Level sets regularization with application to optimization problems

Downloads

DOI:

https://doi.org/10.4067/S0719-06462020000100137

Abstract

Given a coupling function \(c\) and a non empty subset of â„, we define a closure operator. We are interested in extended real-valued functions  whose  sub-level sets are closed for this operator.  Since  this class of functions is closed under pointwise suprema, we introduce a regularization for  extended real-valued functions. By decomposition of the  closure operator using polarity scheme, we recover the  regularization by bi-conjugation. We apply our results to derive a strong duality for a minimization problem.

Keywords

Duality , regularization , level sets , c-elementary functions , polarity , conjugacy
  • Moussa Barro Département de Mathématiques, UFR des Sciences et Techniques, Université Nazi BONI, Burkina Faso.
  • Sado Traoré Département de Mathématiques, UFR des Sciences et Techniques, Université Nazi BONI, Burkina Faso.
  • Pages: 137–154
  • Date Published: 2020-04-18
  • Vol. 22 No. 1 (2020)

Crouzeix, J-P.: Contributions à ` l‘étude des fonctions quasiconvexes. Thesis. University of Clermont-Ferrand, France (1977).

Dolecki, S. and Kurcyusz, S.: On Φ-convexity in extremal problems. SIAM J. Control Optim. 16, 277–300 (1978).

Elias, L. M. and Martínez-Legaz, J.E.: A simplified conjugation scheme for lower semi-continuous functions. Optimization, 65(4):751–763 (2016).

Fenchel, W.: A remark on convex sets and polarity. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 82–89 (1952).

Flores-Bazán, F.: On a notion of subdifferentiability for non-convex functions. Optimization, 33(1):1–8 (1995).

Guillaume, S. and Volle, M.: Level set relaxation, epigraphical relaxation and conditioning in optimization. Positivity, 19:769–795 (2015).

Martínez-Legaz, J.: Generalized Convex Duality and its Economic Applications. Non-convex Optimization and Its Application, Handbook of generalized convexity and generalized mono- tonicity. Springer, New York, (2005).

Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl., 49: pp. 109–154 (1970).

Penot, J.P.: What is quasiconvex analysis? Optimization, 47: pp. 35–110, (2000).

Penot, J.P.: Conjugacies adapted to lower semicontinuous functions. Optimization, 64(3): pp. 473– 494 (2015).

Penot, J.P. and Volle, M.: On quasi-convex duality. Math. Oper. Res, 15: pp. 4597–625, (1990).

Penot, J.P. and Volle,M.: Surrogate programming and multipliers in quasi-convex programming. SIAM J. Control Optim, 42(6): pp. 1994–2003, (2004).

Rockafellar, R.T.: Conjugate Duality and optimization, SIAM (1974).

Rubinov, A.: Abstract Convexity and Global Optimization. Non-convex Optimization and Its Application. Springer US, (2000).

Singer, I.: Abstract convex analysis. Canadian Mathematics Series of Monographs and Texts. A wiley Interscience,(1997).

Volle, M.: Conjugaison par tranches. Annali di Matematica pura ed applicata, CXXXIX(IV): pp. 279–312, (1985).

Volle, M.: Conjugaison par tranche et dualit Ìe de toland. Optimization, 18(5): pp. 633–642 (1987).

Similar Articles

1 2 3 4 5 6 7 8 9 10 11 12 > >> 

You may also start an advanced similarity search for this article.

Downloads

Download data is not yet available.

Published

2020-04-18

How to Cite

[1]
M. Barro and S. . Traoré, “Level sets regularization with application to optimization problems”, CUBO, vol. 22, no. 1, pp. 137–154, Apr. 2020.

Issue

Vol. 22 No. 1 (2020)

Section

Articles

Similar Articles

1 2 3 4 5 6 7 8 9 10 11 12 > >> 

You may also start an advanced similarity search for this article.