We give a characterization of those functions which can be obtained as the indirect utility funct... more We give a characterization of those functions which can be obtained as the indirect utility function associated with the utility function of a consumer. This permits to formulate the duality between direct and indirect utility functions in the most general possible setting, which exhibits a perfect symmetry.
A quantal response specifies choice probabilities that are smooth, increasing functions of expect... more A quantal response specifies choice probabilities that are smooth, increasing functions of expected payoffs. A quantal response equilibrium has the property that the choice distributions match the belief distributions used to calculate expected payoffs. This stochastic generalization of the Nash equilibrium provides strong empirical restrictions that are generally consistent with data from laboratory experiments with human subjects. We define the concept of regular quantal response equilibrium and discuss several applications from the recent literature.
We show that generalized convexity appears quite naturally in some models of mathematical economi... more We show that generalized convexity appears quite naturally in some models of mathematical economics, specially in the consumer's behaviour theory. Résumé. Le but de cet exposé est de montrer,à travers l'exemple de la théorie du consommateur, que la convexité généralisée est une notion essentielle et toutà fait naturelle enéconomie mathématique.
Lecture Notes in Economics and Mathematical Systems, 1977
In the course of the last few years, the concept of conjugacy for the convex functions has permit... more In the course of the last few years, the concept of conjugacy for the convex functions has permitted the construction of an harmonious theory of duality for convex programs. In order to establish a similar theory for quasiconvex programs, it is necessary to define the notions of conjugacy which are appropriate to quasiconvex functions. That is the purpose of this paper.
First and second order characterizations of generalized convexity, and more recently, first order... more First and second order characterizations of generalized convexity, and more recently, first order characterizations of generalized monotonicity have been the object of many papers. We present a state of the art on these questions.
ABSTRACT This research concerns the conditions that ensure the quasimonotonicity of the separable... more ABSTRACT This research concerns the conditions that ensure the quasimonotonicity of the separable operator F(x 1 ,x 2 ,⋯,x p )=(F 1 (x 1 ),F 2 (x 2 ),⋯,F p (x p )), where for i=1,2,⋯,p, C i is an open convex subset of ℝ n i and F i :C i →ℝ n i is a continuous nonnull operator. It is shown, in particular, that all F i , except perhaps one, are monotone. The conditions are given in terms of the monotonicity indices of the operators F i , a concept introduced in this paper.
Lecture Notes in Economics and Mathematical Systems, 1990
The purpose of this paper is to introduce and analyze the convergence of a new interval-type algo... more The purpose of this paper is to introduce and analyze the convergence of a new interval-type algorithm for generalized fractional programming; This new algorithm has the advantage of being easier to implement than earlier algorithms of this type, especialy for nonlinear problem. The numerical results indicate that it is as efficient as other interval-type and Dinkelbach-type algorithms for these problems.
For a system of nondifferentiable convex inequalities and linear equalities, the best bound is gi... more For a system of nondifferentiable convex inequalities and linear equalities, the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector. Applications to stability theory are then obtained.
Necessary and sufficient conditions for convexity of lower semi-continuous quasiconvex functions ... more Necessary and sufficient conditions for convexity of lower semi-continuous quasiconvex functions are given. By applying these results to positively homogeneous functions it is shown that if f is a quadratic form which is quasiconvex on a convex C then f is convexifiable, that is, there exists a strictly increasing function k such that k ∘ f is convex.
First-order criteria for pseudomonotonicity and quasimonotonicity are given for differentiable ma... more First-order criteria for pseudomonotonicity and quasimonotonicity are given for differentiable maps on open convex sets. These criteria are proved stronger than those introduced earlier.
The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained ... more The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+√5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.
It is shown how the Schur complement theory can be used for the derivation of criteria for the de... more It is shown how the Schur complement theory can be used for the derivation of criteria for the definiteness (or semidefiniteness) of the restriction of a quadratic form to the null space of a matrix.
Journal of Optimization Theory and Applications, 1993
This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduc... more This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.
We give a characterization of those functions which can be obtained as the indirect utility funct... more We give a characterization of those functions which can be obtained as the indirect utility function associated with the utility function of a consumer. This permits to formulate the duality between direct and indirect utility functions in the most general possible setting, which exhibits a perfect symmetry.
A quantal response specifies choice probabilities that are smooth, increasing functions of expect... more A quantal response specifies choice probabilities that are smooth, increasing functions of expected payoffs. A quantal response equilibrium has the property that the choice distributions match the belief distributions used to calculate expected payoffs. This stochastic generalization of the Nash equilibrium provides strong empirical restrictions that are generally consistent with data from laboratory experiments with human subjects. We define the concept of regular quantal response equilibrium and discuss several applications from the recent literature.
We show that generalized convexity appears quite naturally in some models of mathematical economi... more We show that generalized convexity appears quite naturally in some models of mathematical economics, specially in the consumer's behaviour theory. Résumé. Le but de cet exposé est de montrer,à travers l'exemple de la théorie du consommateur, que la convexité généralisée est une notion essentielle et toutà fait naturelle enéconomie mathématique.
Lecture Notes in Economics and Mathematical Systems, 1977
In the course of the last few years, the concept of conjugacy for the convex functions has permit... more In the course of the last few years, the concept of conjugacy for the convex functions has permitted the construction of an harmonious theory of duality for convex programs. In order to establish a similar theory for quasiconvex programs, it is necessary to define the notions of conjugacy which are appropriate to quasiconvex functions. That is the purpose of this paper.
First and second order characterizations of generalized convexity, and more recently, first order... more First and second order characterizations of generalized convexity, and more recently, first order characterizations of generalized monotonicity have been the object of many papers. We present a state of the art on these questions.
ABSTRACT This research concerns the conditions that ensure the quasimonotonicity of the separable... more ABSTRACT This research concerns the conditions that ensure the quasimonotonicity of the separable operator F(x 1 ,x 2 ,⋯,x p )=(F 1 (x 1 ),F 2 (x 2 ),⋯,F p (x p )), where for i=1,2,⋯,p, C i is an open convex subset of ℝ n i and F i :C i →ℝ n i is a continuous nonnull operator. It is shown, in particular, that all F i , except perhaps one, are monotone. The conditions are given in terms of the monotonicity indices of the operators F i , a concept introduced in this paper.
Lecture Notes in Economics and Mathematical Systems, 1990
The purpose of this paper is to introduce and analyze the convergence of a new interval-type algo... more The purpose of this paper is to introduce and analyze the convergence of a new interval-type algorithm for generalized fractional programming; This new algorithm has the advantage of being easier to implement than earlier algorithms of this type, especialy for nonlinear problem. The numerical results indicate that it is as efficient as other interval-type and Dinkelbach-type algorithms for these problems.
For a system of nondifferentiable convex inequalities and linear equalities, the best bound is gi... more For a system of nondifferentiable convex inequalities and linear equalities, the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector. Applications to stability theory are then obtained.
Necessary and sufficient conditions for convexity of lower semi-continuous quasiconvex functions ... more Necessary and sufficient conditions for convexity of lower semi-continuous quasiconvex functions are given. By applying these results to positively homogeneous functions it is shown that if f is a quadratic form which is quasiconvex on a convex C then f is convexifiable, that is, there exists a strictly increasing function k such that k ∘ f is convex.
First-order criteria for pseudomonotonicity and quasimonotonicity are given for differentiable ma... more First-order criteria for pseudomonotonicity and quasimonotonicity are given for differentiable maps on open convex sets. These criteria are proved stronger than those introduced earlier.
The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained ... more The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+√5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.
It is shown how the Schur complement theory can be used for the derivation of criteria for the de... more It is shown how the Schur complement theory can be used for the derivation of criteria for the definiteness (or semidefiniteness) of the restriction of a quadratic form to the null space of a matrix.
Journal of Optimization Theory and Applications, 1993
This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduc... more This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.
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Papers by J. Crouzeix