Robust Linear Semi-infinite Optimization

  • Miguel A. Goberna
  • Marco A. López
Chapter
Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)

Abstract

For many finite optimization problems, numerical methods can be compared from the complexity point of view, i.e., computing upper bounds on the number of iterations, arithmetic operations, etc., necessary to get an optimal solution, or an \(\varepsilon\)-optimal solution, in terms of the size of the problem. This methodology can hardly be applied in LSIO because it is not evident how to define the size of the triplet \(\left (a,b,c\right )\) representing the data of a problem like ( 1.1) despite the seminal results on the complexity of the interior point constraint generation algorithm in [182, 192]. On the other hand, the robust counterpart of an uncertain LSIO problem seldom enjoys the strong assumptions which are necessary to apply reduction or feasible point methods.

References

  1. 1.
    Adler, I., Monteiro, R.: A geometric view of parametric linear programming. Algorithmica 8, 161–176 (1992)MATHMathSciNetGoogle Scholar
  2. 2.
    Altinel, I.K., Çekyay, B.Ç., Feyzioğlu, O., Keskin, M.E., Özekici, S.: Mission-based component testing for series systems. Ann. Oper. Res. 186, 1–22 (2011)MATHMathSciNetGoogle Scholar
  3. 3.
    Amaya, J., Bosch, P., Goberna, M.A.: Stability of the feasible set mapping of linear systems with an exact constraint set. Set-Valued Anal. 16, 621–635 (2008)MATHMathSciNetGoogle Scholar
  4. 4.
    Amaya, J., Goberna, M.A.: Stability of the feasible set of linear systems with an exact constraints set. Math. Methods Oper. Res. 63, 107–121 (2006)MATHMathSciNetGoogle Scholar
  5. 5.
    Anderson, E.J., Lewis, A.S.: An extension of the simplex algorithm for semi-infinite linear programming. Math. Program. A 44, 247–269 (1989)MATHMathSciNetGoogle Scholar
  6. 6.
    Anderson, E.J., Goberna, M.A., López, M.A.: Simplex-like trajectories on quasi-polyhedral convex sets. Math. Oper. Res. 26, 147–162 (2001)MATHMathSciNetGoogle Scholar
  7. 7.
    Auslender, A., Ferrer, A., Goberna, M.A., López, M.A.: Comparative study of RPSALG algorithm for convex semi-infinite programming. Departamento de Estadística e Investigación Operativa, Universidad de Alicante, Spain. PreprintGoogle Scholar
  8. 8.
    Auslender, A., Goberna, M.A., López, M.A.: Penalty and smoothing methods for convex semi-infinite programming. Math. Oper. Res. 34, 303–319 (2009)MATHMathSciNetGoogle Scholar
  9. 9.
    Azé, D., Corvellec, J.-N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)MATHMathSciNetGoogle Scholar
  10. 10.
    Balayadi, A., Sonntag, Y., Zălinescu, C.: Stability of constrained optimization problems. Nonlinear Anal. 28, 1395–1409 (1997)MATHMathSciNetGoogle Scholar
  11. 11.
    Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Birkhäuser, Basel (1983)MATHGoogle Scholar
  12. 12.
    Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)MATHMathSciNetGoogle Scholar
  13. 13.
    Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970)MathSciNetGoogle Scholar
  14. 14.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  15. 15.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. A 88, 411–424 (2000)MATHMathSciNetGoogle Scholar
  16. 16.
    Ben-Tal, A., Nemirovski, A.: Robust optimization—methodology and applications. Math. Program. B 92, 453–480 (2002)MATHMathSciNetGoogle Scholar
  17. 17.
    Ben-Tal, A., Teboulle, M.: Expected utility, penalty functions and duality in stochastic nonlinear programming. Manag. Sci. 30, 1445–1466 (1986)MathSciNetGoogle Scholar
  18. 18.
    Bennett, K.P., Parrado-Hernández, E.: The interplay of optimization and machine learning research. J. Mach. Learn. Res. 7, 1265–1281 (2006)MATHMathSciNetGoogle Scholar
  19. 19.
    Berkelaar, A., Roos, C., Terlaky, T.: The optimal set and optimal partition approach to linear and quadratic programming. In: Gal, T., Greenberg, H. (eds.) Recent Advances in Sensitivity Analysis and Parametric Programming, pp.1–44. Kluwer, Dordrecht (1997)Google Scholar
  20. 20.
    Bertsimas, D., Brown, D.B., Caramanis, C: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011)MATHMathSciNetGoogle Scholar
  21. 21.
    Betró, B.: An accelerated central cutting plane algorithm for linear semi-infinite programming. Math. Program. A 101, 479–495 (2004)MATHGoogle Scholar
  22. 22.
    Betrò, B.: Numerical treatment of Bayesian robustness problems. Int. J. Approx. Reason. 50, 279–288 (2009)MATHGoogle Scholar
  23. 23.
    Betrò, B., Bodini, A.: Generalized moment theory and Bayesian robustness analysis for hierarchical mixture models. Ann. Inst. Stat. Math. 58, 721–738 (2006)MATHGoogle Scholar
  24. 24.
    Bhattacharjee, B., Green, W.H., Jr., Barton, P.I.: Interval methods for semiinfinite programs. Comput. Optim. Appl. 30, 63–93 (2005)MATHMathSciNetGoogle Scholar
  25. 25.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)MATHGoogle Scholar
  26. 26.
    Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)MATHGoogle Scholar
  27. 27.
    Boţ, R.I., Jeyakumar, V., Li, G.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)MathSciNetGoogle Scholar
  28. 28.
    Box, E.P.: Robustness in the strategy of scientific model building. In: Launer, R.L., Wilkinson, G.N. (eds.) Robustness in Statistics, pp. 201–236. Academic, New York (1979)Google Scholar
  29. 29.
    Brosowski, B.: Parametric Semi-infinite Optimization. Peter Lang, Frankfurt am Main (1982)MATHGoogle Scholar
  30. 30.
    Brosowski, B.: On the continuity of the optimum set in parametric semi-infinite programming. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations, pp. 23–48, Marcel Dekker, New York (1983)Google Scholar
  31. 31.
    Brosowski, B.: Parametric semi-infinite linear programming I. Continuity of the feasible set and the optimal value. Math. Program. Study 21, 18–42 (1984)MATHMathSciNetGoogle Scholar
  32. 32.
    Cadenas, J.M., Verdegay, J.L.: A primer on fuzzy optimization models and methods. Iran. J. Fuzzy Syst. 3, 1–22 (2006)MATHMathSciNetGoogle Scholar
  33. 33.
    Campi, M.C., Garatti, S.: The exact feasibility of randomized solutions of uncertain convex programs. SIAM J. Optim. 19, 1211–1230 (2008)MATHMathSciNetGoogle Scholar
  34. 34.
    Cánovas, M.J., Dontchev, A.L., López, M.A., Parra, J.: Metric regularity of semi-infinite constraint systems. Math. Program. B 104, 329–346 (2005)MATHGoogle Scholar
  35. 35.
    Cánovas, M.J., Dontchev, A.L., López, M.A., Parra, J.: Isolated calmness of solution mappings in convex semi-infinite optimization. J. Math. Anal. Appl. 350, 892–837 (2009)Google Scholar
  36. 36.
    Cánovas, M.J., Gómez-Senent, F.J., Parra, J.: Stability of systems of linear equations and inequalities: distance to ill-posedness and metric regularity. Optimization 56, 1–24 (2007)MATHMathSciNetGoogle Scholar
  37. 37.
    Cánovas, M.J., Gómez-Senent, F.J., Parra, J.: On the Lipschitz modulus of the argmin mapping in linear semi-infinite optimization. Set-Valued Anal. 16, 511–538 (2008)MATHMathSciNetGoogle Scholar
  38. 38.
    Cánovas, M.J., Gómez-Senent, F.J., Parra, J.: Regularity modulus of arbitrarily perturbed linear inequality systems. J. Math. Anal. Appl. 343, 315–327 (2008)MATHMathSciNetGoogle Scholar
  39. 39.
    Cánovas, M.J., Hantoute, A., López, M.A., Parra, J.: Lipschitz modulus of the optimal set mapping in convex optimization via minimal subproblems. Pac. J. Optim. 4, 411–422 (2008)MATHMathSciNetGoogle Scholar
  40. 40.
    Cánovas, M.J., Hantoute, A., López, M.A., Parra, J.: Stability of indices in KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl. 139, 485–500 (2008)MathSciNetGoogle Scholar
  41. 41.
    Cánovas, M.J., Hantoute, A., López, M.A., Parra, J.: Lipschitz modulus in convex semi-infinite optimization via d.c. functions. ESAIM Control Optim. Calc. Var. 15, 763–781 (2009)Google Scholar
  42. 42.
    Cánovas, M.J., Hantoute, A., Parra, J., Toledo, J.: Calmness of the argmin mapping in linear semi-infinite optimization. J. Optim. Theory Appl. doi:10.1007/s10957-013-0371-z (in press)Google Scholar
  43. 43.
    Cánovas, M.J., Klatte, D., López, M.A., Parra, J.: Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18, 717–732 (2007)MATHMathSciNetGoogle Scholar
  44. 44.
    Cánovas, M.J., Kruger, A.Y., López, M.A., Parra, J., Théra, M.: Calmness modulus of linear semi-infinite programs. SIAM J. Optim., in pressGoogle Scholar
  45. 45.
    Cánovas, M.J., López, M.A., Parra, J.: Upper semicontinuity of the feasible set mapping for linear inequality systems. Set-Valued Anal. 10, 361–378 (2002)MATHMathSciNetGoogle Scholar
  46. 46.
    Cánovas, M.J., López, M.A., Parra, J.: Stability of linear inequality systems in a parametric setting. J. Optim. Theory Appl. 125, 275–297 (2005)MATHMathSciNetGoogle Scholar
  47. 47.
    Cánovas, M.J., López, M.A., Parra, J.: On the continuity of the optimal value in parametric linear optimization: stable discretization of the Lagrangian dual of nonlinear problems. Set-Valued Anal. 13, 69–84 (2005)MATHMathSciNetGoogle Scholar
  48. 48.
    Cánovas, M.J., López, M.A., Parra, J., Todorov, M.I.: Stability and well-posedness in linear semi-infinite programming. SIAM J. Optim. 10, 82–89 (1999)MATHMathSciNetGoogle Scholar
  49. 49.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math. Program. A 103, 95–126 (2005)MATHGoogle Scholar
  50. 50.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to solvability/unsolvability in linear optimization. SIAM J. Optim. 16, 629–649 (2006)MATHMathSciNetGoogle Scholar
  51. 51.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Ill-posedness with respect to the solvability in linear optimization. Linear Algebra Appl. 416, 520–540 (2006)MATHMathSciNetGoogle Scholar
  52. 52.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to ill-posedness in linear optimization via the Fenchel-Legendre conjugate. J. Optim. Theory Appl. 130, 173–183 (2006)MATHMathSciNetGoogle Scholar
  53. 53.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Lipschitz continuity of the optimal value via bounds on the optimal set in linear semi-infinite optimization. Math. Oper. Res. 31, 478–489 (2006)MATHMathSciNetGoogle Scholar
  54. 54.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Sufficient conditions for total ill-posedness in linear semi-infinite optimization. Eur. J. Oper. Res. 181, 1126–1136 (2007)MATHGoogle Scholar
  55. 55.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Error bounds for the inverse feasible set mapping in linear semi-infinite optimization via a sensitivity dual approach. Optimization 56, 547–563 (2007)MATHMathSciNetGoogle Scholar
  56. 56.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to ill-posedness for linear inequality systems under block perturbations: convex and infinite-dimensional cases. Optimization 60, 925–946 (2011)MATHMathSciNetGoogle Scholar
  57. 57.
    Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Calmness of the feasible set mapping for linear inequality systems. Centro de Investigación Operativa, Universidad Miguel Hernández de Elche. PreprintGoogle Scholar
  58. 58.
    Cánovas, M.J., Mordukhovich, B., López, M.A., Parra, J.: Variational analysis in semi-infinite and infinite programming, I: stability of linear inequality systems of feasible solutions. SIAM J. Optim. 20, 1504–1526 (2009)MATHGoogle Scholar
  59. 59.
    Charnes, A., Cooper, W.W., Kortanek, K.O.: Duality, Haar programs, and finite sequence spaces. Proc. Natl. Acad. Sci. USA 48, 783–786 (1962)MATHMathSciNetGoogle Scholar
  60. 60.
    Charnes, A., Cooper, W.W., Kortanek, K.O.: Duality in semi-infinite programs and some works of Haar and Carathéodory. Manag. Sci. 9, 209–228 (1963)MATHMathSciNetGoogle Scholar
  61. 61.
    Coelho, C.J., Galvao, R.K.H., de Araujo, M.C.U., Pimentel, M.F., da Silva, E.C.: A linear semi-infinite programming strategy for constructing optimal wavelet transforms in multivariate calibration problems. J. Chem. Inform. Comput. Sci. 43, 928–933 (2003)Google Scholar
  62. 62.
    da Silva, A.R.: On parametric infinite optimization. Int. Ser. Numer. Math. 72, 83–95 (1984)Google Scholar
  63. 63.
    Daniel, J.W.: Remarks on perturbations in linear inequalities. SIAM J. Numer. Anal. 12, 770–772 (1975)MATHMathSciNetGoogle Scholar
  64. 64.
    Daniilidis, A., Goberna, M.A., López, M.A., Lucchetti, R.: Lower semicontinuity of the solution set mapping of linear systems relative to their domains. Set-Valued Var. Anal. 21, 67–92 (2013)MATHMathSciNetGoogle Scholar
  65. 65.
    Daum, S., Werner, R.: A novel feasible discretization method for linear semi-infinite programming applied to basket options pricing. Optimization 60, 1379–1398 (2011)MATHMathSciNetGoogle Scholar
  66. 66.
    Davidson, M.R.: Stability of the extreme point set of a polyhedron. J. Optim. Theory Appl. 90, 357–380 (1996)MATHMathSciNetGoogle Scholar
  67. 67.
    Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)MATHMathSciNetGoogle Scholar
  68. 68.
    Dentcheva, D., Ruszczyński, A.: Semi-infinite probabilistic optimization: first-order stochastic dominance constraint. Optimization 53, 583–601 (2004)MATHMathSciNetGoogle Scholar
  69. 69.
    Dentcheva, D., Ruszczyński, A.: Portfolio optimization with stochastic dominance constraints. J. Bank. Finance 30, 433–451 (2006)Google Scholar
  70. 70.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)MATHGoogle Scholar
  71. 71.
    Dinh, N., Goberna, M.A., López, M.A.: From linear to convex systems: consistency, Farkas Lemma and applications. J. Convex Anal. 13, 279–290 (2006)Google Scholar
  72. 72.
    Dinh, N., Goberna, M.A., López, M.A., Son, T.Q.: New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim. Calc. Var. 13, 580–597 (2007)MATHMathSciNetGoogle Scholar
  73. 73.
    Dinh, N., Goberna, M.A., López, M.A.: On the stability of the optimal value and the optimal set in optimization problems. J. Convex Anal. 19, 927–953 (2012)MATHMathSciNetGoogle Scholar
  74. 74.
    Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Am. Math. Soc. 355, 493–517 (2003)MATHMathSciNetGoogle Scholar
  75. 75.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mapping. A View from Variational Analysis. Springer, New York (2009)Google Scholar
  76. 76.
    Dubois, D., Kerre, E., Mesiar, R., Prade, H.: Fuzzy interval analysis. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, pp. 483–581. Kluwer, Dordrecht (2000)Google Scholar
  77. 77.
    Dubois, D., Prade, H.: The mean value of a fuzzy number. Fuzzy Sets Syst. 24, 279–300 (1988)MathSciNetGoogle Scholar
  78. 78.
    Epelman, M., Freund, R.M.: Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system. Math. Program. A 88, 451–485 (2000)MATHMathSciNetGoogle Scholar
  79. 79.
    Fang, S.C., Hu, C.F., Wang, H.F., Wu, S.Y.: Linear programming with fuzzy coefficients in constraints. Comput. Math. Appl. 37, 63–76 (1999)MATHMathSciNetGoogle Scholar
  80. 80.
    Feyzioglu, O., Altinel, I.K., Ozekici, S.: The design of optimum component test plans for system reliability. Comput. Stat. Data Anal. 50, 3099–3112 (2006)MATHMathSciNetGoogle Scholar
  81. 81.
    Feyzioglu, O., Altinel, I.K., Ozekici, S.: Optimum component test plans for phased-mission systems. Eur. J. Oper. Res. 185, 255–265 (2008)MATHMathSciNetGoogle Scholar
  82. 82.
    Fischer, T.: Contributions to semi-infinite linear optimization. Meth. Verf. Math. Phys. 27, 175–199 (1983)Google Scholar
  83. 83.
    Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM Optim. 18, 1187–1208 (2007)MATHMathSciNetGoogle Scholar
  84. 84.
    Freund, R.M., Vera, J.R.: Some characterizations and properties of the “distance to ill-posedness”. Math. Program. A 86, 225–260 (1999)MATHMathSciNetGoogle Scholar
  85. 85.
    Gal, T.: Postoptimal Analyses, Parametric Programming, and Related Topics: Degeneracy, Multicriteria Decision Making, Redundancy, 2nd edn. Walter de Gruyter, New York (1995)Google Scholar
  86. 86.
    Gayá, V.E., López, M.A., Vera de Serio, V.N.: Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems. Optimization 52, 693–713 (2003)MATHMathSciNetGoogle Scholar
  87. 87.
    Gauvin, J.: Formulae for the sensitivity analysis of linear programming problems. In: Lassonde, M. (ed.) Approximation, Optimization and Mathematical Economics, pp. 117–120. Physica-Verlag, Berlin (2001)Google Scholar
  88. 88.
    Ghaffari Hadigheh, A., Terlaky, T.: Sensitivity analysis in linear optimization: invariant support set intervals. Eur. J. Oper. Res. 169, 1158–1175 (2006)MATHMathSciNetGoogle Scholar
  89. 89.
    Ghaffari Hadigheh, A., Romanko, O., Terlaky, T.: Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors. Algorithmic Oper. Res. 2, 94–111 (2007)MATHMathSciNetGoogle Scholar
  90. 90.
    Glashoff, K., Gustafson, S.-A.: Linear Optimization and Approximation. Springer, Berlin (1983)MATHGoogle Scholar
  91. 91.
    Goberna, M.A.: Linear semi-infinite optimization: recent advances. In: Rubinov, A., Jeyakumar, V. (eds.) Continuous Optimization: Current Trends and Applications, pp. 3–22. Springer, Berlin (2005)Google Scholar
  92. 92.
    Goberna, M.A.: Post-optimal analysis of linear semi-infinite programs. In: Chinchuluun, A., Pardalos, P.M., Enkhbat, R., Tseveendorj, I. (eds.) Optimization and Optimal Control: Theory and Applications, pp. 23–54. Springer, Berlin (2010)Google Scholar
  93. 93.
    Goberna, M.A., Gómez, S., Guerra, F., Todorov, M.I.: Sensitivity analysis in linear semi-infinite programming: perturbing cost and right-hand-side coefficients. Eur. J. Oper. Res. 181, 1069–1085 (2007)MATHGoogle Scholar
  94. 94.
    Goberna, M.A., Jeyakumar, V., Dinh, N.: Dual characterizations of set containments with strict convex inequalities. J. Global Optim. 34, 33–54 (2006)MATHMathSciNetGoogle Scholar
  95. 95.
    Goberna, M.A., Jeyakumar, V., Li, G., López, M.A.: Robust linear semi-infinite programming duality under uncertainty. Math. Program. B 139, 185–203 (2013)MATHGoogle Scholar
  96. 96.
    Goberna, M.A., Jeyakumar, V., Li, G.Y., Vicente-Pérez, J.: Robust solutions of uncertain multi-objective linear semi-infinite programming. School of Mathematics, UNSW, Sydney. PreprintGoogle Scholar
  97. 97.
    Goberna, M.A., Jeyakumar, V., Li, G.Y., Vicente-Pérez, J.: Robust solutions to multi-objective linear programs with uncertain data. School of Mathematics, UNSW, Sydney. PreprintGoogle Scholar
  98. 98.
    Goberna, M.A., Jornet, V.: Geometric fundamentals of the simplex method in semi-infinite programming. OR Spektrum 10, 145–152 (1988)MATHMathSciNetGoogle Scholar
  99. 99.
    Goberna, M.A., Larriqueta, M., Vera de Serio, V.: On the stability of the boundary of the feasible set in linear optimization. Set-Valued Anal. 11, 203–223 (2003)Google Scholar
  100. 100.
    Goberna, M.A., Larriqueta, M., Vera de Serio, V.: On the stability of the extreme point set in linear optimization. SIAM J. Optim. 15, 1155–1169 (2005)Google Scholar
  101. 101.
    Goberna, M.A., López, M.A.: Topological stability of linear semi-infinite inequality systems. J. Optim. Theory Appl. 89, 227–236 (1996)MATHMathSciNetGoogle Scholar
  102. 102.
    Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, Chichester (1998)MATHGoogle Scholar
  103. 103.
    Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17, 730–743 (1996)MATHMathSciNetGoogle Scholar
  104. 104.
    Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. II: upper semicontinuity of the solution set mapping. SIAM J. Optim. 7, 1138–1151 (1997)MATHGoogle Scholar
  105. 105.
    Goberna, M.A., López, M.A., Todorov, M.I.: On the stability of the feasible set in linear optimization. Set-Valued Anal. 9, 75–99 (2001)MATHMathSciNetGoogle Scholar
  106. 106.
    Goberna, M.A., López, M.A., Todorov, M.I.: On the stability of closed-convex-valued mappings and the associated boundaries. J. Math. Anal. Appl. 306, 502–515 (2005)MATHMathSciNetGoogle Scholar
  107. 107.
    Goberna, M.A., López, M.A., Volle, M.: Primal attainment in convex infinite optimization duality. J. Convex Anal. 21, in press (2014) (unknown DOI)Google Scholar
  108. 108.
    Goberna, M.A., Martínex-Legaz, J.E., Vera de Serio, V.N.: On the Voronoi mapping. Department of Statistics and Operations Research, University of Alicante, Spain. PreprintGoogle Scholar
  109. 109.
    Goberna, M.A., Rodríguez, M.M.L., Vera de Serio, V.N.: Voronoi cells of arbitrary sets via linear inequality systems. Linear Algebra Appl. 436, 2169–2186 (2012)Google Scholar
  110. 110.
    Goberna, M.A., Terlaky, T., Todorov, M.I.: Sensitivity analysis in linear semi-infinite programming via partitions. Math. Oper. Res. 35, 14–25 (2010)MATHMathSciNetGoogle Scholar
  111. 111.
    Goberna, M.A., Todorov M.I.: Primal, dual and primal-dual partitions in continuous linear optimization. Optimization 56, 617–628 (2007)MATHMathSciNetGoogle Scholar
  112. 112.
    Goberna, M.A., Todorov, M.I.: Generic primal-dual solvability in continuous linear semi-infinite programming. Optimization, 57, 1–10 (2008)MathSciNetGoogle Scholar
  113. 113.
    Goberna, M.A., Todorov, M.I.: Primal-dual stability in continuous linear optimization. Math. Program. B 116, 129–146 (2009)MATHMathSciNetGoogle Scholar
  114. 114.
    Goberna, M.A., Todorov, M.I., Vera de Serio, V.N.: On the stability of the convex hull of set-valued mappings. SIAM J. Optim. 17, 147–158 (2006)Google Scholar
  115. 115.
    Goberna, M.A., Todorov, M.I., Vera de Serio, V.N.: On stable uniqueness in linear semi-infinite optimization. J. Global Optim. 53, 347–361 (2012)Google Scholar
  116. 116.
    Goberna, M.A., Vera de Serio, V.N.: On the stability of Voronoi cells. Top 20, 411–425 (2012)MATHMathSciNetGoogle Scholar
  117. 117.
    Goldfarb, D., Scheinberg, K.: On parametric semidefinite programming. Appl. Numer. Math. 29, 361–377 (1999)MATHMathSciNetGoogle Scholar
  118. 118.
    Gol’shtein, E.G.: Theory of Convex Programming. Translations of Mathematical Monographs, vol. 36. American Mathematical Society, Providence (1972)Google Scholar
  119. 119.
    Greenberg, H.: The use of the optimal partition in a linear programming solution for postoptimal analysis. Oper. Res. Lett. 15, 179–185 (1994)MATHMathSciNetGoogle Scholar
  120. 120.
    Greenberg, H.J.: Matrix sensitivity analysis from an interior solution of a linear program. INFORMS J. Comput. 11, 316–327 (1999)MATHMathSciNetGoogle Scholar
  121. 121.
    Greenberg, H.J.: Simultaneous primal-dual right-hand-side sensitivity analysis from a strict complementary solution of a linear program. SIAM J. Optim. 10, 427–442 (2000)MATHMathSciNetGoogle Scholar
  122. 122.
    Greenberg, H.J., Holder, A., Roos, C., Terlaky, T.: On the dimension of the set of rim perturbations for optimal partition invariance. SIAM J. Optim. 9, 207–216 (1998)MATHMathSciNetGoogle Scholar
  123. 123.
    Greenberg, H.J., Pierskalla, W.P.: Stability theory for infinitely constrained mathematical programs. J. Optim. Theory Appl. 16, 409–428 (1975)MATHMathSciNetGoogle Scholar
  124. 124.
    Guddat, J., Jongen, H.Th., Rückmann, J.-J.: On stability and stationary points in nonlinear optimization. J. Aust. Math. Soc. B 28, 36–56 (1986)MATHGoogle Scholar
  125. 125.
    Guo, P., Huang G.H.: Interval-parameter semi-infinite fuzzy-stochastic mixed-integer programming approach for environmental management under multiple uncertainties. Waste Manag. 30, 521–531 (2010)Google Scholar
  126. 126.
    Guo, P., Huang, G.H., He, L.: ISMISIP: an inexact stochastic mixed integer linear semi-infinite programming approach for solid waste management and planning under uncertainty. Stoch. Environ. Res. Risk Assess. 22, 759–775 (2008)MathSciNetGoogle Scholar
  127. 127.
    Hantoute, A., López, M.A.: Characterization of total ill-posedness in linear semi-infinite optimization. J. Comput. Appl. Math. 217, 350–364 (2008)MATHMathSciNetGoogle Scholar
  128. 128.
    Hantoute, A., López, M.A., Zălinescu, C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19, 863–882 (2008)MATHMathSciNetGoogle Scholar
  129. 129.
    He, L., Huang, G.H.: Optimization of regional waste management systems based on inexact semi-infinite programming. Can. J. Civil Eng. 35, 987–998 (2008)Google Scholar
  130. 130.
    He, L., Huang, G.H., Lu, H.: Bivariate interval semi-infinite programming with an application to environmental decision-making analysis. Eur. J. Oper. Res. 211, 452–465 (2011)MATHMathSciNetGoogle Scholar
  131. 131.
    Helbig, S.: Stability in disjunctive linear optimization I: continuity of the feasible set. Optimization 21, 855–869 (1990)MATHMathSciNetGoogle Scholar
  132. 132.
    Hettich, R., Zencke, P.: Numerische Methoden der Approximation und der Semi-Infiniten Optimierung. Teubner, Stuttgart (1982)MATHGoogle Scholar
  133. 133.
    Hirabayashi, R., Jongen, H.Th., Shida, M.: Stability for linearly constrained optimization problems. Math. Program. A 66, 351–360 (1994)MATHMathSciNetGoogle Scholar
  134. 134.
    Hiriart-Urrity, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I, II. Springer, New York (1993)Google Scholar
  135. 135.
    Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)Google Scholar
  136. 136.
    Hogan, W.W.: The continuity of the perturbation function of a convex program. Oper. Res. 21, 351–352 (1973)MATHMathSciNetGoogle Scholar
  137. 137.
    Homem-de-Mello, T., Mehrotra, S.: A cutting-surface method for uncertain linear programs with polyhedral stochastic dominance constraints. SIAM J. Optim. 20, 1250–1273 (2009)MATHMathSciNetGoogle Scholar
  138. 138.
    Hu, C.F., Fang, S.C.: A relaxed cutting plane algorithm for solving fuzzy inequality systems. Optimization 45, 89–106 (1999)MATHMathSciNetGoogle Scholar
  139. 139.
    Hu, H.: Perturbation analysis of global error bounds for systems of linear inequalities. Math. Program. B 88, 277–284 (2000)MATHGoogle Scholar
  140. 140.
    Hu, H., Wang, Q.: On approximate solutions of infinite systems of linear inequalities. Linear Algebra Appl. 114/115, 429–438 (1989)Google Scholar
  141. 141.
    Hu, J., Homem-de-Mello, T., Mehrotra, S.: Sample average approximation of stochastic dominance constrained programs. Math. Program. A 133, 171–201 (2012)MATHMathSciNetGoogle Scholar
  142. 142.
    Huang, G.H., He, L., Zeng, G.M., Lu, H.W.: Identification of optimal urban solid waste flow schemes under impacts of energy prices. Environ. Eng. Sci. 25, 685–695 (2008)Google Scholar
  143. 143.
    Huy, N.Q., Yao, J.-C.: Semi-infinite optimization under convex function perturbations: Lipschitz stability. J. Optim. Theory Appl. 148, 237–256 (2011)MATHMathSciNetGoogle Scholar
  144. 144.
    Ioffe, A.D.: On stability estimates for the regularity of maps. In: Brezis, H., Chang, K.C., Li, S.J., Rabinowitz, P. (eds.) Topological Methods, Variational Methods, and Their Applications, pp. 133–142. World Scientific, River Edge (2003)Google Scholar
  145. 145.
    Ioffe, A.D., Lucchetti, R.: Typical convex program is very well posed. Math. Program. B 104, 483–499 (2005)MATHMathSciNetGoogle Scholar
  146. 146.
    Jansen, B., de Jong, J.J., Roos, C., Terlaky, T.: Sensitivity analysis in linear programming: just be careful! Eur. J. Oper. Res. 101, 15–28 (1997)MATHGoogle Scholar
  147. 147.
    Jansen, B., Roos, C., Terlaky, T.: An interior point approach to postoptimal and parametric analysis in linear programming. Technical Report, Eotvös University, Budapest, Hungary (1992)Google Scholar
  148. 148.
    Jansen, B., Roos, C., Terlaky, T., Vial, J.-Ph.: Interior-point methodology for linear programming: duality, sensitivity analysis and computational aspects. Technical Report 93-28, Delft University of Technology, Faculty of Technical Mathematics and Computer Science, Delft (1993)Google Scholar
  149. 149.
    Jaume, D., Puente, R.: Representability of convex sets by analytical linear inequality systems. Linear Algebra Appl. 380, 135–150 (2004)MATHMathSciNetGoogle Scholar
  150. 150.
    Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)MATHMathSciNetGoogle Scholar
  151. 151.
    Jongen, H.Th., Rückmann, J.-J.: On stability and deformation in semi-infinite optimization. In: Reemtsen, R., Rückmann, J.J. (eds.) Semi-infinite Programming, pp. 29–67. Kluwer, Boston (1998)Google Scholar
  152. 152.
    Jongen, H.Th., Twilt, F., Weber, G.-H.: Semi-infinite optimization: structure and stability of the feasible set. J. Optim. Theory Appl. 72, 529–552 (1992)MATHMathSciNetGoogle Scholar
  153. 153.
    Jongen, H.Th., Weber, G.-H.: Nonlinear optimization: characterization of structural stability. J. Global Optim. 1, 47–64 (1991)MATHMathSciNetGoogle Scholar
  154. 154.
    Juárez, E.L., Todorov, M.I.: Characterization of the feasible set mapping in one class of semi-infinite optimization problems. Top 12, 135–147 (2004)MATHMathSciNetGoogle Scholar
  155. 155.
    Karimi, A., Galdos, G.: Fixed-order H controller design for nonparametric models by convex optimization. Automatica 46, 1388–1394 (2010)MATHMathSciNetGoogle Scholar
  156. 156.
    Klatte, D., Henrion, R.: Regularity and stability in nonlinear semi-infinite optimization. In: Reemtsen, R., Rückmann, J.J. (eds.) Semi-infinite Programming, pp. 69–102. Kluwer, Boston (1998)Google Scholar
  157. 157.
    Klatte, D., Kummer, B.: Stability properties of infima and optimal solutions of parametric optimization problems. In: Demyanov, V.F., Pallaschke, D. (eds.) Nondifferentiable Optimization: Motivations and Applications, pp. 215–229. Springer, Berlin (1985)Google Scholar
  158. 158.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer, Dordrecht (2002)MATHGoogle Scholar
  159. 159.
    Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. B 117, 305–330 (2009)MATHMathSciNetGoogle Scholar
  160. 160.
    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic, New York (1980)Google Scholar
  161. 161.
    Kortanek, K.O.: Constructing a perfect duality in infinite programming. Appl. Math. Optim. 3, 357–372 (1976/1977)Google Scholar
  162. 162.
    Kortanek, K.O., Medvedev, V.G.: Building and Using Dynamic Interest Rate Models. Wiley, Chichester (2001)MATHGoogle Scholar
  163. 163.
    Krabs, W.: Optimization and Approximation. Wiley, New York (1979)MATHGoogle Scholar
  164. 164.
    Krishnan, K., Mitchel, J.E.: A semidefinite programming based polyhedral cut and price approach for the maxcut problem. Comput. Optim. Appl. 33, 51–71 (2006)MATHMathSciNetGoogle Scholar
  165. 165.
    Kruger, A., Ngai, H.V., Thera, M.: Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim. 20, 3280–3296 (2010)MATHMathSciNetGoogle Scholar
  166. 166.
    Larriqueta, M., Vera de Serio, V.N.: On metric regularity and the boundary of the feasible set in linear optimization. Set-Valued Var. Anal. doi:10.1007/s11228-013-0241-8 (in press)Google Scholar
  167. 167.
    Leibfritz, F., Maruhn, J.H.: A successive SDP-NSDP approach to a robust optimization problem in finance. Comput. Optim. Appl. 44, 443–466 (2009)MATHMathSciNetGoogle Scholar
  168. 168.
    León, T., Liern, V., Marco, P., Segura, J.V., Vercher, E.: A downside risk approach for the portfolio selection problem with fuzzy returns. Fuzzy Econ. Rev. 9, 61–77 (2008)Google Scholar
  169. 169.
    León, T., Sanmatías, S., Vercher, E.: On the numerical treatment of linearly constrained semi-infinite optimization problems. Eur. J. Oper. Res. 121, 78–91 (2000)MATHGoogle Scholar
  170. 170.
    León, T., Vercher, E.: Optimization under uncertainty and linear semi-infinite programming: a survey. In: Goberna, M.A., López, M.A. (eds.) Semi-infinite Programming: Recent Advances, pp. 327–348. Kluwer, Dordrecht (2001)Google Scholar
  171. 171.
    León, T., Vercher, E.: Solving a class of fuzzy linear programs by using semi-infinite programming techniques. Fuzzy Sets Syst. 6, 235–252 (2004)Google Scholar
  172. 172.
    Levy, A.B., Poliquin, R.A.: Characterizing the single-valuedness of multifunctions. Set-Valued Anal. 5, 351–364 (1997)MATHMathSciNetGoogle Scholar
  173. 173.
    Li, H., Huang, G.H., Lu, H.: Bivariate interval semi-infinite programming with an application to environmental decision-making analysis. Eur. J. Oper. Res. 211, 452–465 (2011)MATHGoogle Scholar
  174. 174.
    Li, W.: The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. SIAM J. Optim. 187, 15–40 (1993)MATHGoogle Scholar
  175. 175.
    Li, C., Ng, K.F.: On constraint qualification for an infinite system of convex inequalities in a Banach Space. SIAM J. Optim. 15, 488–512 (2005)MATHMathSciNetGoogle Scholar
  176. 176.
    López, M.A.: Stability in linear optimization and related topics. A personal tour. Top 20, 217–244 (2012)MATHGoogle Scholar
  177. 177.
    López, M.A, Mira, J.A., Torregrosa, G.: On the stability of infinite-dimensional linear inequality systems. Numer. Funct. Anal. Optim. 19, 1065–1077 (1985–1986)Google Scholar
  178. 178.
    López, M.A., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MATHGoogle Scholar
  179. 179.
    López, M.A., Vera de Serio, V.: Stability of the feasible set mapping in convex semi-infinite programming. In: Goberna, M.A., López, M.A. (eds.) Semi-infinite Programming. Recent Advances, pp. 101–120. Kluwer, Dordrecht (2001)Google Scholar
  180. 180.
    Lucchetti, R.: Convexity and Well-Posed Problems. Springer, New York (2006)MATHGoogle Scholar
  181. 181.
    Lucchetti, R., Viossat, Y.: Stable correlated equilibria: the zero-sum case. Milano Politecnico, 2011. PreprintGoogle Scholar
  182. 182.
    Luo, Z.-Q., Roos, C., Terlaky, T.: Complexity analysis of a logarithmic barrier decomposition method for semi-infinite linear programming. Appl. Numer. Math. 29, 379–394 (1999)MATHMathSciNetGoogle Scholar
  183. 183.
    Luo, Z.-Q., Tseng, P.: Perturbation analysis of a condition number for linear systems. SIAM J. Matrix Anal. Appl. 15, 636–660 (1994)MATHMathSciNetGoogle Scholar
  184. 184.
    Mangasarian, O.L., Wild, E.W.: Nonlinear knowledge in kernel approximation. IEEE Trans. Neural Netw. 18, 300–306 (2007)Google Scholar
  185. 185.
    Mangasarian, O.L., Wild, E.W.: Nonlinear knowledge-based classification. IEEE Trans. Neural Netw. 19, 1826–1832 (2008)Google Scholar
  186. 186.
    Maruhn, J.H.: Robust Static Super-Replication of Barrier Options. De Gruyter, Berlin (2009)MATHGoogle Scholar
  187. 187.
    Mira, J.A., Mora, G.: Stability of linear inequality systems measured by the Hausdorff metric. Set-Valued Anal. 8, 253–266 (2000)MATHMathSciNetGoogle Scholar
  188. 188.
    Monteiro, R., Mehotra, S.: A generalized parametric analysis approach and its implication to sensitivity analysis in interior point methods. Math. Program. A 72, 65–82 (1996)MATHGoogle Scholar
  189. 189.
    Mordukhovich, B.S.: Coderivative analysis of variational systems. J. Global Optim. 28, 347–362 (2004)MATHMathSciNetGoogle Scholar
  190. 190.
    Norbedo, S., Zang, Z.Q., Claesson, I.: A semi-infinite quadratic programming algorithm with applications to array pattern synthesis. IEEE Trans. Circuits Syst. II Analog Digital Signal Process. 48, 225–232 (2001)Google Scholar
  191. 191.
    Ochoa, P.D., Vera de Serio, V.N.: Stability of the primal-dual partition in linear semi-infinite programming. Optimization 61, 1449–1465 (2012)MATHMathSciNetGoogle Scholar
  192. 192.
    Oskoorouchi, M.R., Ghaffari, H.R., Terlaky, T., Aleman, D.M.: An interior point constraint generation algorithm for semi-infinite optimization with health-care application. Oper. Res. 59, 1184–1197 (2011)MATHMathSciNetGoogle Scholar
  193. 193.
    Ozogur, S., Weber, G.W.: On numerical optimization theory of infinite kernel learning. J. Global Optim. 48, 215–239 (2010)MathSciNetGoogle Scholar
  194. 194.
    Ozogur, S., Weber, G.W.: Infinite kernel learning via infinite and semi-infinite programming. Optim. Methods Softw. 25, 937–970 (2010)MathSciNetGoogle Scholar
  195. 195.
    Parks, M.L., Jr., Soyster, A.L.: Semi-infinite and fuzzy set programming. In: Fiacco, A.V., Kortanek, K.O. (eds.) Semi-infinite Programming and Applications, pp. 219–235. Springer, New York (1983)Google Scholar
  196. 196.
    Peña, J., Vera, J.C., Zuluaga, L.F.: Static-arbitrage lower bounds on the prices of basket options via linear programming. Quant. Finance 10, 819–827 (2010)MATHMathSciNetGoogle Scholar
  197. 197.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)MATHGoogle Scholar
  198. 198.
    Puente, R.: Cyclic convex bodies and optimization moment problems. Linear Algebra Appl. 426, 596–609 (2007)MATHMathSciNetGoogle Scholar
  199. 199.
    Puente, R., Vera de Serio, V.N.: Locally Farkas-Minkowski linear inequality systems. Top 7, 103–121 (1999)MATHMathSciNetGoogle Scholar
  200. 200.
    Renegar, J.: Some perturbation theory for linear programming. Math. Program. A 65, 73–91 (1994)MATHMathSciNetGoogle Scholar
  201. 201.
    Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Program. A 70, 279–351 (1995)MATHMathSciNetGoogle Scholar
  202. 202.
    Robinson, S.M.: Stability theory for systems of inequalities. Part I: linear systems. SIAM J. Numer. Anal. 12, 754–769 (1975)MATHGoogle Scholar
  203. 203.
    Robinson, S.M.: Stability theory for systems of inequalities. Part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)MATHGoogle Scholar
  204. 204.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Study 14, 206–214 (1981)MATHGoogle Scholar
  205. 205.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  206. 206.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)Google Scholar
  207. 207.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)MATHGoogle Scholar
  208. 208.
    Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear optimization: An Interior Point Approach. Wiley, Chichester (1997)MATHGoogle Scholar
  209. 209.
    Rubinstein, G.S.: A comment on Voigt’s paper “a duality theorem for linear semi-infinite programming”. Optimization 12, 31–32 (1981)Google Scholar
  210. 210.
    Shapiro, A.: Directional differentiability of the optimal value function in convex semi-infinite programming. Math. Program. A 70, 149–157 (1995)MATHGoogle Scholar
  211. 211.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. MOS-SIAM Series on Optimization. SIAM, Philadelphia (2009)Google Scholar
  212. 212.
    Sharkey, T.C.: Infinite linear programs. In: Cochran, J.J. (ed.) Wiley Encyclopedia of Operations Research and Management Science, pp. 1–11. Wiley, New York (2010)Google Scholar
  213. 213.
    Sonnenburg, S., Rätsch, G., Schäfer, C., Schölkopf, B.: Large scale multiple kernel learning. J. Mach. Learn. Res. 7, 1531–1565 (2006)MATHMathSciNetGoogle Scholar
  214. 214.
    Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)MATHMathSciNetGoogle Scholar
  215. 215.
    Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003)MATHMathSciNetGoogle Scholar
  216. 216.
    Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. J. Cybern. 3, 37–46 (1974)MathSciNetGoogle Scholar
  217. 217.
    Tian, Y., Shi, Y., Liu, X.: Recent advances on support vector machines research. Technol. Econ. Dev. Econ. 18, 5–33 (2012)Google Scholar
  218. 218.
    Tichatschke, R.: Lineare Semi-Infinite Optimierungsaufgaben und ihre Anwendungen in der Approximationstheorie. Wissenschaftliche Schriftenreihe der Technischen Hochschule, Karl-Marx-Stadt (1981)Google Scholar
  219. 219.
    Tichatschke, R., Hettich, R., Still, G.: Connections between generalized, inexact and semi-infinite linear programming. Math. Methods Oper. Res. 33, 367–382 (1989)MATHMathSciNetGoogle Scholar
  220. 220.
    Todorov, M.I.: Generic existence and uniqueness of the solution set to linear semi-infinite optimization problems. Numer. Funct. Anal. Optim. 8, 27–39 (1985/1986)Google Scholar
  221. 221.
    Toledo, F.J.: Some results on Lipschitz properties of the optimal values in semi-infinite programming. Optim. Methods Softw. 23, 811–820 (2008)MATHMathSciNetGoogle Scholar
  222. 222.
    Tuy, H.: Stability property of a system of inequalities. Math. Oper. Stat. Ser. Opt. 8, 27–39 (1977)Google Scholar
  223. 223.
    Vaz, A., Fernandes, E., Gomes, M.: SIPAMPL: semi-infinite programming with AMPL. ACM Trans. Math. Softw. 30, 47–61 (2004)MATHMathSciNetGoogle Scholar
  224. 224.
    Vercher, E.: Portfolios with fuzzy returns: selection strategies based on semi-infinite programming. J. Comput. Appl. Math. 217, 381–393 (2008)MATHMathSciNetGoogle Scholar
  225. 225.
    Vercher, E., Bermúdez, J.D.: Fuzzy portfolio selection models: a numerical study. In: Doumpos, M., Zopounidis, C., Pardalos, P.M. (eds.) Financial Decision Making Using Computational Intelligence, pp. 245–272. Springer, New York (2012)Google Scholar
  226. 226.
    Voigt, I., Weis, S.: Polyhedral Voronoi cells. Contr. Beiträge Algebra Geom. 51, 587–598 (2010)MATHMathSciNetGoogle Scholar
  227. 227.
    Wan, Z., Meng, F.-Z., Hao, A.-Y., Wang, Y.-L.: Optimization of the mixture design for alumina sintering with fuzzy ingredients. Hunan Daxue Xuebao/J. Hunan Univ. Nat. Sci. 36, 55–58 (2009)Google Scholar
  228. 228.
    Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I) & (II). Fuzzy Sets Syst. 118, 375–385, 387–405 (2001)MATHMathSciNetGoogle Scholar
  229. 229.
    Wu, D., Han, J.-Y., Zhu, J.-H.: Robust solutions to uncertain linear complementarity problems. Acta Math. Appl. Sin., Engl. Ser. 27, 339–352 (2011)Google Scholar
  230. 230.
    Yildirim, E.A.: Unifying optimal partition approach to sensitivity analysis in conic optimization. J. Optim. Theory Appl. 122, 405–423 (2004)MATHMathSciNetGoogle Scholar
  231. 231.
    Yiu, K.F., Xioaqi, Y., Nordholm, S., Teo, K.L.: Near-field broadband beamformer design via multidimensional semi-infinite linear programming techniques. IEEE Trans. Speech Audio Process. 11, 725–732 (2003)Google Scholar
  232. 232.
    Zălinescu, C.: On the differentiability of the support function. J. Global Optim. 57, 719–731 (2013)MathSciNetGoogle Scholar
  233. 233.
    Zălinescu, C.: Relations between the convexity of a set and the differentiability of its support function. arXiv:1301.0810 [math.FA] (2013). http://arxiv.org/abs/1301.0810
  234. 234.
    Zencke, P., Hettich, R.: Directional derivatives for the value-function in semi-infinite programming. Math. Program. A 38, 323–340 (1987)MATHMathSciNetGoogle Scholar
  235. 235.
    Zimmermann, H.J.: Description and optimization of fuzzy systems. Int. J. Gen. Syst. 2, 209–215 (1976)MATHGoogle Scholar
  236. 236.
    Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequality on Banach spaces. SIAM J. Optim. 14, 757–772 (2003)MathSciNetGoogle Scholar
  237. 237.
    Zhu, Y., Huang, G.H., Li, Y.P., He, L., Zhang, X.X.: An interval full-infinite mixed-integer programming method for planning municipal energy systems—a case study of Beijing. Appl. Energy 88, 2846–2862 (2011)Google Scholar
  238. 238.
    Zopounidis, C., Doumpos, M.: Multicriteria decision systems for financial problems. Top 21, 241–261 (2013)MATHMathSciNetGoogle Scholar

Copyright information

© Miguel A. Goberna, Marco A. Lopez 2014

Authors and Affiliations

  • Miguel A. Goberna
    • 1
  • Marco A. López
    • 1
  1. 1.Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain

Personalised recommendations