On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization

Unternehmensforschung - Tập 92 - Trang 199-228 - 2020
Gulcin Dinc Yalcin1, Refail Kasimbeyli1
1Department of Industrial Engineering, Faculty of Engineering, Eskisehir Technical University, Eskisehir, Turkey

Tóm tắt

In this paper, zero duality gap conditions in nonconvex optimization are investigated. It is considered that dual problems can be constructed with respect to the weak conjugate functions, and/or directly by using an augmented Lagrangian formulation. Both of these approaches and the related strong duality theorems are studied and compared in this paper. By using the weak conjugate functions approach, special cases related to the optimization problems with equality and inequality constraints are studied and the zero duality gap conditions in terms of objective and constraint functions, are established. Illustrative examples are provided.

Tài liệu tham khảo

Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York Azimov AY, Gasimov RN (1999) On weak conjugacy, weak subdifferentials and duality with zero gap in nonconvex optimization. Int J Appl Math 1:171–192 Azimov AY, Gasimov RN (2002) Stability and duality of nonconvex problems via augmented Lagrangian. Cybern Syst Anal 3:120–130 Bagirov AM, Ozturk G, Kasimbeyli R (2019) A sharp augmented Lagrangian-based method in constrained nonconvex optimization. Optim Methods Softw 34(3):462–488 Balder EJ (1977) An extension of duality–stability relations to nonconvex optimization problems. SIAM J Control Optim 15:329–343 Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms, 3rd edn. Wiley, Hoboken Burachik RS, Rubinov A (2007) Abstract convexity and augmented Lagrangians. SIAM J Optim 18:413–436 Burachik RS, Iusem AN, Melo JG (2010a) A primal dual modified subgradient algorithm with sharp Lagrangian. J Glob Optim 46:347–361 Burachik RS, Kaya CY, Mammadov M (2010b) An inexact modified algorithm for nonconvex optimization. Comput Optim Appl 45:1–24 Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York Dolecki S, Kurcyusz S (1978) On \(\phi \)-convexity in extremal problems. SIAM J Control Optim 16:277–300 Dolgopolik MV (2015) A unifying theory of exactness of linear penalty functions. Optimization 65:1167–1202 Dolgopolik MV (2017) Existence of augmnented Lagrange multipliers: reduction to exact penalty functions and localization principle. Math Program Ser A 166:297–326 Ekeland I, Temam R (1976) Convex analysis and variational problems. Elsevier North-Holland, Amsterdam Ernst E, Volle M (2013) Zero duality gap for convex programs: a generalization of the Clark–Duffin theorem. J Optim Theory Appl 158:668–686 Ernst E, Volle M (2016) Zero duality gap and attainment with possibly non-convex data. J Convex Anal 23:615–629 Flores-Bazan F, Mastroeni G (2015) Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J Optim 25:647–676 Flores-Bazan F, Echegaray W, Flores-Bazan F, Ocana E (2017) Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap. J Glob Optim 69:823–845 Gasimov RN (1992) Duality in nonconvex optimization. Ph.D. dissertation, Department of Operations Research and Mathematical Modeling, Baku State University, Baku Gasimov RN (2002) Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J Glob Optim 24:187–203 Gasimov RN, Rubinov AM (2004) On augmented Lagrangians for optimization problems with a single constraint. J Glob Optim 28(2):153–173 Goberna MA, Lopez MA, Volle M (2014) Primal attainment in convex infinite optimization duality. J Convex Anal 21:1043–1064 Goh CJ, Yang XQ (2001) A nonlinear Lagrangian theory for nonconvex optimization. J Optim Theory Appl 109:99–121 Huang XX, Yang XQ (2003) A unified augmented Lagrangian approach to duality and exact penalization. Math Oper Res 28:520–533 Huang XX, Yang XQ (2005) Further study on augmented Lagrangian duality. J Glob Optim 31:193–210 Ioffe AD (1979) Necessary and sufficient conditions for a local minimum. III: second-order conditions and augmented duality. SIAM J Control Optim 17:266–288 Jeyakumar V, Huy NQ, Li G (2009) Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization. Optim Eng 10:491–503 Kasimbeyli N, Kasimbeyli R (2017) A representation theorem for Bishop–Phelps cones. Pac J Optim 13(1):55–74 Kasimbeyli R (2009) Radial epiderivatives and set-valued optimization. Optimization 58(5):521–534 Kasimbeyli R (2010) A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J Optim 20(3):1591–1619 Kasimbeyli R, Karimi M (2019) Separation theorems for nonconvex sets and application in optimization. Oper Res Lett 47:569–573 Kasimbeyli R, Mammadov M (2009) On weak subdifferentials, directional derivatives and radial epiderivatives for nonconvex functions. SIAM J Optim 20(2):841–855 Kasimbeyli R, Mammadov M (2011) Optimality conditions in nonconvex optimization via weak subdifferentials. Nonlinear Anal 74:2534–2547 Kasimbeyli R, Ustun O, Rubinov AM (2009) The modified subgradient algorithm based on feasible values. Optimization 58(5):535–560 Li D (1995) Zero duality gap for a class of nonconvex optimization problems. J Optim Theory Appl 85:309–324 Pallaschke D, Rolewicz S (1997) Foundation of mathematical optimization. Kluwer Academic Publishers, Dordrecht Polik I, Terlaky T (2007) A survey of the S-lemma. SIAM Rev 49(3):371–418 Polyak BT (1998) Convexity of quadratic transformations and its use in control and optimization. SIAM Rev 99(3):553–583 Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton Rockafellar RT (1974) Conjugate duality and optimization. SIAM, Philadelphia Rockafellar RT (1993) Lagrange multipliers and optimality. SIAM Rev 35:183–238 Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin Rubinov AM (2000) Abstract convexity and global optimization. Kluwer Academic Publishers, Dordrecht Rubinov AM, Glover BM, Yang XQ (1999a) Decreasing functions with applications to penalization. SIAM J Optim 10:289–313 Rubinov AM, Glover BM, Yang XQ (1999b) Modified Lagrangian and penalty functions in continuous optimization. Optimization 46:327–351 Rubinov AM, Huang XX, Yang XQ (2002) The zero duality gap property and lower semicontinuity of the perturbation function. Math Oper Res 27(4):775–791 Rubinov AM, Yang XQ, Bagirov AM, Gasimov RN (2003) Lagrange-type functions, in constrained optimization. J Math Sci 115(4):2437–2505 Sharikov E (2009) On generalized conjugations and subdifferentials in abstract convex analysis. Optimization 58:599–610 Singer I (1997) Abstract convex analysis. Wiley, New York Wang C, Liu Q, Qu B (2017) Global saddle points of nonlinear augmented Lagrangian functions. J Glob Optim 68:125–146 Yang XQ, Huang XX (2001) A nonlinear Lagrangian approach to constrained optimization problems. SIAM J Optim 11(4):1119–1144