Γ-robust linear complementarity problems

References

• A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton.
   Google Scholar
• A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, Adjustable robust solutions of uncertain linear programs, Math. Program. 99 (2004), pp. 351–376. doi: 10.1007/s10107-003-0454-y
   Web of Science ®Google Scholar
• D. Bertsimas, D.B. Brown, and C. Caramanis, Theory and applications of robust optimization, SIAM Rev. 53 (2011), pp. 464–501. doi: 10.1137/080734510
   Web of Science ®Google Scholar
• D. Bertsimas and M. Sim, Robust discrete optimization and network flows, Math. Program. 98 (2003), pp. 49–71. doi: 10.1007/s10107-003-0396-4
   Web of Science ®Google Scholar
• D. Bertsimas and M. Sim, The price of robustness, Oper. Res. 52 (2004), pp. 35–53. doi: 10.1287/opre.1030.0065
   Web of Science ®Google Scholar
• J.R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer Science & Business Media, New York.
   Google Scholar
• E. Çelebi, V. Krebs, and M. Schmidt, Γ-robust electricity market equilibrium models with transmission and generation investments, Tech. rep., 2020. Available at http://www.optimization-online.org/DB_HTML/2020/01/7586.html.
   Google Scholar
• X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res. 30 (2005), pp. 1022–1038. doi: 10.1287/moor.1050.0160
   Web of Science ®Google Scholar
• X. Chen, R.J.B. Wets, and Y. Zhang, Stochastic variational inequalities: Residual minimization smoothing sample average approximations, SIAM J. Optim. 22 (2013), pp. 649–673. doi: 10.1137/110825248
   Web of Science ®Google Scholar
• X. Chen, C. Zhang, and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Math. Program. 117 (2009), pp. 51–80. doi: 10.1007/s10107-007-0163-z
   Web of Science ®Google Scholar
• R.W. Cottle, J.S. Pang, and R.E. Stone, The Linear Complementarity Problem, Society for Industrial and Applied Mathematics, 2009.
   Google Scholar
• S. Dafermos, Traffic equilibrium and variational inequalities, Transp. Sci. 14 (1980), pp. 42–54. doi: 10.1287/trsc.14.1.42
   Web of Science ®Google Scholar
• M. Fischetti and M. Monaci, Light robustness, in Robust and Online Large-Scale Optimization: Models and Techniques for Transportation Systems, R.K. Ahuja, R.H. Möhring, and C.D. Zaroliagis, eds., Springer, Berlin, Heidelberg, 2009, pp. 61–84.
   Google Scholar
• M. Frank and P. Wolfe, An algorithm for quadratic programming, Nav. Res. Logist. Q. 3 (1956), pp. 95–110. doi: 10.1002/nav.3800030109
   Google Scholar
• S.A. Gabriel, Solving discretely constrained mixed complementarity problems using a median function, Optim. Eng. 18 (2017), pp. 631–658. doi: 10.1007/s11081-016-9341-2
   Web of Science ®Google Scholar
• S.A. Gabriel, A.J. Conejo, C. Ruiz, and S. Siddiqui, Solving discretely constrained, mixed linear complementarity problems with applications in energy, Comput. Oper. Res. 40 (2013), pp. 1339–1350. doi: 10.1016/j.cor.2012.10.017
   Web of Science ®Google Scholar
• S.A. Gabriel, S.A. Siddiqui, A.J. Conejo, and C. Ruiz, Solving discretely-constrained Nash–Cournot games with an application to power markets, Networks Spatial Econ. 13 (2013), pp. 307–326. doi: 10.1007/s11067-012-9182-2
   Web of Science ®Google Scholar
• J. Gao, Uncertain bimatrix game with applications, Fuzzy Optim. Decision Making 12 (2013), pp. 65–78. doi: 10.1007/s10700-012-9145-6
   Web of Science ®Google Scholar
• D. Huppmann and S. Siddiqui, An exact solution method for binary equilibrium problems with compensation and the power market uplift problem, Eur. J. Oper. Res. 266 (2018), pp. 622–638. doi: 10.1016/j.ejor.2017.09.032
   Web of Science ®Google Scholar
• P. Kall and S.W. Wallace, Stochastic Programming, Wiley-Interscience Series in Systems and Optimization, Wiley, New York, 1994.
   Google Scholar
• A. Kramer, V. Krebs, and M. Schmidt, Strictly and Γ-robust counterparts of electricity market models: Perfect competition and Nash–Cournot equilibria. Available at http://www.optimization-online.org/DB_HTML/2018/07/6709.html.
   Google Scholar
• C. Lemke and J. Howson Jr., Equilibrium points of bimatrix games, J. Soc. Ind. Appl. Math. 12 (1964), pp. 413–423. doi: 10.1137/0112033
   Google Scholar
• C. Liebchen, M. Lübbecke, R. Möhring, and S. Stiller, The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications, Springer, Berlin, Heidelberg, 2009, pp. 1–27.
   Google Scholar
• G.H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems, Optim. Methods Softw. 21 (2006), pp. 551–564. doi: 10.1080/10556780600627610
   Web of Science ®Google Scholar
• Z.Q. Luo and S. Zhang, On extensions of the Frank-Wolfe theorems, Comput. Optim. Appl. 13 (1999), pp. 87–110. doi: 10.1023/A:1008652705980
   Web of Science ®Google Scholar
• O.L. Mangasarian, A simple characterization of solution sets of convex programs, Oper. Res. Lett. 7 (1988), pp. 21–26. doi: 10.1016/0167-6377(88)90047-8
   Web of Science ®Google Scholar
• J. Mather and E. Munsing, Robust Cournot-Bertrand equilibria on power networks, in American Control Conference (ACC), Seattle, 2017, pp. 2747–2754.
   Google Scholar
• J.F. Nash, Equilibrium points in n-person games, Proc. Natl. Acad. Sci. U.S.A. 36 (1950), pp. 48–49. Available at http://www.jstor.org/stable/88031. doi: 10.1073/pnas.36.1.48
   PubMed Web of Science ®Google Scholar
• J.F. Nash, Non-cooperative games, Ann. Math. 54 (1951), pp. 286–295. doi: 10.2307/1969529
   Web of Science ®Google Scholar
• R. Reemtsen and J.J. Rückmann, Semi-infinite Programming, Vol. 25, Springer Science & Business Media, 1998.
   Google Scholar
• S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim. 11 (2001), pp. 918–936. doi: 10.1137/S1052623499361233
   Web of Science ®Google Scholar
• M. Sim, Robust optimization, PhD thesis, Massachusetts Institute of Technology, 2004.
   Google Scholar
• A.L. Soyster, Technical note–convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res. 21 (1973), pp. 1154–1157. doi: 10.1287/opre.21.5.1154
   Web of Science ®Google Scholar
• D. Wu, J.Y. Han, and J.H. Zhu, Robust solutions to uncertain linear complementarity problems, Acta Mathematicae Applicatae Sinica 27 (2011), pp. 339–352. doi: 10.1007/s10255-010-0033-y
   Web of Science ®Google Scholar
• Y. Xie and U.V. Shanbhag, On robust solutions to uncertain linear complementarity problems and their variants, SIAM J. Optim. 26 (2016), pp. 2120–2159. doi: 10.1137/15M1010427
   Web of Science ®Google Scholar
• Y. Xie and U.V. Shanbhag, On robust solutions to uncertain monotone linear complementarity problems (LCPs) and their variants, in 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), Los Angeles, pp. 2834–2839.
   Google Scholar