Skip to main content

Conjugacy in Convex Analysis

  • Chapter
Fundamentals of Convex Analysis

Part of the book series: Grundlehren Text Editions ((TEXTEDITIONS))

Abstract

In classical real analysis, the gradient of a differentiable function f : ℝn → ℝ. plays a key role - to say the least. Considering this gradient as a mapping xs(x) = ∇f(x) from (some subset X of) ℝn to (some subset S of) ℝn, an interesting object is then its inverse: to a given sS, associate the xX such that s =f(x). This question may be meaningless: not all mappings are invertible! but could for example be considered locally, taking for X x S a neighborhood of some (x 0, s 0 = ∇f(x 0)), with ∇2 f continuous and invertible at x 0 (use the local inverse theorem).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
USD 9.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hiriart-Urruty, JB., Lemaréchal, C. (2001). Conjugacy in Convex Analysis. In: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56468-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56468-0_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42205-1

  • Online ISBN: 978-3-642-56468-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics