Revisiting optimization algorithms for maximum likelihood estimation

Parmi les méthodes d’estimation de paramètres de loi de probabilité en statistique, le maximum de vraisemblance est une des techniques les plus populaires, comme, sous des conditions l´egères, les estimateurs ainsi produits sont consistants et asymptotiquement efficaces. Les problèmes de maximum de vraisemblance peuvent être traités comme des problèmes de programmation non linéaires, éventuellement non convexe, pour lesquels deux grandes classes de méthodes de résolution sont les techniques de région de confiance et les méthodes de recherche linéaire. En outre, il est possible d’exploiter la structure de ces problèmes pour tenter d’accélerer la convergence de ces méthodes, sous certaines hypothèses. Dans ce travail, nous revisitons certaines approches classiques ou récemment d´eveloppées en optimisation non linéaire, dans le contexte particulier de l’estimation de maximum de vraisemblance. Nous développons également de nouveaux algorithmes pour résoudre ce problème, reconsidérant différentes techniques d’approximation de hessiens, et proposons de nouvelles méthodes de calcul de pas, en particulier dans le cadre des algorithmes de recherche linéaire. Il s’agit notamment d’algorithmes nous permettant de changer d’approximation de hessien et d’adapter la longueur du pas dans une direction de recherche fixée. Finalement, nous évaluons l’efficacité numérique des méthodes proposées dans le cadre de l’estimation de modèles de choix discrets, en particulier les modèles logit mélangés.

Lagrangian-informed mixed integer programming reformulations

La programmation linéaire en nombres entiers est une approche robuste qui permet de résoudre rapidement de grandes instances de problèmes d'optimisation discrète. Toutefois, les problèmes gagnent constamment en complexité et imposent parfois de fortes limites sur le temps de calcul. Il devient alors nécessaire de développer des méthodes spécialisées afin de résoudre approximativement ces problèmes, tout en calculant des bornes sur leurs valeurs optimales afin de prouver la qualité des solutions obtenues. Nous proposons d'explorer une approche de reformulation en nombres entiers guidée par la relaxation lagrangienne. Après l'identification d'une forte relaxation lagrangienne, un processus systématique permet d'obtenir une seconde formulation en nombres entiers. Cette reformulation, plus compacte que celle de Dantzig et Wolfe, comporte exactement les mêmes solutions entières que la formulation initiale, mais en améliore la borne linéaire: elle devient égale à la borne lagrangienne. L'approche de reformulation permet d'unifier et de généraliser des formulations et des méthodes de borne connues. De plus, elle offre une manière simple d'obtenir des reformulations de moins grandes tailles en contrepartie de bornes plus faibles. Ces reformulations demeurent de grandes tailles. C'est pourquoi nous décrivons aussi des méthodes spécialisées pour en résoudre les relaxations linéaires. Finalement, nous appliquons l'approche de reformulation à deux problèmes de localisation. Cela nous mène à de nouvelles formulations pour ces problèmes; certaines sont de très grandes tailles, mais nos méthodes de résolution spécialisées les rendent pratiques.

MA336exam99pdf

Exam questions and solutions in PDF

MA336exam00pdf

Exam questions and solutions in PDF

MA336exam98

Exam questions and solutions in LaTex

MA336exam99

Exam questions and solutions in LaTex

MA336exam01

Exam questions and solutions in LaTex

MA336exam01pdf

Exam questions and solutions in PDF

MA336exam00

Exam questions and solutions in LaTex

MA336exam98pdf

Exam questions and solutions in PDF

Farming systems methodology for efficient resource management at the farm level: a review from an Indian perspective

Farming systems research is a multi-disciplinary holistic approach to solve the problems of small farms. Small and marginal farmers are the core of the Indian rural economy Constituting 0.80 of the total farming community but possessing only 0.36 of the total operational land. The declining trend of per capita land availability poses a serious challenge to the sustainability and profitability of farming. Under such conditions, it is appropriate to integrate land-based enterprises such as dairy, fishery, poultry, duckery, apiary, field and horticultural cropping within the farm, with the objective of generating adequate income and employment for these small and marginal farmers Under a set of farm constraints and varying levels of resource availability and Opportunity. The integration of different farm enterprises can be achieved with the help of a linear programming model. For the current review, integrated farming systems models were developed, by Way Of illustration, for the marginal, small, medium and large farms of eastern India using linear programming. Risk analyses were carried out for different levels of income and enterprise combinations. The fishery enterprise was shown to be less risk-prone whereas the crop enterprise involved greater risk. In general, the degree of risk increased with the increasing level of income. With increase in farm income and risk level, the resource use efficiency increased. Medium and large farms proved to be more profitable than small and marginal farms with higher level of resource use efficiency and return per Indian rupee (Rs) invested. Among the different enterprises of integrated farming systems, a chain of interaction and resource flow was observed. In order to make fanning profitable and improve resource use efficiency at the farm level, the synergy among interacting components of farming systems should be exploited. In the process of technology generation, transfer and other developmental efforts at the farm level (contrary to the discipline and commodity-based approaches which have a tendency to be piecemeal and in isolation), it is desirable to place a whole-farm scenario before the farmers to enhance their farm income, thereby motivating them towards more efficient and sustainable fanning.

Modeling small-scale dairy farms in central Mexico using multi-criteria programming

Milk supply from Mexican dairy farms does not meet demand and small-scale farms can contribute toward closing the gap. Two multi-criteria programming techniques, goal programming and compromise programming, were used in a study of small-scale dairy farms in central Mexico. To build the goal and compromise programming models, 4 ordinary linear programming models were also developed, which had objective functions to maximize metabolizable energy for milk production, to maximize margin of income over feed costs, to maximize metabolizable protein for milk production, and to minimize purchased feedstuffs. Neither multicriteria approach was significantly better than the other; however, by applying both models it was possible to perform a more comprehensive analysis of these small-scale dairy systems. The multi-criteria programming models affirm findings from previous work and suggest that a forage strategy based on alfalfa, rye-grass, and corn silage would meet nutrient requirements of the herd. Both models suggested that there is an economic advantage in rescheduling the calving season to the second and third calendar quarters to better synchronize higher demand for nutrients with the period of high forage availability.

Evaluating the sustainable intensification of arable farms

Sustainable Intensification (SI) of agriculture has recently received widespread political attention, in both the UK and internationally. The concept recognises the need to simultaneously raise yields, increase input use efficiency and reduce the negative environmental impacts of farming systems to secure future food production and to sustainably use the limited resources for agriculture. The objective of this paper is to outline a policy-making tool to assess SI at a farm level. Based on the method introduced by Kuosmanen and Kortelainen (2005), we use an adapted Data Envelopment Analysis (DEA) to consider the substitution possibilities between economic value and environmental pressures generated by farming systems in an aggregated index of Eco-Efficiency. Farm level data, specifically General Cropping Farms (GCFs) from the East Anglian River Basin Catchment (EARBC), UK were used as the basis for this analysis. The assignment of weights to environmental pressures through linear programming techniques, when optimising the relative Eco-Efficiency score, allows the identification of appropriate production technologies and practices (integrating pest management, conservation farming, precision agriculture, etc.) for each farm and therefore indicates specific improvements that can be undertaken towards SI. Results are used to suggest strategies for the integration of farming practices and environmental policies in the framework of SI of agriculture. Paths for improving the index of Eco-Efficiency and therefore reducing environmental pressures are also outlined.

Utilização de modelos de programação não-linear para sistematização de terras para irrigação em áreas regulares

Objetivou-se com este trabalho, desenvolver modelos de programação não-linear para sistematização de terras, aplicáveis para áreas com formato regular e que minimizem a movimentação de terra, utilizando o software GAMS para o cálculo. Esses modelos foram comparados com o Método dos Quadrados Mínimos Generalizado, desenvolvido por Scaloppi & Willardson (1986), sendo o parâmetro de avaliação o volume de terra movimentado. Concluiu-se que, ambos os modelos de programação não-linear desenvolvidos nesta pesquisa mostraram-se adequados para aplicação em áreas regulares e forneceram menores valores de movimentação de terra quando comparados com o método dos quadrados mínimos.

Modelo de programação linear para seleção de pulverizadores agrícolas de barras

A seleção de pulverizadores agrícolas que se adaptem às necessidades da propriedade, é um processo trabalhoso, sendo uma das etapas mais importantes dentro do processo produtivo. O objetivo do presente trabalho foi o de desenvolver e utilizar um modelo de programação linear para auxiliar na seleção de pulverizadores agrícolas de barras, baseado no menor custo horário do equipamento. Foram utilizadas as informações técnicas referentes a 20 modelos de pulverizadores disponíveis no mercado, sendo quatro autopropelidos, oito de arrasto e oito do tipo montado. A análise de sensibilidade dos componentes dos custos operacionais mostrou que as taxas de reparo e depreciação foram os fatores que mais interferiram na variação do custo horário do conjunto trator-pulverizador. O modelo matemático desenvolvido facilitou a realização da análise de sensibilidade que foi processada em um tempo muito pequeno.