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
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TL;DR: In this paper, the authors studied a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous, and showed that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards.
Abstract: This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.
27 citations
01 Jan 1984
TL;DR: In this paper, the relative importance of some assumptions in recent results concerning the continuity and directional differentiability of the optimal value function in a nonlinear programming problem is discussed, and examples and counterexamples are provided.
Abstract: This short paper illustrates by examples and counterexamples the relative importance of some assumptions in recent results concerning the continuity and the directional differentiability of the optimal value function in a nonlinear programming problem.
25 citations
25 citations
"Optimization in economies with nonc..." refers background in this paper
...For example, starting in the 1950s with the work of Farrell [19], Rothenberg [58], Koopmans [36], Reiter [51] and others, emphasized the potential importance of nonconvexities in general equilibrium theory....
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TL;DR: In this paper, the Clarke results on generalized gradients were used to prove that the value function has left and right derivatives with respect to the initial capital stock, without requiring supermodularity assumptions.
Abstract: We consider an optimal growth (multi-sector) model with nonconvex technology. Using the Clarke results on generalized gradients, we prove that the value function has left and right derivatives with respect to the initial capital stock, without requiring supermodularity assumptions.
24 citations
TL;DR: A new class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium decision processes for a large class of infinite horizon economies with capital, and provides new conditions for preserving complementarity under maximization, and new generalized envelope theorems for nonconcave dynamic programming problems.
Abstract: Using lattice programming and order theoretic fixpoint theory, we develop a powerful class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium for a large class of infinite horizon economies with capital. The class of economies is large and includes situations where the second welfare theorem fails as in models with public policy, valued fiat money, various forms of market imperfections (e.g., monopolistic competition), production externalities, and various other nonconvexities in the production sets. The methods can be easily adapted to construct symmetric Markov equilibrium in models with many agents and market incompleteness. As our methods are constructive, we prove they have important implications for characterizing the structure of numerical approximations to extremal Markovian equilibrium within the class. Of independent interest is our new approach to characterizing dynamic complementarities. We apply recent generalized envelope theorems found in the literature on nonsmooth analysis to characterize equilibrium value functions in our dynamic programming setting. Our fixed point algorithms are sharp, and are able to distinguish sufficient conditions under which Markovian equilibrium form a complete lattice of Lipschitz continuous, uniformily continuous and semi-continuous monotone functions as well as unique differentiable equilibrium. We develop a new collection of partial orders that allow us to conduct extensive monotone comparative dynamics on the space of economies. We conclude with a discussion of how the methods can be extended to economies with ordinal (as opposed to cardinal) forms of complementary.
23 citations