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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134 citations
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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TL;DR: The nature of economic dynamics in a one-sector optimal growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous is analyzed, showing that under certain conditions, any optimal path from a given initial capital stock converges to a small neighborhood of the golden rule capital stock.
56 citations
"Optimization in economies with nonc..." refers background in this paper
...for the case of one-sector growth, Dechert and Nishimura [15], Amir, Mirman, and Perkins [2], Hopenhayn and Prescott [27], Nishimura, and Rudnicki, and Stachurski [48], and Kamihigashi and Roy[30] [31] have studied the structure of "nonclassical" optimal growth models....
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TL;DR: In this article, the authors establish necessary and sufficient optimality conditions for weak efficiency and firm efficiency by using Hadamard directional derivatives and scalarizing the multiobjective problem under consideration via signed distances.
Abstract: We establish both necessary and sufficient optimality conditions for weak efficiency and firm efficiency by using Hadamard directional derivatives and scalarizing the multiobjective problem under consideration via signed distances. For the first-order conditions, the data of the problem need not even be continuous; for the second-order conditions, we assume only that the first-order derivatives of the data are calm. We include examples showing the advantages of our results over some recent papers in the literature.
51 citations
Additional excerpts
...From ([34], Theorem 3....
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TL;DR: In particular, the authors showed that the strict Spence-mirrlees condition may hold even though the strict single-crossing condition fails, and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizers is nondecreasing in such problems.
Abstract: MILGROM AND SHANNON (1994) clarify the relationship between order-theoretic methods for comparative statics and more traditional differential techniques by developing relationships between the differential Spence-Mirrlees single crossing property and the order-theoretic single crossing property. Both conditions are central for monotone comparative statics analysis in a number of settings. In particular, Milgrom and Shannon show that the order-theoretic single crossing property is necessary and sufficient for the set of optimal choices to be nondecreasing in certain choice problems, and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizers is nondecreasing in such problems. Milgrom and Shannon assert that under appropriate conditions the Spence-Mirrlees condition is equivalent to their single crossing property, and that the strict versions are also equivalent. In this note, however, we give counterexamples which show that their strict single crossing property may hold even though the strict Spence-Mirrlees condition fails. In fact, we show that the strict single crossing property may hold even though the strict Spence-Mirrlees condition holds only on a set of arbitrarily small measure. We also give a correct statement of the relationship between the Spence-Mirrlees condition and the single crossing property. These counterexamples explain the discrepancy between the monotonicity conclusions that Milgrom and Shannon (1994) derive from the strict single crossing property and the strict monotonicity conclusions that Edlin and Shannon (1998) derive from the strict Spence-Mirrlees condition. In Section 3 we also use these counterexamples to illustrate the fact that the strict single crossing property can allow both pooling and separating equilibria while the strict Spence-Mirrlees condition eliminates the possibility of pooling equilibria. The elimination of pooling equilibria in signalling and screening models is more subtle than Edlin and Shannon's (1998) strict monotonicity conclusions because agents need not face a differentiable constraint.
45 citations
TL;DR: In this paper, a model of social learning in a population of myopic, memoryless agents is considered, where agents are placed at integer points on an infinite line, and each observes the outcomes and technology choices of the two adjacent agents as well as his own outcome.
Abstract: In this paper, we consider a model of social learning in a population of myopic, memoryless agents. The agents are placed at integer points on an infinite line. Each time period, they perform experiments with one of two technologies, then each observes the outcomes and technology choices of the two adjacent agents as well as his own outcome. Two learning rules are considered; it is shown that under the first, where an agent changes his technology only if he has had a failure (a bad outcome), the society converges with probability 1 to the better technology. In the other, where agents switch on the basis of the neighbourhood averages, convergence occurs if the better technology is sufficiently better. The results provide a surprisingly optimistic conclusion about the diffusion of the better technology through imitation, even under the assumption of extremely boundedly rational agents.
44 citations
"Optimization in economies with nonc..." refers background in this paper
...Chatterjee and Xu [6] consider myopic agents and place them at the integer points of the real line, i....
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01 Jan 1998
TL;DR: In particular, this article showed that the strict Spence-mirrlees condition may hold even though the strict single-crossing condition fails, and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizers is nondecreasing in such problems.
Abstract: MILGROM AND SHANNON (1994) clarify the relationship between order-theoretic methods for comparative statics and more traditional differential techniques by developing relationships between the differential Spence-Mirrlees single crossing property and the order-theoretic single crossing property. Both conditions are central for monotone comparative statics analysis in a number of settings. In particular, Milgrom and Shannon show that the order-theoretic single crossing property is necessary and sufficient for the set of optimal choices to be nondecreasing in certain choice problems, and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizers is nondecreasing in such problems. Milgrom and Shannon assert that under appropriate conditions the Spence-Mirrlees condition is equivalent to their single crossing property, and that the strict versions are also equivalent. In this note, however, we give counterexamples which show that their strict single crossing property may hold even though the strict Spence-Mirrlees condition fails. In fact, we show that the strict single crossing property may hold even though the strict Spence-Mirrlees condition holds only on a set of arbitrarily small measure. We also give a correct statement of the relationship between the Spence-Mirrlees condition and the single crossing property. These counterexamples explain the discrepancy between the monotonicity conclusions that Milgrom and Shannon (1994) derive from the strict single crossing property and the strict monotonicity conclusions that Edlin and Shannon (1998) derive from the strict Spence-Mirrlees condition. In Section 3 we also use these counterexamples to illustrate the fact that the strict single crossing property can allow both pooling and separating equilibria while the strict Spence-Mirrlees condition eliminates the possibility of pooling equilibria. The elimination of pooling equilibria in signalling and screening models is more subtle than Edlin and Shannon's (1998) strict monotonicity conclusions because agents need not face a differentiable constraint.
42 citations