Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures

Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9(3): 203–228

Auslender A, Teboulle M (2003) Asymptotic cones and functions in optimization and variational inequalities. Springer, New York

Barrieu P, El Karoui N (2004) Optimal derivatives design under dynamic risk measures. Math Finance. Contemp Math 351: 13–25

Ben-Tal A, Teboulle M (1986) Expected utility, penalty functions and duality in stochastic nonlinear programming. Manag Sci 32(11): 1445–1466

Ben-Tal A, Teboulle M (2007) An old-new concept of convex risk measures: the optimized certainty equivalent. Math Finance 17(3): 449–476

Borwein JM, Jeyakumar V, Lewis AS, Wolkowicz H (1988) Constrained approximation via convex programming, Preprint. University of Waterloo, Canada

Borwein JM, Lewis AS (1992) Partially finite convex programming, part I: quasi relative interiors and duality theory. Math Program 57: 15–48

Boţ RI (2010) Conjugate duality in convex optimization, Lecture notes in economics and mathematical systems, vol 637. Springer, Berlin

Boţ RI, Csetnek ER (2010) Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relations to some classical statements. Optimization. doi: 10.1080/02331934.2010.505649

Boţ RI, Csetnek ER, Moldovan A (2008a) Revisiting some duality theorems via the quasirelative interior in convex optimization. J Opt Theory Appl 139(1): 67–84

Boţ RI, Csetnek ER, Wanka G (2008b) Regularity conditions via quasi-relative interior in convex programming. SIAM J Opt 19(1): 217–233

Boţ RI, Lorenz N, Wanka G. (2010) Dual representations for convex risk measures via conjugate duality. J Opt Theory Appl 144(2): 185–203

Boţ RI, Lorenz N, Wanka G (2009) Optimality conditions for portfolio optimization problems with convex deviation measures as objective functions. Taiwan J Math 13(2A): 515–533

Cheridito P, Li T (2008) Dual characterizations of properties of risk measures on Orlicz hearts. Math Finan Econ 2: 29–55

Cheridito P, Li T (2009) Risk measures on Orlicz hearts. Math Finance 19: 189–214

Filipović D, Kupper M (2007) Monotone and cash-invariant convex functions and hulls. Insur Math Econ 41(1): 1–16

Filipović D, Svindland G (2007) Convex risk measures on L p , Preprint, www.math.lmu.de/~filipo/PAPERS/crmlp

Föllmer H, Schied A (2002) Stochastic finance, a introduction in discrete time. Walter de Gruyter, Berlin

Fritelli M, Rosazza Gianin E (2002) Putting order in risk measures. J Bank Finance 26(7): 1473–1486

Giannessi F (2005) Constrained optimization and image space analysis, vol 1. Separation of sets and optimality conditions. Mathematical concepts and methods in science and engineering, vol 49. Springer, New York

Limber MA, Goodrich RK (1993) Quasi interiors, Lagrange multipliers, and L p spectral estimation with lattice bounds. J Opt Theory Appl 78(1): 143–161

Pflug G.Ch (2007) Subdifferential representation of risk measures. Math Program 108(2–3): 339–354

Pflug G.Ch, Römisch W (2007) Modeling, measuring, and managing risk. World Scientific Publishing, Singapore

Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3): 21–42

Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Finance 26(7): 1443–1471

Rockafellar RT, Uryasev S, Zabarankin M (2006a) Optimality conditions in portofolio analysis with general deviation measures. Math Program 108(2–3, Ser. B): 515–540

Rockafellar RT, Uryasev S, Zabarankin M (2006b) Generalized deviations in risk analysis. Finance Stoch 10(1): 51–74

Rockafellar RT, Wets R.J-B (1998) Variational analysis, fundamental principles in mathematical sciences. Vol.317. Springer, Berlin

Ruszczynski A, Shapiro A (2006) Optimization of convex risk functions. Math Oper Res 31(3): 433–452

Zălinescu C (2002) Convex analysis in general vector spaces. World Scientific Publishing, Singapore