Optimality conditions in convex optimization : a finite-dimensional view


    • Dhara, Anulekha
    • Dutta, Joydeep


Optimality conditions in convex optimization : a finite-dimensional view

Anulekha Dhara, Joydeep Dutta

CRC Press, Taylor & Francis Group, c2012

  • : hardback

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Includes bibliographical references and index



Optimality Conditions in Convex Optimization explores an important and central issue in the field of convex optimization: optimality conditions. It brings together the most important and recent results in this area that have been scattered in the literature-notably in the area of convex analysis-essential in developing many of the important results in this book, and not usually found in conventional texts. Unlike other books on convex optimization, which usually discuss algorithms along with some basic theory, the sole focus of this book is on fundamental and advanced convex optimization theory. Although many results presented in the book can also be proved in infinite dimensions, the authors focus on finite dimensions to allow for much deeper results and a better understanding of the structures involved in a convex optimization problem. They address semi-infinite optimization problems; approximate solution concepts of convex optimization problems; and some classes of non-convex problems which can be studied using the tools of convex analysis. They include examples wherever needed, provide details of major results, and discuss proofs of the main results.


What Is Convex Optimization? Introduction Basic concepts Smooth Convex Optimization Tools for Convex Optimization Introduction Convex Sets Convex Functions Subdifferential Calculus Conjugate Functions -Subdifferential Epigraphical Properties of Conjugate Functions Basic Optimality Conditions using the Normal Cone Introduction Slater Constraint Qualification Abadie Constraint Qualification Convex Problems with Abstract Constraints Max-Function Approach Cone-Constrained Convex Programming Saddle Points, Optimality, and Duality Introduction Basic Saddle Point Theorem Affine Inequalities and Equalities and Saddle Point Condition Lagrangian Duality Fenchel Duality Equivalence between Lagrangian and Fenchel Duality: Magnanti's Approach Enhanced Fritz John Optimality Conditions Introduction Enhanced Fritz John Conditions Using the Subdifferential Enhanced Fritz John Conditions under Restrictions Enhanced Fritz John Conditions in the Absence of Optimal Solution Enhanced Dual Fritz John Optimality Conditions Optimality without Constraint Qualification Introduction Geometric Optimality Condition: Smooth Case Geometric Optimality Condition: Nonsmooth Case Separable Sublinear Case Sequential Optimality Conditions and Generalized Constraint Qualification Introduction Sequential Optimality: Thibault's Approach Fenchel Conjugates and Constraint Qualification Applications to Bilevel Programming Problems Representation of the Feasible Set and KKT Conditions Introduction Smooth Case Nonsmooth Case Weak Sharp Minima in Convex Optimization Introduction Weak Sharp Minima and Optimality Approximate Optimality Conditions Introduction -Subdifferential Approach Max-Function Approach -Saddle Point Approach Exact Penalization Approach Ekeland's Variational Principle Approach Modified -KKT Conditions Duality-Based Approach to -Optimality Convex Semi-Infinite Optimization Introduction Sup-Function Approach Reduction Approach Lagrangian Regular Point Farkas-Minkowski Linearization Noncompact Scenario: An Alternate Approach Convexity in Nonconvex Optimization Introduction Maximization of a Convex Function Minimization of d.c. Functions Bibliography Index

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  • ISBN
    • 9781439868225
  • 出版国コード
  • タイトル言語コード
  • 本文言語コード
  • 出版地
    Boca Raton
  • ページ数/冊数
    xviii, 426 p.
  • 大きさ
    25 cm
  • 分類
  • 件名