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: Tools and guidelines for investigating Lipschitz continuity of the value functions in MDP’s, using the Hausdorff metric and the Kantorovich metric for measuring the influence of the constraint set and the transition law, respectively are presented.
Abstract: We present tools and guidelines for investigating Lipschitz continuity of the value functions in MDP’s, using the Hausdorff metric and the Kantorovich metric for measuring the influence of the constraint set and the transition law, respectively. The methods are explained by examples. Additional topics include an application to the the discretization algorithm of Bertsekas (1975).
66 citations
01 Mar 1977
TL;DR: An approach to the solution of max-min problems which takes into account the peculiarities of both the external (max) and the internal (min) operations is considered.
Abstract: An approach to the solution of max-min problems which takes into account the peculiarities of both the external (max) and the internal (min) operations is considered.
The solution allows us to develop a set of methods for the solution of different kinds of max-min problems, including multistage max-min problems, max-min problems with linked constraints, etc.
61 citations
TL;DR: Many economic models are now explicitly dynamic and stochastic, and their state variables evolve in line with the decisions and actions of individual economic agents.
58 citations
"Optimization in economies with nonc..." refers methods in this paper
...For nonclassical stochastic growth models, Nishimura and Stachurski [47] develops a Foster-Lyapunov method to characterize conditions under which optimal dynamics in growth models with nonconvexities are stochastically stable....
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TL;DR: Under minimal assumptions it is proved that the value function is continuously differentiable in concave dynamic programs.
57 citations