Optimality Conditions in Convex Optimization A Finite-Dimensional View

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?IntroductionBasic conceptsSmooth Convex OptimizationTools for Convex OptimizationIntroductionConvex SetsConvex FunctionsSubdifferential CalculusConjugate Functionse-SubdifferentialEpigraphical Properties of Conjugate FunctionsBasic Optimality Conditions using the Normal ConeIntroduction Slater Constraint QualificationAbadie Constraint QualificationConvex Problems with Abstract ConstraintsMax-Function Approach Cone-Constrained Convex ProgrammingSaddle Points, Optimality, and DualityIntroductionBasic Saddle Point TheoremAffine Inequalities and Equalities and Saddle Point ConditionLagrangian DualityFenchel DualityEquivalence between Lagrangian and Fenchel Duality: Magnanti’s ApproachEnhanced Fritz John Optimality ConditionsIntroduction Enhanced Fritz John Conditions Using the SubdifferentialEnhanced Fritz John Conditions under RestrictionsEnhanced Fritz John Conditions in the Absence of Optimal SolutionEnhanced Dual Fritz John Optimality ConditionsOptimality without Constraint QualificationIntroduction Geometric Optimality Condition: Smooth CaseGeometric Optimality Condition: Nonsmooth CaseSeparable Sublinear CaseSequential Optimality Conditions and Generalized Constraint QualificationIntroductionSequential Optimality: Thibault’s ApproachFenchel Conjugates and Constraint QualificationApplications to Bilevel Programming ProblemsRepresentation of the Feasible Set and KKT ConditionsIntroductionSmooth CaseNonsmooth CaseWeak Sharp Minima in Convex OptimizationIntroductionWeak Sharp Minima and OptimalityApproximate Optimality ConditionsIntroductione-Subdifferential ApproachMax-Function Approache-Saddle Point ApproachExact Penalization ApproachEkeland’s Variational Principle ApproachModified e-KKT ConditionsDuality-Based Approach to e-OptimalityConvex Semi-Infinite OptimizationIntroduction Sup-Function ApproachReduction ApproachLagrangian Regular PointFarkas–Minkowski LinearizationNoncompact Scenario: An Alternate ApproachConvexity in Nonconvex OptimizationIntroductionMaximization of a Convex FunctionMinimization of d.c. FunctionsBibliographyIndex

Anulekha Dhara earned her Ph.d. in IIT Delhi and subsequently moved to IIT Kanpur for her post-doctoral studies. Currently, she is a post-doctoral fellow in Mathematics at the University of Avignon, France. Her main area of interest is optimization theory. Joydeep Dutta is an Associate Professor of Mathematics at the Indian Institute of Technology, (IIT) Kanpur. His main area of interest is optimization theory and applications.