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Elements of Convex Analysis. Linear Theorems of the Alternative. Tangent Cones

In: Basic Mathematical Programming Theory

Author

Listed:
  • Giorgio Giorgi

    (University of Pavia)

  • Bienvenido Jiménez

    (National University of Distance Education)

  • Vicente Novo

    (National University of Distance Education)

Abstract

Mathematical programming theory is strictly connected with Convex Analysis. We give in the present section the main concepts and definitions regarding convex sets and convex cones. Convex functions and generalized convex functions will be discussed in the next chapter. Geometrically, a set $$S\subset \mathbb {R}^n$$ S ⊂ R n is convex Setconvexif the line segment joining any two points in the set lies entirely in the set. We recall that the (closed) line segment joining the points $$x^{1}$$ x 1 and $$x^{2}$$ x 2 of S, denoted as $$\left[ x^{1},x^{2}\right] $$ x 1 , x 2 .

Suggested Citation

  • Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2023. "Elements of Convex Analysis. Linear Theorems of the Alternative. Tangent Cones," International Series in Operations Research & Management Science, in: Basic Mathematical Programming Theory, chapter 0, pages 23-52, Springer.
    • Handle: RePEc:spr:isochp:978-3-031-30324-1_2
    DOI: 10.1007/978-3-031-30324-1_2
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