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Duality in set-valued optimization

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Song W.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction...........................................................................................................5 1. Preliminaries on convex and set-valued analysis..............................................6 1.1. Convexity of sets...........................................................................................6 1.2. Convexity of set-valued mappings.................................................................9 1.3. Closed convex processes and invex set-valued mappings..........................12 2. Vector optimization problems...........................................................................14 2.1. Characterization for optimal points of a set..................................................14 2.2. Characterization for optimal solutions of an optimization problem................17 3. Lagrangian multiplier rule................................................................................19 3.1. Lagrangian conditions for weak optimality...................................................19 3.2. Lagrangian conditions for optimality.............................................................21 3.3. Lagrangian conditions for invex set-valued mappings.................................28 4. Lagrangian duality...........................................................................................33 4.1. Duality for weak optimality............................................................................34 4.2. Duality for optimality.....................................................................................35 4.3. Duality for invex set-valued mappings..........................................................36 5. Geometric duality.............................................................................................37 5.1. A general duality principle for sets...............................................................37 5.2. A geometric approach to duality...................................................................39 5.3. Linear optimization problems.......................................................................42 6. Conjugate duality.............................................................................................45 6.1. Conjugate mappings and subdifferentials....................................................45 6.2. A general conjugate duality..........................................................................50 6.3. Duality in vector optimization with constraints...............................................55 6.4. The Fenchel type duality..............................................................................59 References...........................................................................................................62 List of symbols......................................................................................................67 Index.....................................................................................................................68

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1 Song W.

1 1998

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Duality in set-valued optimization