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Optimality and Duality for Variational & Bilevel Optimization Problems
No full textOptimization is a fundamental aspect of mathematical modeling and decision-making, playing a crucial role in several areas such as economics, engineering, and operations research. In many practical situations, decisions must be made to optimize multiple conflicting objectives simultaneously, leading to the field of multiobjective optimization. Multiobjective optimization seeks to find solutions that balance these competing objectives, providing a set of optimal trade-offs rather than a single optimal solution. The study of various models in optimization and multiobjective optimization is essential because it allows researchers and practitioners to explore different mathematical formulations and algorithms tailored to specific problem characteristics. By studying these models, we gain insights into how different optimization approaches perform under varying conditions, enabling us to choose the most suitable methods for addressing complex decision-making challenges in diverse application domains.
The main objective of this thesis is to address three significant problems in the field of optimization. One major focus is achieving higher-order duality results for fractional variational problems under generalized convexity conditions. This involves developing and proving advanced duality theorems that can handle complex fractional and variational formulations. Secondly, the thesis aims to derive optimality conditions and duality results for interval-valued variational programming problems, where uncertainty in parameters is represented as intervals. Lastly, using a reformulation approach, the thesis concentrates on obtaining optimality conditions and duality results for bilevel optimization problems.
Keeping these objectives in mind, we will address the optimality and duality of developed mathematical models.
In what follows, the whole work has been examined by dividing it into six chapters.
Chapter 1 provides a brief overview of the theory of variational problems, interval-valued optimization problems, and bilevel optimization problems, tracing their evolution and significance. It presents essential definitions, notations and a comprehensive review of the foundational concepts necessary to understand these areas. This chapter also includes a concise survey of relevant work conducted by other researchers, highlighting key findings and advancements that have contributed to the current body of knowledge. Thus, it provides readers with the necessary background to comprehend the contributions of the subsequent chapters to these fields.
Chapter 2 explores the multiobjective fractional variational problems, emphasizing support functions. It introduces the definitions of higher-order pseudoinvex, higher-order (F,α,ρ,d)-convex, and higher-order (F,α,ρ,d)-pseudoconvex functions, enriching the theoretical framework for optimization problems. Further, Mond-Weir-type primal-dual pairs are formulated under both inequality and cone constraints. A rigorous theoretical analysis is then conducted to establish the relationship between the values of the dual pairs, providing insights into the underlying structure and connections between these models. Finally, the chapter concludes by validating the weak duality theorems, reinforcing the theoretical findings and demonstrating their practical implications for optimization problems.
Chapter 3 investigates multiobjective fractional variational problems involving support functions over cones, introducing the concept of higher-order K-η convexity. Within this context, the study explored duality results that relate to the values of primal and dual problems. To enhance understanding, a numerical example of a functional that is higher-order K-η convex but not first-order K-η convex is included. Various models are further shown to be special cases of our proposed framework under certain parametric values. Additionally, a real-world example is presented to validate the findings of the weak duality theorem, demonstrating the practical relevance of the theoretical framework developed in this work.
Chapter 4 deals with the interval-valued variational problems for both second-order and higher-order objective functions. The chapter introduces definitions of η -bonvexity and higher-order invexity, providing illustrative examples that satisfy these definitions while not conforming to existing ones. Thus, this necessitates a theoretical analysis of interval-valued problems. Furthermore, primal-dual pairs for both second-order and higher-order interval-valued variational objective functions are formulated, along with the governing optimality conditions for the models. The exploration of duality theorems elucidates the relationship between primal and dual problem values, offering insights into the underlying structure and connections between these models. We further discuss under what environment and how the existing model can be derived from the presented model. Numerical examples are presented to demonstrate the effectiveness and applicability of the proposed model. Moreover, an example is employed to validate the weak duality theorem, showcasing the practical relevance of the theoretical framework developed in this research.
Chapter 5 aims to study the robust bilevel programming problems with interval-valued objective functions and constraint-wise uncertainty at the upper-level. By utilizing an optimal value reformulation, the bilevel problem is remodified into a single-level problem, which allows for the application of robust counterpart optimization techniques to handle uncertainty effectively. The study establishes necessary optimality conditions for robust LU-optimal solutions, employing an extended robust nonsmooth Abadie constraint qualification (EACQ) based on convexificators. Furthermore, an example is provided with a detailed explanation to enhance understanding of the theoretical optimality conditions. Additionally, the chapter discusses duality results for the original problem and its Mond-Weir dual.
Chapter 6 focuses on the class of multiobjective interval-valued bilevel optimization problem. Initially, we state a nonsmooth constraint qualification for this class. Following this, necessary and sufficient optimality conditions for the optimization model are developed. Further, the Mond–Weir-type dual model is formulated for considered bilevel interval-valued multiobjective optimization problems, and weak, strong, and converse duality results are established under generalized ∂^*-convexity assumptions. To understand the established necessary and sufficient conditions proposed in the theorem, a detailed discussion is provided through numerical examples
Duality in Mathematical Programming Involving Class of E-Convex Functions
Get PDFMaster of Science -Mathematics & ComputingThe present thesis is organised into four chapters. The additional objective is to clarify the structure underlying generalized convex functions by presenting analogoues to the properties of convex functions. The material on convex functions and their generalizations is intensely large. The overview of the chapters are described below: The first chapter of the dissertation is the introduction with the brief description of basic concepts, definitions of convex functions and and other definitions to be used in subsequent chapter are given, that are used throughout work which is useful for understanding the general concept of convexity. A brief account of the related studies made by various authors in the field and a summary of the thesis has also presented in this chapter.
Chapter 2 is devoted to a class of E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established. In Chapter 3, we have reviewed a class of functions, F-convex functions and Second order F-convex functions as a generalization of convex functions. Under F-convexity, F-concavity, F-pseudoconvexity, F-pseudoconcavity, duality results for pair of Wolfe and Mond-Weir type symmetric dual nonlinear programming problems and second order symmetric duality are established.
Chapter 4, we discussed a class of functions called F-E-convex functions which are generalizations of F-convex and E-convex function. We proved the weak duality and strong duality results for the second order Mond-Weir type dual problem with cone constraints
Optimality and Duality Results for Some Bilevel Programming Problems
No full textThe work exhibited in this thesis is an endeavor to achieve various optimality and duality results for bilevel programming problems. The proposed work encapsulates these results which are weaved into five chapters. The present thesis is assembled into chapters as described below:
Chapter 1 is introductory and consists of definitions, notations and prerequisites of the present work. A brief account of the related work studied by various authors in the field and a summary of the thesis are also presented.
Chapter 2 presents a Wolfe type dual corresponding to a multiobjective bilevel problem. Duality results are developed and with the help of a non-trivial example weak duality the- orem is demonstrated. Further we have studied a multi-objective bilevel problem where both the levels have multiple objectives. By using optimal value reformulation and a scalarization technique we reformulate the problem. We have developed sufficient opti- mality conditions for this model. We have proposed a Mond-Weir type dual corresponding to this model and developed the relevant duality theorems under ∂∗-pseudoconvex and ∂∗-quasiconvex assumptions.
In Chapter 3, we examined a bilevel problem with multiple objectives at both lev- els. With the aid of kth-objective weighted constraint scalarization and objective value function reformulation, the problem is converted into a single-level mathematical pro- gramming problem. The necessary optimality conditions are obtained and an illustrative example is given to validate our result.
In Chapter 4, we have considered a bilevel programming problem with uncertainty at the upper-level constraint. By using robust counterpart approach and optimal value reformulation we transform the robust counterpart bilevel problem into a single-level problem. We have developed the optimality conditions in terms of subdifferentials and convexifactors. Moreover we have considered a multi-objective robust bilevel problem and developed the necessary optimality conditions.
Chapter 5 is devoted to the development of relationship between a fractional multi- objective bilevel programming problem and its Mond-Weir type dual under ∂∗-pseudoconvex and ∂∗-quasiconvex assumptions. An example is given to validate the weak duality the- orem
Some Aspects of Duality in Mathematical Programming Problems
No full textThe work exhibited in this thesis is an endeavor to achieve various duality results for minimax
fractional programming and multiobjective programming problems. The proposed
work encapsulates these results which are weaved into six chapters. The present thesis is
assembled into chapters as described below:
Chapter 1 is introductory and consists of definitions, notations and prerequisites of
the present work. A brief account of the related work studied by various authors in the
field and a summary of the thesis are also presented.
Chapter 2 presents a parametric dual model for nondifferentiable minimax fractional
programming (NMFP) problems. Optimality conditions and duality relations are acquired
using (p, r)-ρ-(η, θ)-invex suppositions. Two types of second-order dual models are proposed
for NMFP problem and usual duality results are developed under second-order B-
(p, r)-invex functions.
In Chapter 3, we present a novel concept of higher-order B-(p, r)-invex functions.
we construct a higher-order dual for NMFP problem and achieve duality results under
higher-order B-(p, r)-invexity. A numerical example is solved for finding optimal solution
of NMFP problem.
In Chapter 4, we develop second-order duality results for nondifferentiable multiobjective
fractional variational problem under second-order (F, α, ρ, d)-pseudoconvexity suppositions.
An illustration showing the existence of second-order (F, α, ρ, d)-pseudoconvex
functions is provided. An example is obtained to validate the theoretical results of weak
duality.
Chapter 5 presents a new pair of higher-order symmetric dual for multiobjective
programming problems involving support functions over arbitrary cones. We construct an
example of a non trivial function that shows the existence of higher-order K-η-convex functions.
Various duality relations are explored under aforesaid assumptions. Some special
cases are also examined to show that this work extends known results of the literature.
In Chapter 6, we propose a mixed type higher-order symmetric dual model for multiobjective
programming problems. Weak, strong and converse duality theorems are established
under higher-order K-(F, α, ρ, d)-convexity assumptions
Duality in Mathematical Programming Under Generalized Convexity
No full textphd, SMCAThe work being presented in the present thesis is devoted to the study of duality
results for some mathematical programming problems under generalized convexity
assumptions. The chapterwise summary of the thesis is as follows:
Chapter 1 is introductory and consist of nonlinear and multiobjective programming
problems, definitions, notations and prerequisites of the present work. A brief
account of the related studies made by various authors in the field and a summary of
the thesis are also presented.
In Chapter 2, we have considered Wolfe type second-order multiobjective symmetric
dual programs involving nondifferentiable functions and appropriate duality
theorems are established using the notion of second-order F-convexity assumptions.
Moreover, an example has been given which is second-order F-convex but not convex.
Further, these symmetric dual programs are generalized over arbitrary cones
and usual duality results are obtained under second-order (F, , , d)-convexity assumptions.
A non-trivial example which shows that second-order (F, , , d)-convex
functions are generalization of second-order F-convex functions has also been exemplified.
iii
iv
In Chapter 3, a new pair of second-order multiobjective symmetric dual programs
in which the objective function is optimized with respect to an arbitrary closed
convex cone is formulated and appropriate duality relations are then obtained under
K- -bonvexity assumptions. We identify a function lying exclusively in the class of
K- -bonvex and not in class of invex function already existing in literature. Self
duality for this pair is also obtained by assuming the functions involved to be skewsymmetric.
Further, we have considered a pair of Mond-Weir type nondifferentiable
multiobjective second-order symmetric dual programs over arbitrary cones, where
each of the objective function contains a square root term with positive semidefinite
matrix in Rn×n. Weak, strong and converse duality results are then established under
K- -bonvexity/second-order K-F-convexity assumptions.
In Chapter 4, we have established duality relations for a pair of second-order
mixed symmetric dual programs involving nondifferentiable functions under secondorder
F-convexity/pseudoconvexity assumptions. Next, we have considered a pair of
mixed second-order symmetric dual programs over cones and obtained duality results
under second-order (F, ) convexity/pseudoconvexity assumptions. This mixed formulation
unifies two second-order symmetric dual formulations exist in the literature.
Several known results [39, 55, 57, 70] are obtained as special cases.
In Chapter 5, we have formulated a pair of second-order multiobjective mixed
symmetric dual programs over arbitrary cones and obtained appropriate duality
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results under second-order invexity/pseudoinvexity assumptions. Further, we construct
a pair of multiobjective second-order mixed nondifferentiable symmetric dual
programs involving the square root of a positive semidefinite quadratic function,
(xTBx)
1
2 . The usual duality results are then established using the notion of secondorder
F-convexity/pseudoconvexity assumptions.
Agarwal et al. [3] extended the results of Chen [43] over arbitrary cones and
proved appropriate duality relations under higher-order K-F-convexity assumptions.
Mond-Weir type duality has been discussed in both the papers. In Chapter 6,
we have studied higher-order Wolfe type multiobjective symmetric dual programs
over arbitrary cones and the duality results are then established under higher-order
(F, , , d)-convexity/pseudo-convexity assumptions. We have also illustrated a nontrivial
example of function lying in the class of higher-order K-(F, , , d)-convex but
not in class of higher-order K-F-convex. Further, we consider the higher-order multiobjective
symmetric nondifferentiable dual programs in which the objective function
is optimized with respect to an arbitrary closed convex cone and proved duality theorems
under higher-order-K-(F, , , d)-convexity assumptions.
In Chapter 7, motivated by Lai et al. [85], Lai and Lee [84] and Antczak [19],
we have discussed sufficient optimality conditions and duality theorems for a nondifferentiable
minimax fractional programming problem with B-(p, r)-invexity. An
example which is B-(1, 1)-invex but not (p, r)-invex is exemplified. We also illustrate
another example which is (−1, 1)-invex but not convex.
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In Chapter 8, we have formulated a pair of multiobjective fractional variational
symmetric dual problems for a class of nondifferentiable functions over arbitrary cones
and achieved duality results under generalized (F, , , d)-convexity assumptions. A
self duality theorem is also obtained by assuming the functions involved to be skewsymmetric.
At the last, an Appendix A has been given, in which we establish a strong
duality theorem for a pair of multiobjective second-order symmetric dual programs.
This removes an omission in an earlier result in Yang et al.
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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