Level sets regularization with application to optimization problems

• Autores: Moussa Barro, Sado Traoré
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 22, Nº. 1, 2020, págs. 137-154
• Idioma: inglés
• DOI: 10.4067/S0719-06462020000100137
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen Dada una función de acoplamiento c y un subconjunto no vacío de R, definimos un operador clausura. Estamos interesados en funciones extendidas a valores reales cuyos conjuntos de sub-nivel son cerrados para este operador. Dado que esta clase de funciones es cerrada bajo supremos puntuales, introducimos una regularización para funciones extendidas a valores reales. Gracias a la descomposición del operador clausura usando el esquema de polaridad, recuperamos la regularización por bi-conjugación. Aplicamos nuestros resultados para derivar una dualidad fuerte para un problema de minimización.

  • English

    Abstract Given a coupling function c and a non empty subset of R, 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.

• Referencias bibliográficas
  • Crouzeix, J-P.. (1977). Contributions l’étude des fonctions quasiconvexes. University of Clermont-Ferrand. France.
  • Dolecki, S.,Kurcyusz, S.. (1978). On φ-convexity in extremal problems. SIAM J. Control Optim.. 16. 277-300
  • Elias, L.M.,Martínez-Legaz, J.E.. (2016). A simplified conjugation scheme for lower semi-continuous functions. Optimization. 65. 751
  • Fenchel, W.. (1952). A remark on convex sets and polarity. Comm. Sém. Math. Univ. Lund. 82
  • Flores-Bazán, F.. (1995). On a notion of subdifferentiability for non-convex functions. Optimization. 33. 1-8
  • Guillaume, S.,Volle, M.. (2015). Level set relaxation, epigraphical relaxation and conditioning in optimization. Positivity. 19. 769
  • Martínez-Legaz, J.. (2005). Handbook of generalized convexity and generalized monotonicity. Springer. New York.
  • Moreau, J.J.. (1970). Inf-convolution, sous-additivité, convexité des fonctions numériques.. J. Math. Pures Appl.. 49. 109
  • Penot, J.P.. (2000). What is quasiconvex analysis?. Optimization. 47. 35-110
  • Penot, J.P.. (2015). Conjugacies adapted to lower semicontinuous functions. Optimization. 64. 473
  • Penot, J.P.,Volle, M.. (1990). On quasi-convex duality.. Math. Oper. Res,. 15. 4597
  • Penot, J.P.,Volle, M.. (2004). Surrogate programming and multipliers in quasi-convex programming. SIAM J. Control Optim. 42. 1994-2003
  • Rockafellar, R.T.. (1974). Conjugate Duality and optimization. SIAM.
  • Rubinov, A.. (2000). Abstract Convexity and GlobalOptimization . Nonconvex Optimization and Its Application. Springer. US.
  • Singer, I.. (1997). Abstract convex analysis. wiley Interscience.
  • Volle, M. (1985). Conjugaison par tranches. Annali di Matematica pura ed applicata. 279-312
  • Volle, M.. (1987). Conjugaison par tranche et dualité de toland. Optimization. 18. 633