A General Backwards Calculus of Variations via Duality

We prove Euler-Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality conditions for the product and the quotient of nabla variational functionals.Comment: Submitted to Optimization Letters 03-June-2010; revised 01-July-2010; accepted for publication 08-July-201

A Linear Programming Approach to Sequential Hypothesis Testing

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem, depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function with respect to these multipliers coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and an be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.Comment: 25 pages, 4 figures, accepted for publication in Sequential Analysi

A Methodological Note on the Estimation of Programming Models

The paper introduces a general methodological approach for the estimation of constrained optimisation models in agricultural supply analysis. It is based on optimality conditions of the desired programming model and shows a conceptual advantage compared to Positive Mathematical Programming in the context of well posed estimation problems. Moreover, it closes the empirical and methodological gap between programming models and duality based functional models with explicit allocation of fixed factors. Monte Carlo simulations are performed with a maximum entropy estimator to evaluate the functionality of the approach as well as the impact of empirically relevant prior information in small sample situations.Agricultural Supply Analysis, Programming Models, Maximum Entropy Estimation, Prior Information, Research Methods/ Statistical Methods,

Duality and Geometric Programming

Two main problems arise from the use of the Transcendental Logarithmic form: 1. For practical and estimation purposes, the authors take the approximating function as the true function and include any possible source of error in the error term of the regression equation. This implies that there is no way of telling whether the results are affected by stochastic or approximation error. 2. The Cobb-Douglas and the CES production function have the property of "self duality", i.e., both the production and the cost forms are members of the same family of functional forms. This makes irrelevant the choice of representation of the technology by the production or cost functions. The Transcendental Logarithmic Form when taken as, the true form for the primal (dual) problem and then taken again as the true form of the dual (primal), makes one of the selections arbitrary since the form is not self-dual. This point is treated by Burgess [9] who shows with empirical results the consequences of choosing the cost or the production Transcendental Logarithmic form as a representation of the underlying technology. This paper is addressed to the possible solution of these two problems while still being able to work with more general production functions. We propose for the consideration of the economists interested in the Theory of Production, the Geometric Programming (GP) method of solving cost minimization problems which is extensively used in engineering. The similarities observed in both fields also indicate the possible benefits of closer communication among them. In the coming sections, we give an introduction to GP and illustrate with examples using the Cobb-Douglas, CES, and a more general explicit production function

A Class of Nondifferentiable Multiobjective Control Problems

Optimality conditions are derived for a class of nondifferentiable multiobjective control problems having a nondifferentiable term in each component of vector-valued integrand of objective functional. Using Karush-Kuhn-Tucker type optimality conditions, we formulate Mond-Weir type dual to the nondifferentiable control problem and derive duality results extensively under generalized invexity. Finally, it is indicated that our duality results can be considered as dynamic generalizations of those of nondifferentiable nonlinear programming problems recently obtained