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(Convex) Level Sets Integration

Author

Listed:
  • Jean-Pierre Crouzeix

    (Université Blaise Pascal)

  • Andrew Eberhard

    (RMIT University)

  • Daniel Ralph

    (University of Cambridge)

Abstract

The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or “bundles” of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations.

Suggested Citation

  • Jean-Pierre Crouzeix & Andrew Eberhard & Daniel Ralph, 2016. "(Convex) Level Sets Integration," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 865-886, December.
  • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-015-0795-8
    DOI: 10.1007/s10957-015-0795-8
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    References listed on IDEAS

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    1. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-973, July.
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    Cited by:

    1. Suliman Al-Homidan & Nicolas Hadjisavvas & Loai Shaalan, 2018. "Transformation of Quasiconvex Functions to Eliminate Local Minima," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 93-105, April.
    2. Jean-Pierre Crouzeix, 2022. "On Quasiconvex Functions Which are Convexifiable or Not," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 66-80, June.

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