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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11 Apr 1983
TL;DR: In this paper, generalized convexity and concavity properties of the optimal value function f* for the general parametric optimization problem P(e) of the form min x sub f (x,e) s.t. x epsilon R(e).
Abstract: : This paper considers generalized convexity and concavity properties of the optimal value function f* for the general parametric optimization problem P(e) of the form min x sub f (x,e) s.t. x epsilon R(e). Many results on convexity and concavity characterizations of f* were presented by the authors in a previous paper. Such properties of f* and the solution map S* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. The authors give sufficient conditions for several types of generalized convexity and concavity of f*, in terms of respective generalized convexity and concavity assumptions on f and convexity and concavity assumptions on the feasible region point-to-set map R. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. (Author)
95 citations
TL;DR: In this article, convexity and concavity properties of the optimal value function f* are considered for the general parametric optimization problemP(ź) of the form minxf(x, ź), s.t.x źR (ź), such properties off* and the solution set mapS* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization.
Abstract: Convexity and concavity properties of the optimal value functionf* are considered for the general parametric optimization problemP(ź) of the form minxf(x, ź), s.t.x źR(ź). Such properties off* and the solution set mapS* form an important part of the theoretical basis for sensitivity, stability, and parametric analysis in mathematical optimization. Sufficient conditions are given for several standard types of convexity and concavity off*, in terms of respective convexity and concavity assumptions onf and the feasible region point-to-set mapR. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. To the authors' knowledge, this is the most comprehensive compendium of such results to date. Many new results are given.
92 citations
TL;DR: In this paper, the authors consider a production economy with a finite number of heterogeneous, infinitely lived consumers and show that, if the economy is smooth enough, equilibria are locally unique for almost all endowments.
85 citations
TL;DR: In this article, the authors considered a one-sector nonclassical model of optimal economic growth, characterized by a convex-concave production function, and provided, in a dynamic-programming context, a characterization of all local (interior) maximum of the miximand of the Bellman equation.
Abstract: The authors consider a one-sector nonclassical model of optimal economic growth, characterized by a convex-concave production function. They provide, in a dynamic-programming context, a characterization of all local (interior) maximum of the miximand of the Bellman equation. These conditions are the Euler equation and a second order condition, namely, that the marginal propensity to consume is less than one. An example is used to illustrate these conditions. Also, several comparative dynamic results are derived. In particular, it is shown that the maximum and minimum selections out of the optimal consumption correspondence shift down as the discount factor increases. Copyright 1991 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
80 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, general properties of Dini derivatives of functions of one and several variables are studied and some applications of this topic to the study of generalized convexity and generalized optimality conditions for mathematical programming problems.
Abstract: This paper, published in two parts, is mainly concerned with general properties of Dini derivatives of functions of one and several variables and with some applications of this topic to the study of generalized convexity and generalized optimality conditions for mathematical programming problems.
68 citations