Mathematics / Statistics

Saba Naser Majeed

Department of Mathematics, College of Education for Pure Sciences Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq

 

Received: November 10, 2022
Accepted: December 27, 2022
Publication Date: March 23, 2023

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.


Download Citation: ||https://doi.org/10.6180/jase.202312_26(12).0004  


Among a variety of approaches introduced in the literature to establish duality theory, Fenchel duality was of great importance in convex analysis and optimization. In this paper we establish some conditions to obtain classical strong Fenchel duality for evenly convex optimization problems defined in infinite dimensional spaces. The objective function of the primal problem is a family of (possible) infinite even convex functions. The strong duality conditions we present are based on the consideration of the epigraphs of the c-conjugate of the dual objective functions and the ε-c-subdifferential of the primal objective functions.


Keywords: evenly convex set and function, c-conjugate function, ε - c-subdifferentiability of a function, Fenchel duality


  1. [1] W. Fenchel, (1952) “A remark on convex sets and polarity" Meddelanden Lunds Universitets Matematiska Seminarium: 82–89.
  2. [2] M. A. Goberna, V. Jornet, and M. M. Rodriguez, (2003) “On linear systems containing strict inequalities" Linear Algebra and its Applications 360: 151–171. DOI: 10.1016/S0024-3795(02)00445-7.
  3. [3] M. A. Goberna and M. M. Rodriguez, (2006) “Analyzing linear systems containing strict inequalities via evenly convex hulls" European journal of operational research 169(3): 1079–1095. DOI: 10.1016/j.ejor.2003.12.028.
  4. [4] V. Klee, E. Maluta, C. Zanco, et al., (2007) “Basic properties of evenly convex sets" Journal of Convex Analysis 14(1): 137.
  5. [5] M. M. Rodriguez and J. Vicente-Perez, (2011) “On evenly convex functions" J. Convex Anal 18: 721–736.
  6. [6] J. E. Martinez-Legaz and J. Vicente-Perez, (2011) “The e-support function of an e-convex set and conjugacy for e-convex functions" Journal of mathematical analysis and applications 376(2): 602–612. DOI: 10.1016/j .jmaa.2010.10.058.
  7. [7] M. D. Fajardo, J. Vicente-Perez, and M. Rodriguez, (2012) “Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization" Top 20(2): 375–396. DOI: 10.1007/s11750-011-0208-6.
  8. [8] U. Passy and E. Z. Prisman, (1984) “Conjugacy in quasi-convex programming" Mathematical Programming 30(2): 121–146. DOI: 10.1007/BF02591881.
  9. [9] R. T. Rockafellar. Convex analysis. 18. Princeton university press, 1970.
  10. [10] J. J. Moreau, (1970) “Inf-convolution, sous-additivite, convexite des fonctions numeriques" HAL 1970:
  11. [11] J. E. Martinez-Legaz. “Generalized convex duality and its economic applicatons”. In: Handbook of generalized convexity and generalized monotonicity. Springer, 2005, 237–292.
  12. [12] N. Bourbaki. General Topology: Chapters 1–4. 18. Springer Science & Business Media, 2013.
  13. [13] L. Guoyin and N. K. Fu, (2008) “On extension of Fenchel duality and its application" SIAM Journal on Optimization 19(3): 1489–1509. DOI: 10 . 1137/080716803.
  14. [14] P. Wolfe, (1961) “A duality theorem for non-linear programming" Quarterly of applied mathematics 19(3): 239–244.
  15. [15] O. Mangasarian, (1962) “Duality in nonlinear programming" Quarterly of Applied Mathematics 20(3): 300–302.
  16. [16] S. Mishra, (1997) “On sufficiency and duration in nonsmooth multiobjective programming" Opsearch 34(4): 221–231.
  17. [17] S. Mishra, (1997) “Generalized fractional programming problems containing locally subdifferentiable and ρ-univex functions" Optimization 41(2): 135–158. DOI: 10.1080/02331939708844331.
  18. [18] S. K. Mishra, S. Wang, and K. K. Lai, (2006) “Optimality and duality for a multi-objective programming problem involving generalized d-type-I and related n-set functions" European journal of operational research 173(2): 405–418. DOI: 10.1016/j.ejor.2005.02.062.

Latest Articles

Multi-level Semantic Fusion Network for Few-shot Multimedia Image Recognition in Education Management
Multi-level Semantic Fusion Network for Few-shot Multimedia Image Recognition in Education ManagementVolume 28, Issue 2 (In Press)

Read more: ...

River bank erosion prediction using multivariable linear regression
River bank erosion prediction using multivariable linear regressionVolume 27, Issue 12 (In Press)

Read more: ...

Optimization of Ultrasound-Assisted Pectin Extraction from Durian Rind
Optimization of Ultrasound-Assisted Pectin Extraction from Durian RindVolume 28, Issue 2 (In Press)

Read more: ...

Short-time prediction model for cross-section passenger flow in urban transit trains using GCN-AM-BiLSTM
Short-time prediction model for cross-section passenger flow in urban transit trains using GCN-AM-BiLSTMVolume 28, Issue 2 (In Press)

Read more: ...

Response Prediction Analysis of RC Frame Structures Using Support Vector Machine Algorithm
Response Prediction Analysis of RC Frame Structures Using Support Vector Machine AlgorithmVolume 28, Issue 2 (In Press)

Read more: ...

Using the non-dominated sorting genetic algorithm II for multi-objective optimization of acoustic performance in sandwich panels with cellular honeycomb core
Using the non-dominated sorting genetic algorithm II for multi-objective optimization of acoustic performance in sandwich panels with cellular honeycomb coreVolume 28, Issue 2 (In Press)

Read more: ...

Analyzing Glass Configurations For Energy Efficiency In Building Envelopes: A Comparative
Analyzing Glass Configurations For Energy Efficiency In Building Envelopes: A ComparativeVolume 28, Issue 2 (In Press)

Read more: ...

Assessment of the optimal gold recovery rate through heap leach extraction process from hydrothermal alteration zone ore
Assessment of the optimal gold recovery rate through heap leach extraction process from hydrothermal alteration zone oreVolume 28, Issue 2 (In Press)

Read more: ...

Research on the Time-Varying Relationship between Macroeconomic Variables and the Stock Market Volatility Based on TVP-VAR Model
Research on the Time-Varying Relationship between Macroeconomic Variables and the Stock Market Volatility Based on TVP-VAR ModelVolume 28, Issue 2 (In Press)

Read more: ...