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Foreword Preface Introduction: From Convex Analysis to Abstract Convex Analysis 0.1 Abstract Convexity of Sets 0.1a Inner Approaches 0.1b Intersectional and Separational Approaches 0.1c Approaches via Convexity Systems and Hull Operators 0.2 Abstract Convexity of Functions 0.3 Abstract Convexity of Elements of Complete Lattices 0.4 Abstract Quasi-Convexity of Functions 0.5 Dualities 0,6 Abstract Conjugations 0.7 Abstract Subdifferentials 0.8 Some Applications of Abstract Convex Analysis to Optimization Theory 0.Sa Applications to Abstract Lagrangian Duality 0.8b Applications to Abstract Surrogate Duality Chapter One Abstract Convexity of Elements of a Complete Lattice 1.1 The Main (Supremal) Approach: M-Convexity of Elements of a Complete Lattice E, Where M c E 1.2 lnfimal and Supremal Generators and M-Convexity 1.3 An Equivalent Approach: Convexity Systems 1.4 Another Equivalent Approach: Convexity with Respect to a Hull Operator Chapter Two Abstract Convexity of Subsets of a Set 2.1 M-Convexity of Subsets of a Set X, Where M c 2x 2.2 Some Particular Cases 2.2a Convex Subsets of a Linear Space X 2.2b Closed Convex Subsets of a Locally Convex Space X 2.2c Evenly Convex Subsets of a Locally Convex Space X 2.2d Closed Affine Subsets of a Locally Convex Space X 2.2e Evenly Coaffine Subsets of a Locally Convex Space X 2.2f Spherically Convex Subsets of a Metric Space X 2.2g Closed Subsets of a Topological Space X 2.2h Order Ideals and Order Convex Subsets of a Poset X 2.2i Parametrizations of Families ¡õ(ÊýÀí»¯¹«Ê½) Where X Is a Set 2.3 An Equivalent Approach, via Separation by Functions: W-Convexity of Subsets of a Set X, Where ¡õ(ÊýÀí»¯¹«Ê½) 2.4 A Particular Case: Closed Convex Sets Revisited 2.5 Other Concepts of Convexity of Subsets of a Set X, with Respect to a Set of Functions ¡õ(ÊýÀí»¯¹«Ê½) 2.6 (W, ¡õ(ÊýÀí»¯¹«Ê½))-Convexity of Subsets of a Set X, Where W Is a Set and ¡õ(ÊýÀí»¯¹«Ê½)R Is a Coupling Function Chapter Three Abstract Convexity of Functions on a Set 3.1 W-Convexity of Functions on a Set X, Where ¡õ(ÊýÀí»¯¹«Ê½) 3.2 Some Particular Cases 3.2a C(X* + R), Where X Is a Locally Convex Space 3.2b C(X*), Where X Is a Locally Convex Space 3.2c The Case Where X = {0, 1}n and W¡õ(ÊýÀí»¯¹«Ê½) 3.2d The Case Where X = {0, 1}n and W ¡õ(ÊýÀí»¯¹«Ê½) 3.2e ot-Ho1der Continuous Functions with Constant N, Where0 ¡õ(ÊýÀí»¯¹«Ê½) 3.2f Suprema of Ho1der Continuous Functions, Where ¡õ(ÊýÀí»¯¹«Ê½) 3.2g The Case Where ¡õ(ÊýÀí»¯¹«Ê½) 3.3 (W, ¡úo)-ConvexityofFunctions on a Set X, Where W Is a Set and ¡õ(ÊýÀí»¯¹«Ê½)R Is a Coupling Function Chapter Four Abstract Quasi-Convexity of Functions on a Set 4.1 M-Quasi-Convexity of Functions on a Set X, Where ¡õ(ÊýÀí»¯¹«Ê½) 4.2 Some Particular Cases 4.2a Quasi-Convex Functions on a Linear Space X 4.2b Lower Semicontinuous Quasi-Convex Functions on a Locally Convex Space X 4.2c Evenly Quasi-Convex Functions on a Locally Convex Space X 4.2d Evenly Quasi-Coaffine Functions on a Locally Convex Space X 4.2e Lower Semicontinuous Functions on a Topological Space X 4.2f Nondecreasing Functions on a Poset X 4.3 An Equivalent Approach: W-Quasi-Convexity of Functions on a ¡õ(ÊýÀí»¯¹«Ê½) 4.4 Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X, Where W ¡õ(ÊýÀí»¯¹«Ê½) 4.5 Some Particular Cases 4.5a Lower Semicontinuous Quasi-Convex Functions Revisited 4.5b Evenly Quasi-Convex Functions Revisited 4.5c Evenly Quasi-Coaffine Functions Revisited 4.6 (W, ¡ú0)-Quasi-Convexity of Functions on a Set X, Where W Is a Set and ¡õ(ÊýÀí»¯¹«Ê½) : X x W ¡ú R Is a Coupling Function 4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on a Set X, with Respect to Convexity Systems/3 c 2x and Hull Operators u : 2x ¡ú 2x 4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators on ¡õ(ÊýÀí»¯¹«Ê½) Chapter Five Dualities Between Complete Lattices 5.1 Dualities and lnfimal Generators 5.2 Duals of Dualities 5.3 Relations Between Dualities and M-Convex Hulls 5.4 Partial Order and Lattice Operations for Dualities Chapter Six Dualities Between Families of Subsets 6.1 DualitiesA :2x ¡ú 2w, Where X and W Are Two Sets 6.2 Some Particular Cases 6.2a Some Minkowski-Type Dualities 6.2b Some Dualities Obtained from the Minkowski-Type Dualities AM, by Parametrizing the Family M 6.3 Representations of Dualities A : 2x ¡ú 2w with the Aid of Subsets ¡õ(ÊýÀí»¯¹«Ê½) of X ¡ú W and Coupling Functions ¡õ(ÊýÀí»¯¹«Ê½) : X ¡ú W ¡ú 6.4 Some Particular Cases 6.4a Representations with the Aid of Subsets f2 of X X W 6.4b Representations with the Aid of Coupling Functions ¡õ(ÊýÀí»¯¹«Ê½) Chapter Seven Dualities Between Sets of Functions 7.1 Dualities A ¡õ(ÊýÀí»¯¹«Ê½) Where X and W Are Two Sets 7.2 Representations of Dualities A : Ax ¡ú F, Where X Is a Set and ¡õ(ÊýÀí»¯¹«Ê½) and F Are Complete Lattices 7.3 Dualities A : Ax ¡ú Bw, Where X Is a Set and (A,