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242
Identification and Estimation of Hedonic Models
 Journal of Political Economy
, 2004
"... This paper considers the identification and estimation of hedonic models. We establish that in an additive version of the hedonic model, technology and preferences are generically nonparametrically identified from data on demand and supply in a single hedonic market. The empirical literature that c ..."
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Cited by 145 (4 self)
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This paper considers the identification and estimation of hedonic models. We establish that in an additive version of the hedonic model, technology and preferences are generically nonparametrically identified from data on demand and supply in a single hedonic market. The empirical literature that claims that hedonic models estimated on data from a single market are fundamentally underidentified is based on arbitrary linearizations that do not use all the information in the model. The exact economic model that justifies linear approximations is unappealing. Nonlinearities are generic features of equilibrium in hedonic models and a fundamental and economically motivated source of identification.
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 120 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
Optimization of Convex Risk Functions
, 2004
"... We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functio ..."
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Cited by 95 (15 self)
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We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions.
Optimization Problems with perturbations, A guided tour
 SIAM REVIEW
, 1996
"... This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and app ..."
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Cited by 63 (10 self)
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This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
Nonlinear inverse scale space methods for image restoration
 Communications in Mathematical Sciences
, 2005
"... Abstract. In this paper we generalize the iterated refinement method, introduced by the authors in [8], to a timecontinuous inverse scalespace formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space m ..."
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Cited by 63 (18 self)
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Abstract. In this paper we generalize the iterated refinement method, introduced by the authors in [8], to a timecontinuous inverse scalespace formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noisefree image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for onedimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation. 1
Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems
 Annals of Statistics
, 1988
"... Abstract. We study the asymptotic behavior of the statistical estimators that maximize a not necessarily differentiable criterion function, possibly subject to side constraints (equalities and inequalities). The consistency results generalize those of Wald and Huber. Conditions are also given under ..."
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Cited by 55 (1 self)
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Abstract. We study the asymptotic behavior of the statistical estimators that maximize a not necessarily differentiable criterion function, possibly subject to side constraints (equalities and inequalities). The consistency results generalize those of Wald and Huber. Conditions are also given under which one is still able to obtain asymptotic normality. The analysis brings to the fore the relationship between the problem of finding statistical estimators and that of finding the optimal solutions of stochastic optimization problems with partial information. The last section is devoted to the properties of the saddle points of the associated Lagrangians.
Duality theory for optimal investments under model uncertainty
, 2005
"... Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utili ..."
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Cited by 48 (8 self)
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Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaRλ, concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.
A Minimax Method for Finding Multiple Critical Points and Its Applications to Semilinear PDE
 SIAM J. Sci. Comp
"... Most minimax theorems in critical point theory require one to solve a twolevel global optimization problem and therefore are not for algorithm implementation. The objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points i ..."
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Cited by 33 (16 self)
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Most minimax theorems in critical point theory require one to solve a twolevel global optimization problem and therefore are not for algorithm implementation. The objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way. In this paper, inspired by the numerical works of ChoiMcKenna and DingCostaChen, and the idea to define a solution submanifold, some local minimax theorems are established, which require to solve only a twolevel local optimization problem. Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed. The local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex, non starshaped and multiconnected domains. Numerical solutions are illustrated by their graphics for visualization. In a subsequent paper [20], we establish some convergence results for the algorithm.