Abstract
An evenly convex function on a locally convex space is an extended real-valued function, whose epigraph is the intersection of a family of open halfspaces. In this paper, we consider an infinite-dimensional optimization problem, for which both objective function and constraints are evenly convex, and we recover the classical Lagrange dual problem for it, via perturbational approach. The aim of the paper was to establish regularity conditions for strong duality between both problems, formulated in terms of even convexity.
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Fenchel, W.: A remark on convex sets and polarity. Comm. Sèm. Math. Univ. Lund (Medd. Lunds Algebra Univ. Math. Sem.) Tome Supplémentaire 3, 82–89 (1952)
Daniilidis, A., Martínez-Legaz, J.E.: Characterizations of evenly convex sets and evenly quasiconvex functions. J. Math. Anal. Appl. 273, 58–66 (2002)
Goberna, M.A., Jornet, V., Rodríguez, M.M.L.: On linear systems containing strict inequalities. Linear Algebra Appl. 360, 151–171 (2003)
Goberna, M.A., Rodríguez, M.M.L.: Analyzing linear systems containing strict inequalities via evenly convex hulls. Eur. J. Oper. Res. 169, 1079–1095 (2006)
Klee, V., Maluta, E., Zanco, C.: Basic properties of evenly convex sets. J. Convex Anal. 14, 137–148 (2007)
Rodríguez, M.M.L., Vicente-Pérez, J.: On evenly convex functions. J. Convex Anal. 18, 721–736 (2011)
Martínez-Legaz, J.E., Vicente-Pérez, J.: The e-support function of an e-convex set and conjugacy for e-convex functions. J. Math. Anal. Appl. 376, 602–612 (2011)
Martínez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)
Fajardo, M.D., Vicente-Pérez, J., Rodríguez, M.M.L.: Infimal convolution, c-subdifferentiability and Fenchel duality in evenly convex optimization. Top 20, 375–396 (2012)
Penot, J.-P.: Variational analysis for the consumer theory. J. Optim. Theory Appl. 159, 769–794 (2013)
Fajardo, M.D.: Regularity conditions for strong duality in evenly convex optimization problems. An application to Fenchel duality. J. Convex Anal. 22 (2015)
Goberna, M.A., Jeyakumar, V., López, M.A.: Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities. Nonlinear Anal. Theory Methods Appl. 68, 1184–1194 (2008)
Jeyakumar, V., Dinh, N., Lee, G.M.: A new closed constraint qualification for convex optimization. Applied Mathematics Report AMR 04/8, University of New South Wales (2004)
Boţ, R.I., Wanka, G.: An alternative formulation for a new closed cone constraint qualification. Nonlinear Anal. 64, 1367–1381 (2006)
Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)
Martínez-Legaz, J.E.: Generalized convex duality and its economic applications. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 237–292. Springer, New York (2005)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976)
Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2009)
Rockafellar, R.T.: Conjugate Duality and Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 16. SIAM Publications, Philadelphia (1974)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)
Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. Ser. B 104, 229–233 (2005)
Boţ, R.I.: Conjugate Duality in Convex Optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)
Köthe, G.: Topological Vector Spaces. Springer, Berlin (1969)
Munkres, J.R.: Topology, a First Course. Prentice Hall, New Jersey (1975)
Vicente-Pérez, J.: Nuevos resultados sobre e-convexidad. Ph.D. thesis, University of Murcia (2011)
Acknowledgments
This research was partially supported by MINECO of Spain, Grant MTM2011-29064-C03-02 and by Consellería d’Educació de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012 Grant ACIF-2013-156. The authors wish to thank anonymous referees for their valuable comments and suggestions that have significantly improved the quality of the paper.
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Fajardo, M.D., Rodríguez, M.M.L. & Vidal, J. Lagrange Duality for Evenly Convex Optimization Problems. J Optim Theory Appl 168, 109–128 (2016). https://doi.org/10.1007/s10957-015-0775-z
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DOI: https://doi.org/10.1007/s10957-015-0775-z