In this paper, we derive sufficient condition for global optimality for a nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ > 0 assumptions. We formulate the Wolfe and Mond–Weir type dual models for the problem using convexificators. We establish weak, strong and strict converse duality theorems to relate the semi-infinite mathematical program with equilibrium constraints and the dual models in the framework of convexificators.
Keywords: Nonsmooth analysis, convexificators, duality, constraint qualification
@article{RO_2021__55_S1_S2221_0,
author = {Joshi, Bhuwan Chandra},
title = {Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order $\sigma > 0$ },
journal = {RAIRO. Operations Research},
pages = {S2221--S2240},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
doi = {10.1051/ro/2020081},
mrnumber = {4223154},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2020081/}
}
TY - JOUR AU - Joshi, Bhuwan Chandra TI - Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order $\sigma > 0$ JO - RAIRO. Operations Research PY - 2021 SP - S2221 EP - S2240 VL - 55 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2020081/ DO - 10.1051/ro/2020081 LA - en ID - RO_2021__55_S1_S2221_0 ER -
%0 Journal Article %A Joshi, Bhuwan Chandra %T Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order $\sigma > 0$ %J RAIRO. Operations Research %D 2021 %P S2221-S2240 %V 55 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2020081/ %R 10.1051/ro/2020081 %G en %F RO_2021__55_S1_S2221_0
Joshi, Bhuwan Chandra. Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order $\sigma > 0$. RAIRO. Operations Research, Tome 55 (2021), pp. S2221-S2240. doi: 10.1051/ro/2020081
[1] and , What is Invexity? J. Aust. Math. Soc. Ser. B. 28 (1986) 1–9. | MR | Zbl | DOI
[2] , and , Modeling water allocating institutions based on multiple optimization problems with equilibrium constraints. Environ. Model. Softw. 46 (2013) 196–207. | DOI
[3] , Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, NY (1983). | MR | Zbl
[4] , and , An overview of bilevel optimization. Ann. Oper. Res. 153 (2007) 235–256. | MR | Zbl | DOI
[5] and , Invex Functions and Duality. J. Aust. Math. Soc. Ser. A. 39 (1985) 1–20. | MR | Zbl | DOI
[6] and , Bilevel road pricing: theoretical analysis and optimality conditions. Ann. Oper. Res. 196 (2012) 223–240. | MR | Zbl | DOI
[7] , Convexification and concavification of a positively homogeneous function by the same family of linear functions. Universia di Pisa, Report 3, 208, 802 (1994).
[8] and , Hunting for a smaller convex subdifferential. J. Glob. Optim. 10 (1997) 305–326. | MR | Zbl | DOI
[9] and , Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt (1995). | MR | Zbl
[10] and , Convexifactors, generalized convexity and vector optimization. Optimization 53 (2004) 77–94. | MR | Zbl | DOI
[11] and , Convexifactors, generalized convexity and optimality conditions. J. Optim. Theory Appl. 113 (2002) 41–65. | MR | Zbl | DOI
[12] , and , Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15 (2006) 139–162. | MR | Zbl | DOI
[13] and , Linear Semi-Infinite Optimization. John Wiley & Sons, Chichester (1998). | MR | Zbl
[14] and , editors. Semi-Infinite Programming: Recent Advances. Kluwer, Dordrecht (2001). | Zbl | DOI
[15] , and , On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints. J. Inequal. Appl. 2017 (2017) 1–9. | MR
[16] , On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80 (1981) 545–550. | MR | Zbl | DOI
[17] and , Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48 (1990) 161–220. | MR | Zbl | DOI
[18] and , Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35 (1993) 380–429. | MR | Zbl | DOI
[19] and , Approximate Jacobian matrices for nonsmooth continuous maps and -optimization. SIAM J. Control Optim. 36 (1998) 1815–1832. | MR | Zbl | DOI
[20] and , Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101 (1999) 599–621. | MR | Zbl | DOI
[21] , and , Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians. J. Convex Anal. 5 (1998) 119–132. | MR | Zbl
[22] , Higher order duality in multiobjective fractional programming problem with generalized convexity. Yugosl. J. Oper. Res. 27 (2017) 249–264. | MR | DOI
[23] , and , On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity. J. Oper. Res. Soc. Ch. (2019). DOI: . | DOI | MR
[24] , Optimality conditions for optimistic bilevel programming problem using convexifactors. J. Optim. Theory Appl. 152 (2012) 632–651. | MR | Zbl | DOI
[25] , A multiplier rule for multiobjective programming problems with continuous data. SIAM J. Optim. 13 (2002) 168–178. | MR | Zbl | DOI
[26] and , Semi-infinite programming. Eur. J. Oper. Res. 180 (2007) 491–518. | MR | Zbl | DOI
[27] , and , Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996). | MR | Zbl
[28] , Optimal parameterization of curves for robot trajectory design. IEEE T Automat. Contr. 33 (1988) 209–214. | Zbl | DOI
[29] and , Calcul sous-différentiel pour des fonctions lipschitziennes et nonlipschitziennes. C. R. Math. Acad. Sci. 12 (1984) 269–272. | MR | Zbl
[30] and , Invexity and Optimization, Springer-Verlag, Berlin-Heidelberg (2008). | MR | Zbl
[31] , and , On duality for mathematical programs with vanishing constraints. Ann. Oper. Res. 243 (2016) 249–272. | MR | DOI
[32] and , On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2 (1995) 211–228. | MR | Zbl
[33] and , Constraint qualifications for nonsmooth mathematical programs with equilibrium constraints. Set-Valued Anal. 17 (2009) 65–95. | MR | Zbl | DOI
[34] and , Nondifferentiable multiplier rules for optimization problems with equilibrium constraints. J. Convex Anal. 16 (2009) 187–210. | MR | Zbl
[35] and , Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints. Nonlinear Anal. 72 (2010) 2694–2705. | MR | Zbl | DOI
[36] and , Duality for nonsmooth optimization problems with equilibrium constraints, using convexificators. J. Optim. Theory Appl. 171 (2016) 694–707. | MR | DOI
[37] and , On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints. Oper. Res. Lett. 44 (2016) 148–151. | MR | DOI
[38] and , Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators. Ann. Oper. Res. 269 (2018) 549–564. | MR | DOI
[39] , On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Rev. 29 (1987) 21–89. | MR | DOI
[40] and , Mathematical programs with equilibrium constraints in process engineering. Comput. Chem. Eng. 27 (2003) 1381–1392. | DOI
[41] and , editors. Semi-Infinite Programming. Kluwer, Dordrecht (1998). | MR | DOI
[42] , and , Saddle point optimality criteria for mathematical programming problems with equilibrium constraints. Oper. Res. Lett. 45 (2017) 254–258. | MR | DOI
[43] , Characterizations and applications of generalized invexity and monotonicity in Asplund spaces. Top. 20 (2012) 592–613. | MR | Zbl | DOI
[44] , Bi-level strategies in semi-infinite programming. In: Vol. 71 of Nonconve Optimization and its Applications. Kluwer Academic Publishers, Boston (2003). | MR | Zbl
[45] , How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 223 (2012) 312–320. | MR | Zbl | DOI
[46] and , Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review. Ann. Oper. Res. 34 (1992) 203–218. | MR | Zbl | DOI
[47] and , Optimality and duality results for bilevel programming problem using convexifactors. J. Optim. Theory Appl. 150 (2011) 1–19. | MR | Zbl | DOI
[48] , and , A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints. J Comput. Appl. Math. 217 (2008) 432–447. | MR | Zbl | DOI
[49] and , Air pollution control with semi-infinite programming. Appl. Math. Model. 33 (2009) 1957–1969. | MR | Zbl | DOI
[50] , Application of general semi-infinite programming to lapidary cutting problems. Eur. J. Oper. Res. 191 (2008) 838–854. | MR | Zbl | DOI
[51] , Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307 (2005) 350–369. | MR | Zbl | DOI
Cité par Sources :
