 Global Theory of OptimizationZigang Pan Chapter Chapter 8 in Measure-Theoretic Calculus in Abstract Spaces, 2023, pp 221-256 from Springer Abstract: Abstract This chapter studies the convex optimization problems in real Banach spaces. The concept of a hyperplane is defined first. A closed hyperplane is identified by a set { x ∈ X | f ( x ) = c } $$\lbrace x \in \mathscr {X}\,|\,f(x) = c \rbrace $$ , where f is a non-null bounded linear functional in X ∗ $$\mathscr {X}^*$$ and c ∈ ℝ $$c \in \mathbb {R}$$ . The direct consequence of the Hahn–Banach Theorem is Mazur’s Theorem 8.7 and Eidelheit Separation Theorem 8.9 that establishes the existence of a hyperplane that separates convex sets. For general optimization problems where the optimizing functional is convex but not necessary a norm expression, we introduce the concept of conjugate convex functional and pre-conjugate convex functional. This culminates in Propositions 8.32, 8.33, and 8.34. A step further in this line of study is the Fenchel Duality Theorem 8.35. The key game theory result is obtained in Proposition 8.39. For constrained convex optimization problems, we introduce the concept of positive cone and its dual concept of positive conjugate cone. The chapter ends with presentation of two flavor of the Lagrange multiplier theory (Propositions 8.57 and 8.58), as well as sufficient conditions of optimality. Date: 2023 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-21912-2_8 Ordering information: This item can be ordered from DOI: 10.1007/978-3-031-21912-2_8 Access Statistics for this chapter More chapters in Springer Books from Springer |
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