Duality in mathematical programming.

In this thesis entitled, “Duality in Mathematical Programming”, the emphasis is given on formulation and conceptualization of the concepts of second-order duality, second-order mixed duality, second-order symmetric duality in a variety of nondifferentiable nonlinear programming under suitable second-order convexity/second-order invexity and generalized second-order convexity / generalized second-order invexity. Throughout the thesis nondifferentiablity occurs due to square root function and support functions. A support function which is more general than square root of a positive definite quadratic form. This thesis also addresses second-order duality in variational problems under suitable second-order invexity/secondorder generalized invexity. The duality results obtained for the variational problems are shown to be a dynamic generalization for thesis of nonlinear programming problem.Digital copy of Thesis.University of Kashmir

Symmetric Duality in Mathematical Programming

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Some Aspects of Mathematical Programming in Statictics

The Almighty has created the Universe and things present in it with an order and proper positions and the creation looks unique and perfect. No one can even think much better or imagine to optimize these further. People inspired by these optimum results started thinking about usage of optimization techniques for solving their real life problems. The concept of constraint optimization came into being after World War II and its use spread vastly in all fields. However, in this process, still lots of efforts are needed to uncover the mysteries and unanswered questions, one of the questions always remains live that whether there can be a single method that can solve all types of nonlinear programming problems like Simplex Method solves linear programming problems. In the present thesis, we have tried to proceed in this direction and provided some contributions towards this area. The present thesis has been divided into five chapters, chapter wise summary is given below: Chapter-1 is an introductory one and provides genesis of the Mathematical Programming Problems and its use in Statistics. Relationship of mathematical programming with other statistical measures are also reviewed. Definitions and other pre-requisites are also presented in this chapter. The relevant literature on the topic has been surveyed. Chapter-2 deals with the two dimensional non-linear programming problems. We develop a method that can solve approximately all type of two dimensional nonlinear programming problems of certain class. The method has been illustrated with numerical examples. Chapter-3 is devoted to the study of n-dimensional non-linear programming problems of certain types. We provide a new method based on regression analysis and statistical distributions. The method can solve n-dimensional non-linear programming problems making use of regression analysis/co-efficient of determination. In chapter-4 we introduce a filtration method of mathematical programming. This method divides the constraints into active and non active and try to eliminate the less important constraints (non-active constraints) and solve the problem with only active constraints. This helps to find solution in less iterations and less in time while retaining optimality of the solution. The final chapter-5 deals with an interesting relationship between linear and nonlinear programming problems. Using this relationship, we can solve linear programming problems with the help of non-linear programming problems. This relationship also helps to find a better alternate solutions to the linear programming problems. In the end, a complete bibliography is provided

Some contributions to optimality criteria and duality in Multiobjective mathematical programming.

This thesis entitled, “some contributions to optimality criteria and duality in multiobjective mathematical programming”, offers an extensive study on optimality, duality and mixed duality in a variety of multiobjective mathematical programming that includes nondifferentiable nonlinear programming, variational problems containing square roots of a certain quadratic forms and support functions which are prominent nondifferentiable convex functions. This thesis also deals with optimality, duality and mixed duality for differentiable and nondifferentiable variational problems involving higher order derivatives, and presents a close relationship between the results of continuous programming problems through the problems with natural boundary conditions between results of their counter parts in nonlinear programming. Finally it formulates a pair of mixed symmetric and self dual differentiable variational problems and gives the validation of various duality results under appropriate invexity and generalized invexity hypotheses. These results are further extended to a nondifferentiable case that involves support functions.Digital copy of Thesis.University of Kashmir

On some contributions to size-biased probability distributions.

Statistical distributions and models are used in many applied areas such as economics, engineering, social, health and biological sciences. In this era of inexpensive and faster personnel computers, practitioners of statistics and scientists in various disciplines have no difficulty in fitting a probability model to describe the distributions of a real-life data set. Traditional enviromentric theory and practice have been occupied with randomization and replication. But in environmental and ecological work, observations also fall in the non-experimental, non-replicated and non-random catogries.The problems of model specification and data interpretation then acquire special importance and great concern. The theory of weighted distributions provides a unifying approach for these problems. Weighted distributions take into account the method of ascertainment, by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of those events as observed and recorded. Failure to make such adjustments can lead to incorrect conclusions. The weighted distributions arise when the observations generated from a stochastic process are not given equal chance of being recorded; instead they are recorded according to some weight function. When the weight function depends on the lengths of the units of interest, the resulting distribution is called length biased. More generally, when the sampling mechanism selects units with probability proportional to some measure of the unit size, resulting distribution is called size-biased. Size-biased distributions are a special case of the more general form known as weighted distributions. These distributions arise in practice when observations from a sample are recorded with unequal probability. In Bayesian Statistics, the posterior distribution summarizes the current state of knowledge about all the uncertain quantities including unobservable parameters. In this thesis, the efforts have been made to study the areas.Digital copy of Ph.D thesisUniverity of Kashmir