# Semi-infinite Mathematical Programming Problems involving Generalized Convexity

### Abstract

In this paper, we consider semi-infinite mathematical programming problems withequilibrium constraints (SIMPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPEC. We formulate Wolfe and Mond-Weir type dual models for the SIMPEC under generalized convexity assumptions. Moreover, weak and strong duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators.### References

T. Antczak, On (p, r)-invexity-type nonlinear programming problems, Journal of Mathematical Analysis and Applications, vol. 264, no. 2, pp. 382-397, 2001.

T. Antczak, Generalized B-(p, r)-invexity functions and nonlinear mathematical programming, Numerical Functional Analysis and Optimization, vol. 30, no. 1-2, pp. 1-22, 2009.

T. Antczak, Lipschitz r-invex functions and nonsmooth programming, Numerical Functional Analysis and Optimization, vol. 23, no. 3-4, pp. 265-283, 2002.

M. Alavi Hejazi, N. Movahedian, and S. Nobakhtian, Multiobjective Problems: Enhanced Necessary Conditions and New Constraint Qualifications through Convexificators, Numerical Functional Analysis and Optimization, vol. 39, no. 1, pp. 11-37, 2018.

W. Britz, M. Ferris, and M. Kuhn, Modeling water allocating institutions based on multiple optimization problems with equilibrium constraints, Environmental Modelling & Software, vol. 46, pp. 196-207, 2013.

F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983.

B. Colson, P. Marcotte, and G. Savard, A overview of bilevel optimization, Annals of Operations Research, vol. 153, no. 1, pp. 235-256, 2007.

B.D. Craven, Invex function and constrained local minima, Bulletin of the Australian Mathematical Society, vol. 24, no. 3, pp. 357- 366, 1981.

S. Dempe and A.B. Zemkoho, Bilevel road pricing: Theoretical analysis and optimality conditions, Annals of Operations Research, vol. 196, no. 1, pp. 223-240, 2012.

J. Dutta and S. Chandra, Convexificators, generalized convexity and vector optimization, Optimization, vol. 53, no. 1, pp. 77-94, 2004.

M.A. Goberna and M.A. Lo´pez, Linear Semi-Infinite Optimization, John Wiley & Sons Ltd, Chichester, England 1998.

M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545-550, 1981.

P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems, a survey of theory, algorithms and applications, Mathematical Programming, vol. 48 no. 1-3, pp. 161-220, 1990.

J. Haslinger, P. Neittaanm¨aki, Finite element approximation for optimal shape design: Theory and applications (p. xii). Chichester, Wiley. 1988.

R. Hettich, K.O. Kortanek, Semi-infinite programming: Theory, methods and applications, SIAM Review, vol. 35, no. 3, pp. 380-429, 1993.

A. D.Ioffe, Approximate Subdifferentials and Applications, II, Mathematika. vol. 33, no. 1, pp. 111-128, 1986.

V. Jeyakumar, D.T. Luc, Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators, Journal of Optimization Theory and Applications, vol. 101, no. 3, pp. 599-621, 1999.

B.C. Joshi, S.K. Mishra, and P. Kumar, On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity, Journal of the Operations Research Society of China, vol. 8, no. 4, pp. 619-636, 2020.

B.C. Joshi, Higher order duality in multiobjective fractional programming problem with generalized convexity. Yugoslav Journal of Operations Research, vol. 27, no. 2, pp. 249-264, 2017.

B.C. Joshi and S. K. Mishra, On nonsmooth mathematical programs with equilibrium constraints using generalized convexity, Yugoslav Journal of Operations Research, vol. 29, no. 4, pp. 449-463, 2019.

B.C. Joshi, Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ > 0, RAIRO - Operations Research, vol. 55, no. S2221-S2240, 2021.

B.C. Joshi, R. Mohan, and Pankaj, Generalized invexity and mathematical programs, Yugoslav Journal of Operations Research, vol. 31, no. 4, pp. 455-469, 2020.

B.C. Joshi, Generalized convexity and mathematical programs, Scientific Bulletin-University Politehnica of Bucharest, Series A, vol. 82, no. 4, pp. 151-160, 2020.

B.C. Joshi, On generalized approximate convex functions and variational inequalities, RAIRO-Operations Research, vol 55, pp. S2999-S3008, 2021.

B.C. Joshi, Pankaj, and S. K. Mishra, On nonlinear complementarity problems with applications, Journal of Information and Optimization Sciences, vol. 42, no. 1, pp. 155-171, 2021.

A. Kabgani, M. Soleimani-damaneh, and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators, Mathematical Methods of Operations Research, vol. 86, no. 1, pp. 103-121, 2017.

A. Kabgani, and M. Soleimani-damaneh, Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators, Optimization, vol. 67, no. 2, pp. 217-235, 2018.

O. Kostyukova and T. V. Tchemisova, CQ-free optimality conditions and strong dual formulations for a special conic optimization problem, Statistics, Optimization & Information Computing, vol. 8, no. 3, 668-683, 2020.

V. Laha and S.K. Mishra, On vector optimization problems and vector variational inequalities using convexificators, Optimization, vol. 66. no. 11. pp. 1837-1850, 2017.

M.A. Lo´pez and G. Still, Semi-infinite programming, European Journal of Operational Research, vol. 180, no. 1, pp. 491-518, 2007.

Z.Q. Luo, J.S. Pang, and D. Ralph, Mathematical programs with equilibrium constraints, Cambridge: Cambridge University Press, 1996.

P. Michel and J.P. Penot, A Generalized Derivative for Calm and Stable Functions, Differential and Integral Equations, vol. 5, no. 2, pp. 433-454, 1992.

S.K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex Optimization and Its Applications, Springer, Berlin, 2008.

S.K. Mishra, V. Singh, and V. Laha, On duality for mathematical programs with vanishing constraints, Annals of Operations Research, vol. 243, no. 1, pp. 249-272, 2016.

S.K. Mishra and M. Jaiswal, Optimality conditions and duality for semi-infinite mathematical programming problem with equilibrium constraints, Numerical Functional Analysis and Optimization, vol. 36, no. 4, pp. 460-480, 2015

S.K. Mishra, M. Jaiswal, and L.T.H. An, Duality for nonsmooth semi-infinite programming problems, Optimization Letters, vol. 6, no. 2, pp. 261-271, 2012.

B. Mond and T. Weir, Generalized concavity and duality, generalized concavity in optimization and economics, New York: Academic Press 1981.

Y. Pandey and S.K. Mishra, Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators, Annals of Operations Research, vol. 269, no. 1, pp. 549-564, 2018.

Y. Pandey and S. K. Mishra, On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints, Operations Research Letters, vol. 44, no. 1, pp. 148-151. 2016.

Y. Pandey and S. K. Mishra, Duality for nonsmooth mathematical programming problems with equilibrium constraints using convexificators, Journal of Optimization Theory and Applications, vol. 171, no. 1, pp. 694-707, 2016.

E. Polak, On the mathematical foundation of nondifferentiable optimization in engineering design, SIAM Review, vol. 29, no. 1, pp. 21-89, 1987.

A.U. Raghunathan and L.T. Biegler, Mathematical programs with equilibrium constraints (MPECs) in process engineering, Computers & Chemical Engineering vol. 27, no. 10, pp. 1381-1392, 2003.

S. Sharma and P. Yadav, Nonsmooth Vector Optimization Problem Involving Second-Order Semipseudo, Semiquasi Cone-Convex Functions, Statistics, Optimization & Information Computing, vol. 9, no. 2, pp. 383-398, 2021.

S. K. Suneja, S. Sharma, and M. Kapoor, Second-order optimality and duality in vector optimization over cones, Statistics, Optimization & Information Computing, vol. 4, no. 2, pp. 163-173, 2016.

S. Suh and T.J. Kim, Solving nonlinear bilevel programming models of the equilibrium network design problem: A comparative review, Annals of Operations Research, vol. 34, no. 1, pp. 203-218, 1992.

S.K. Suneja and B. Kohli, Optimality and duality results for bilevel programming problem using convexifactors, Journal of Optimization Theory and Applications, vol. 150, no. 1, pp. 1-19, 2011.

J.S. Treiman, The Linear Nonconvex Generalized Gradient and Lagrange Multipliers, SIAM Journal on Optimization, vol. 5, no. 3, pp. 670-680, 1995.

P. Wolfe, A duality theorem for nonlinear programming, Quarterly of Applied Mathematics, vol. 19, no. 3, pp. 239-244, 1961.

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*11*(3), 640-654. https://doi.org/10.19139/soic-2310-5070-881

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