finite dimension and the relation between nonlinear variations and implicit functions.
Abstract: Introduction. List of Results. Basic Notation. 1. Basic Concepts. 2. Regularity and Consequences. 3. Characterizations of Regularity by Derivatives. 4. Nonlinear Variations and Implicit Functions. 5. Closed Mappings in Finite Dimension. 6. Analysis of Generalized Derivatives. 7. Critical Points and Generalized Kojima-Functions. 8. Parametric Optimization Problems. 9. Derivatives and Regularity of Further Nonsmooth Maps. 10. Newton's Method for Lipschitz Equations. 11. Particular Newton Realizations and Solution Methods. 12. Basic Examples and Exercises. Appendix. Bibliography. Index.
203 citations
TL;DR: This paper argued that nonconvexity is essential for economic growth and pointed out the difference between public goods and the technological advances that are fundamental to economic growth, emphasizing the distinction between two of the fundamental attributes of any economic good: rivalry and excludability.
Abstract: Everyday experience and a simple logical argument show that nonconvexities are essential for understanding growth. Compared to previous statements of this well known argument, the presentation here places more emphasis on the distinction between two of the fundamental attributes of any economic good: rivalry and excludability. It also emphasizes the difference between public goods and the technological advances that are fundamental to economic growth. Like public goods, technological advances are rionrival goods. Hence, they are inextricably linked to nonconvexities. But in contrast to public goods, which are nonexcludable, technological advances generate benefits that are at least partially excludable. Hence, innovations in the technology are for the most part privately provided. This means that nonconvexities are relevant for goods that trade in private markets. Consequently, an equilibrium with price-taking in all markets cannot be sustained. Concluding remarks describe some of the recent equilibrium growth models that do not rely on price-taking and highlight some of the implications of these models.
196 citations
01 Jan 1982
TL;DR: In this article, the marginal function of a mathematical program neither assumed convex in its variables or in its parameters is studied. But the bounds for the Dini directional derivatives and estimates for the Clarke generalized gradient are obtained.
Abstract: This paper consists in a study of the differential properties of the marginal or perturbation function of a mathematical programming problem where a parameter or perturbation vector is present. Bounds for the Dini directional derivatives and estimates for the Clarke generalized gradient are obtained for the marginal function of the mathematical program neither assumed convex in its variables or in its parameters. This study generalizes some previously published results on this subject for the special case of right-hand side parameters or perturbations.
180 citations