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The Legendre Transformation in Modern Optimization

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Optimization and Its Applications in Control and Data Sciences

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 115))

Abstract

The Legendre transform (LET) is a product of a general duality principle: any smooth curve is, on the one hand, a locus of pairs, which satisfy the given equation and, on the other hand, an envelope of a family of its tangent lines.

An application of the LET to a strictly convex and smooth function leads to the Legendre identity (LEID). For strictly convex and three times differentiable function the LET leads to the Legendre invariant (LEINV).

Although the LET has been known for more then 200 years both the LEID and the LEINV are critical in modern optimization theory and methods.

The purpose of the paper is to show the role of the LEID and the LEINV play in both constrained and unconstrained optimization.

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Authors and Affiliations

  1. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Roman A. Polyak

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  1. Roman A. Polyak
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Correspondence to Roman A. Polyak .

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  1. Department of Industrial and Systems Engineering, Ohio University, Athens, Ohio, USA

    Boris Goldengorin

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Polyak, R.A. (2016). The Legendre Transformation in Modern Optimization. In: Goldengorin, B. (eds) Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications, vol 115. Springer, Cham. https://doi.org/10.1007/978-3-319-42056-1_15

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