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Recent contributions to linear semi-infinite optimization: an update. (English) Zbl 1411.90341
Summary: This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.

MSC:
90C34 Semi-infinite programming
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[1] Agmon, S., The relaxation method for linear inequalities, Canadian Journal of Mathematics, 6, 382-392, (1954) · Zbl 0055.35001
[2] Ahmed, F.; Dür, M.; Still, G., Copositive programming via semi-infinite optimization, Journal of Optimization Theory and Applications, 159, 322-340, (2013) · Zbl 1282.90203
[3] Aliprantis, C., & Border, K. (2005). Infinite dimensional analysis: A Hitchhiker’s guide (3rd ed.). Berlin: Springer. · Zbl 1156.46001
[4] Altinel, IK; Çekyay, BÇ; Feyzioğlu, O.; Keskin, ME; Özekici, S., Mission-based component testing for series systems, Annals of Operations Research, 186, 1-22, (2011) · Zbl 1227.62086
[5] Altinel, IK; Çekyay, BÇ; Feyzioğlu, O.; Keskin, ME; Özekici, S., The design of mission-based component test plans for series connection of subsystems, IIE Transactions, 45, 1202-1220, (2013)
[6] Anderson, EJ; Goberna, MA; López, MA, Simplex-like trajectories on quasi-polyhedral convex sets, Mathematics of Operations Research, 26, 147-162, (2001) · Zbl 1073.90527
[7] Anderson, EJ; Lewis, AS, An extension of the simplex algorithm for semi-infinite linear programming, Mathematical Programming, 44A, 247-269, (1989) · Zbl 0682.90058
[8] Anderson, E. J., & Nash, P. (1987). Linear programming in infinite-dimensional spaces: Theory and applications. Chichester: Wiley.
[9] Astaf’ev, NN; Ivanov, AV; Trofimov, SP, The set of target vectors of a semi-infinite linear programming problem with a duality gap. (Russian) Tr, Inst. Mat. Mekh., 22, 43-52, (2016)
[10] Audy, J-F; D’Amours, S.; Rönnqvist, M., An empirical study on coalition formation and cost/savings allocation, International Journal of Production Economics, 136, 13-27, (2012)
[11] Auslender, A.; Ferrer, A.; Goberna, MA; López, MA, Comparative study of RPSALG algorithms for convex semi-infinite programming, Computational Optimization and Applications, 60, 59-87, (2015) · Zbl 1316.90054
[12] Auslender, A.; Goberna, MA; López, MA, Penalty and smoothing methods for convex semi-infinite programming, Mathematics of Operations Research, 34, 303-319, (2009) · Zbl 1218.90199
[13] Azé, D.; Corvellec, J-N, Characterizations of error bounds for lower semicontinuous functions on metric spaces, ESAIM: Control Optimisation and Calculus of Variations, 10, 409-425, (2004) · Zbl 1085.49019
[14] Badikov, S.; Jacquier, A.; Liu, DQ; Roome, P., No-arbitrage bounds for the forward smile given marginals, Quantitative Finance, 17, 1243-1256, (2017) · Zbl 1402.91752
[15] Barragán, A.; Hernández, LA; Todorov, MI, New primal-dual partition of the space of linear semi-infinite continuous optimization problems, Comptes rendus de l’Academie bulgare des Sciences, 69, 1263-1274, (2016) · Zbl 1374.90390
[16] Basu, A.; Martin, K.; Ryan, CT, On the sufficiency of finite support duals in semi-infinite linear programming, Operations Research Letters, 42, 16-20, (2014) · Zbl 1408.90170
[17] Basu, A.; Martin, K.; Ryan, CT, A Unified approach to semi-infinite linear programs and duality in convex programming, Mathematics of Operations Research, 40, 146-170, (2015) · Zbl 1308.90185
[18] Basu, A.; Martin, K.; Ryan, CT, Strong duality and sensitivity analysis in semi-infinite linear programming, Mathematical Programming, 161A, 451-485, (2017) · Zbl 1364.90322
[19] Ben-Tal, A., Ghaoui, L. E., & Nemirovski, A. (2009). Robust optimization. Princeton: Princeton University Press. · Zbl 1221.90001
[20] Benavoli, A.; Piga, D., A probabilistic interpretation of set-membership filtering: Application to polynomial systems through polytopic bounding, Automatica, 70, 158-172, (2016) · Zbl 1339.93107
[21] Betró, B., An accelerated central cutting plane algorithm for linear semi-infinite programming, Mathematical Programming, 101A, 479-495, (2004) · Zbl 1073.90053
[22] Betró, B.; Ruggeri, F. (ed.); Kenett, RS (ed.); Faltin, F. (ed.), Bayesian robustness: Theory and computation, 203-207, (2007), Chichester
[23] Betró, B., Numerical treatment of Bayesian robustness problems, International Journal of Approximate Reasoning, 50, 279-288, (2009) · Zbl 1185.62056
[24] Bisbos, CD; Ampatzis, AT, Shakedown analysis of spatial frames with parameterized load domain, Engineering Structures, 303, 119-3128, (2008)
[25] Blado, D.; Hu, W.; Toriello, A., Semi-infinite relaxations for the dynamic knapsack problem with stochastic item sizes, SIAM Journal on Optimization, 26, 1625-1648, (2016) · Zbl 1346.90606
[26] Bodirsky, M.; Jonsson, P.; Oertzen, T., Essential convexity and complexity of semi-algebraic constraints, Logical Methods in Computer Science, 8, 4-25, (2012) · Zbl 1253.68143
[27] Boţ, RI; Csetnek, ER; Wanka, G., Sequential optimality conditions in convex programming via perturbation approach, Journal of Convex Analysis, 15, 149-164, (2008) · Zbl 1144.90018
[28] Brosowski, B. (1982). Parametric semi-infinite optimization. Frankfurt am Main: Peter Lang.
[29] Brosowski, B., Parametric semi-infinite linear programming I. Continuity of the feasible set and the optimal value, Mathematical Programming Study, 21, 18-42, (1984) · Zbl 0547.90091
[30] Cánovas, MJ; Dontchev, AL; López, MA; Parra, J., Isolated calmness of solution mappings in convex semi-infinite optimization, Journal of Mathematical Analysis and Applications, 350, 892-837, (2009) · Zbl 1163.49028
[31] Cánovas, M. J., Hall, J. A. J., López, M. A., & Parra, J. Calmness of partially perturbed linear systems with an application to the central path. Mathematical Methods of Operations Research (submitted).
[32] Cánovas, MJ; Hantoute, A.; Parra, J.; Toledo, FJ, Calmness of the argmin mapping in linear semi-infinite optimization, Journal of Optimization Theory and Applications, 160, 111-126, (2014) · Zbl 1306.90161
[33] Cánovas, MJ; Hantoute, A.; Parra, J.; Toledo, FJ, Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization, Optimization Letters, 9, 513-521, (2015) · Zbl 1332.90319
[34] Cánovas, MJ; Hantoute, A.; Parra, J.; Toledo, FJ, Calmness modulus of fully perturbed linear programs, Mathematical Programming, 158A, 267-290, (2016) · Zbl 1370.90267
[35] Cánovas, MJ; Henrion, R.; López, MA; Parra, J., Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming, Journal of Optimization Theory and Applications, 169, 925-952, (2016) · Zbl 1342.90198
[36] Cánovas, M. Josefa; Henrion, René; López, Marco A.; Parra, Juan, Indexation Strategies and Calmness Constants for Uncertain Linear Inequality Systems, 831-843, (2018), Cham
[37] Cánovas, MJ; Henrion, R.; Parra, J.; Toledo, FJ, Critical objective size and calmness modulus in linear programming, Set-Valued and Variational Analysis, 24, 565-579, (2016) · Zbl 1355.90097
[38] Cánovas, MJ; Kruger, AY; López, MA; Parra, J.; Théra, MA, Calmness modulus of linear semi-infinite programs, SIAM Journal on Optimization, 24, 29-48, (2014) · Zbl 1374.90391
[39] Cánovas, MJ; López, MA; Parra, J.; Toledo, FJ, Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems, Mathematical Programming, 103A, 95-126, (2005) · Zbl 1070.65026
[40] Cánovas, MJ; López, MA; Parra, J.; Toledo, FJ, Distance to solvability/unsolvability in linear optimization, SIAM Journal on Optimization, 16, 629-649, (2006) · Zbl 1097.65066
[41] Cánovas, MJ; López, MA; Parra, J.; Toledo, FJ, Ill-posedness with respect to the solvability in linear optimization, Linear Algebra and Its Applications, 416, 520-540, (2006) · Zbl 1097.65067
[42] Cánovas, MJ; López, MA; Parra, J.; Toledo, FJ, Error bounds for the inverse feasible set mapping in linear semi-infinite optimization via a sensitivity dual approach, Optimization, 56, 547-563, (2007) · Zbl 1136.90040
[43] Cánovas, MJ; López, MA; Parra, J.; Toledo, FJ, Calmness of the feasible set mapping for linear inequality systems, Set-Valued and Variational Analysis, 22, 375-389, (2014) · Zbl 1297.90163
[44] Cánovas, M. J., Parra, J., & Toledo, F. J. Lipschitz modulus of fully perturbed linear programs. Pacific Journal of Optimization (to appear).
[45] Cánovas, MJ; Parra, J.; Rückmann, J.; Toledo, FJ, Point-based neighborhoods for sharp calmness constants in linear programming, Set-Valued and Variational Analysis, 25, 757-772, (2017) · Zbl 1387.90247
[46] Chan, TCY; Mar, PhA, Stability and Continuity in Robust Optimization, SIAM Journal on Optimization, 27, 817-841, (2017) · Zbl 06724131
[47] Charnes, A.; Cooper, WW; Kortanek, KO, Duality, Haar programs, and finite sequence spaces, Proceedings of the National Academy of Sciences of the United States of America, 48, 783-786, (1962) · Zbl 0105.12804
[48] Charnes, A.; Cooper, WW; Kortanek, KO, Duality in semi-infinite programs and some works of Haar and Carathéodory, Management Science, 9, 209-228, (1963) · Zbl 0995.90615
[49] Charnes, A.; Cooper, WW; Kortanek, KO, On representations of semi-infinite programs which have no duality gaps, Management Science, 12, 113-121, (1965) · Zbl 0143.42304
[50] Charnes, A.; Cooper, WW; Kortanek, KO, On the theory of semi-infinite programming and a generalization of the Kuhn-Tucker saddle point theorem for arbitrary convex functions, Naval Research Logistics Quarterly, 16, 41-51, (1969) · Zbl 0169.22201
[51] Cho, H.; Kim, KK; Lee, K., Computing lower bounds on basket option prices by discretizing semi-infinite linear programming, Optimization Letters, 10, 1629-1644, (2016) · Zbl 1414.91366
[52] Chu, YC, Generalization of some fundamental theorems on linear inequalities, Acta Mathematica Sinica, 16, 25-40, (1966)
[53] Chuong, TD; Jeyakumar, V., An exact formula for radius of robus feasibility of uncertain linear programs, Journal of Optimization Theory and Applications, 173, 203-226, (2017) · Zbl 1373.49035
[54] Chuong, TD; Jeyakumar, V., A generalized Farkas lemma with a numerical certificate and linear semi-infinite programs with SDP duals, Linear Algebra and Its Applications, 515, 38-52, (2017) · Zbl 1352.90060
[55] Chuong, T. D., & Jeyakumar, V. Semi-infinite convex quadratic programming with geometric index sets: Exact second-order cone duals, preprint, School of Mathematics, University of New South Wales, Sydney.
[56] Clarke, FH, A new approach to Lagrange multipliers, Mathematics of Operations Research, 1, 165-174, (1976)
[57] Correa, R.; Hantoute, A.; López, MA, Weaker conditions for subdifferential calculus of convex functions, Journal of Functional Analysis, 271, 1177-1212, (2016) · Zbl 1351.26022
[58] Cozad, A.; Sahinidis, NV; Miller, DC, A combined first-principles and data-driven approach to model building, Computers & Chemical Engineering, 73, 116-127, (2015)
[59] Cozman, FG; Polpo de Campos, C., Kuznetsov independence for interval-valued expectations and sets of probability distributions: Properties and algorithms, International Journal of Approximate Reasoning, 55, 666-682, (2014) · Zbl 1316.68174
[60] Daniilidis, A.; Goberna, MA; López, MA; Lucchetti, R., Stability in linear optimization under perturbations of the left-hand side coefficients, Set-Valued and Variational Analysis, 23, 737-758, (2015) · Zbl 1330.49023
[61] Dantzig, GB; Lenstra, JK (ed.); etal., Linear programming, 19-31, (1991), Amsterdam
[62] Daum, S.; Werner, R., A novel feasible discretization method for linear semi-infinite programming applied to basket options pricing, Optimization, 60, 1379-1398, (2011) · Zbl 1230.91189
[63] Dinh, N.; Goberna, MA; López, MA; Song, TQ, New Farkas-type constraint qualifications in convex infinite programming. ESAIM: Control, Optimisation and Calculus of Variations, 13, 580-597, (2007) · Zbl 1126.90059
[64] Dinh, N.; Goberna, MA; López, MA; Volle, M., Convex inequalities without constraint qualification nor closedness condition, and their applications in optimization, Set-Valued and Variational Analysis, 18, 423-445, (2010) · Zbl 1231.90381
[65] Dolgin, Y.; Zeheb, E., Model reduction of uncertain systems retaining the uncertainty structure, Systems & Control Letters, 54, 771-779, (2005) · Zbl 1129.93338
[66] Dontchev, A. L., & Rockafellar, R. T. (2009). Implicit functions and solution mappings: A view from variational analysis. New York: Springer.
[67] Duffin, R.; Karlovitz, LA, An infinite linear program with a duality gap, Management Science, 12, 122-134, (1965) · Zbl 0133.42604
[68] Dür, M.; Jargalsaikhan, B.; Still, G., Genericity results in linear conic programming—A tour d’horizon, Mathematics of Operations Research, 42, 77-94, (2016) · Zbl 1359.90093
[69] Eberhard, A., Roshchina, V., & Sang, T. (2017). Outer limits of subdifferentials for min-max type functions. Optimization. https://doi.org/10.1080/02331934.2017.1398750.
[70] Elbassioni, K.; Makino, K.; Najy, W., A multiplicative weights update algorithm for packing and covering semi-infinite linear programs. Approximation and online algorithms, Lecture Notes in Computer Science, 10138, 78-91, (2017) · Zbl 1430.90534
[71] Fabian, M.; Henrion, R.; Kruger, AY; Outrata, J., Error bounds: Necessary and sufficient conditions, Set-Valued Analysis, 18, 121-149, (2010) · Zbl 1192.49018
[72] Fajardo, MD; López, MA, Locally Farkas-Minkowski systems in convex semi-infinite programming, Journal of Optimization Theory and Applications, 103, 313-335, (1999) · Zbl 0945.90069
[73] Fajardo, MD; López, MA, Some results about the facial geometry of convex semi-infinite systems, Optimization, 55, 661-684, (2006) · Zbl 1134.90509
[74] Fajardo, MD; López, MA; Puente, R., Linear representations and quasipolyhedrality of a finite-valued convex function, Optimization, 57, 215-237, (2008) · Zbl 1191.90086
[75] Fang, DH; Li, C.; Ng, KF, Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM Journal on Optimization, 20, 1311-1332, (2009) · Zbl 1206.90198
[76] Fang, DH; Li, C.; Ng, KF, Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming, Nonlinear Analysis, 73, 1143-1159, (2010) · Zbl 1218.90200
[77] Faybusovich, L.; Mouktonglang, T.; Tsuchiya, T., Numerical experiments with universal barrier functions for cones of Chebyshev systems, Computational Optimization and Applications, 41, 205-223, (2008) · Zbl 1169.90424
[78] Feng, S., The research on the multiple kernel learning-based face recognition in pattern matching, The Open Automation and Control Systems Journal, 7, 1796-1801, (2015)
[79] Ferrer, A.; Goberna, MA; González-Gutiérrez, E.; Todorov, MI, A comparative study of relaxation algorithms for the linear semi-infinite feasibility problem, Annals of Operations Research, 258, 587-612, (2017) · Zbl 1382.90107
[80] Feyzioglu, O.; Altinel, IK; Ozekici, S., Optimum component test plans for phased-mission systems, European Journal of Operational Research, 185, 255-265, (2008) · Zbl 1137.90446
[81] Fischer, T., Contributions to semi-infinite linear optimization, Meth Verf Math Phys, 27, 175-199, (1983)
[82] Gaitsgory, V.; Rossomakhine, S., Averaging and linear programming in some singularly perturbed problems of optimal control, Applied Mathematics and Optimization, 71, 195-276, (2015) · Zbl 1317.49042
[83] Gao, Sarah Yini; Sun, Jie; Wu, Soon-Yi, A semi-infinite programming approach to two-stage stochastic linear programs with high-order moment constraints, Optimization Letters, 12, 1237-1247, (2016) · Zbl 1401.90137
[84] Gisbert, M. J.; Cánovas, M. J.; Parra, J.; Toledo, F. J., Calmness of the Optimal Value in Linear Programming, SIAM Journal on Optimization, 28, 2201-2221, (2018) · Zbl 1401.90231
[85] Ghate, A.; Sharma, D.; Smith, RL, A shadow simplex method for infinite linear programs, Operations Research, 58, 865-877, (2010) · Zbl 1228.90151
[86] Glashoff, K., & Gustafson, S. A. (1983). Linear optimization and approximation. Berlin: Springer.
[87] Goberna, MA; Rubinov, A. (ed.); Jeyakumar, V. (ed.), Linear semi-infinite optimization: Recent advances, 3-22, (2005), New York · Zbl 1115.90060
[88] Goberna, M. A. (2005). Linear semi-infinite programming: A guided tour. Lima: IMCA Monoghaphs.
[89] Goberna, MA; Gómez, S.; Guerra-Vázquez, F.; Todorov, MI, Sensitivity analysis in linear semi-infinite programming: Perturbing cost and right-hand-side coefficients, European Journal of Operational Research, 181, 1069-1085, (2007) · Zbl 1121.90121
[90] Goberna, MA; González, E.; Martinez-Legaz, JE; Todorov, MI, Motzkin decomposition of closed convex sets, Journal of Mathematical Analysis and Applications, 364, 209-221, (2010) · Zbl 1188.90195
[91] Goberna, MA; Guerra-Vázquez, F.; Todorov, MI, Constraint qualifications in linear vector semi-infinite optimization, European Journal of Operational Research, 227, 12-21, (2013) · Zbl 1292.90298
[92] Goberna, MA; Guerra-Vázquez, F.; Todorov, MI, Constraint qualifications in convex vector semi-infinite optimization, European Journal of Operational Research, 249, 32-40, (2016) · Zbl 1346.90783
[93] Goberna, MA; Hiriart-Urruty, JB; López, MA, Best approximate solutions of inconsistent linear inequality systems, Vietnam Journal of Mathematics, 46, 271-284, (2018) · Zbl 1391.65096
[94] Goberna, MA; Iusem, A.; Martínez-Legaz, JE; Todorov, MI, Motzkin decomposition of closed convex sets via truncation, Journal of Mathematical Analysis and Applications, 400, 35-47, (2013) · Zbl 1262.52002
[95] Goberna, MA; Jeyakumar, V.; Li, G.; Linh, N., Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constrains, Operations Research Letters, 44, 67-73, (2016) · Zbl 1408.90235
[96] Goberna, MA; Jeyakumar, V.; Li, G.; López, MA, Robust linear semi-infinite programming duality under uncertainty, Mathematical Programming, 139B, 185-203, (2013) · Zbl 1282.90204
[97] Goberna, MA; Jeyakumar, V.; Li, G.; Vicente-Pérez, J., Robust solutions of multi-objective linear semi-infinite programs under constraint data uncertainty, SIAM Journal on Optimization, 24, 1402-1419, (2014) · Zbl 1325.90081
[98] Goberna, MA; Jeyakumar, V.; Li, G.; Vicente-Pérez, J., Robust solutions to multi-objective linear programs with uncertain data, European Journal of Operational Research, 242, 730-743, (2015) · Zbl 1341.90124
[99] Goberna, MA; Jeyakumar, V.; Li, G.; Vicente-Pérez, J., Guaranteeing highly robust weakly efficient solutions for uncertain multi-objective convex programs, European Journal of Operational Research, 70, 40-50, (2018) · Zbl 1403.90606
[100] Goberna, MA; Lancho, A.; Todorov, MI; Vera de Serio, VN, On implicit active constraints in linear semi-infinite programs with unbounded coefficients, Applied Mathematics and Optimization, 63, 239-256, (2011) · Zbl 1220.90142
[101] Goberna, MA; López, MA, Optimal value function in semi-infinite programming, Journal of Optimization Theory and Applications, 59, 261-279, (1988) · Zbl 0628.90046
[102] Goberna, MA; López, MA, Topological stability of linear semi-infinite inequality systems, Journal of Optimization Theory and Applications, 89, 227-236, (1998) · Zbl 0866.90128
[103] Goberna, M. A., & López, M. A. (1998b). Linear semi-infinite optimization. Chichester: Wiley.
[104] Goberna, M. A., & López, M. A. (2014). Post-optimal analysis in linear semi-infinite optimization. Springer, New York: Springer Briefs.
[105] Goberna, M. A., & López, M. A. (2017). Recent contributions to linear semi-infinite optimization. 4OR, 15, 221-264. · Zbl 1374.90392
[106] Goberna, MA; López, MA; Pastor, JT, Farkas-Minkowski systems in semi-infinite programming, Applied Mathematics and Optimization, 7, 295-308, (1980) · Zbl 0438.90054
[107] Goberna, MA; López, MA; Todorov, MI, Stability theory for linear inequality systems, SIAM Journal on Matrix Analysis and Applications, 17, 730-743, (1996) · Zbl 0864.15009
[108] Goberna, MA; López, MA; Todorov, MI, Stability theory for linear inequality systems II: Upper semicontinuity of the solution set mapping, SIAM Journal on Optimization, 7, 1138-1151, (1997) · Zbl 0897.15006
[109] Goberna, MA; López, MA; Todorov, MI, On the stability of the feasible set in linear optimization, Set-Valued Analysis, 9, 75-99, (2001) · Zbl 1039.90077
[110] Goberna, MA; López, MA; Todorov, MI, Extended active constraints in linear optimization with applications, SIAM Journal on Optimization, 14, 608-619, (2003) · Zbl 1046.90039
[111] Goberna, MA; López, MA; Volle, M., Primal attainment in convex infinite optimization duality, Journal of Convex Analysis, 21, 1043-1064, (2014) · Zbl 1327.90205
[112] Goberna, MA; López, MA; Volle, M., Modified Lagrangian duality for the supremum of convex functions, Pacific Journal of Optimization, 13, 501-514, (2017)
[113] Goberna, MA; Kanzi, N., Optimality conditions in convex multi-objective SIP, Mathematical Programming, 164A, 167-191, (2017) · Zbl 1387.90234
[114] Goberna, MA; Ridolfi, A.; Vera de Serio, VN, Stability of the duality gap in linear optimization, Set-Valued and Variational Analysis, 25, 617-636, (2017) · Zbl 1373.90073
[115] Goberna, MA; Terlaky, T.; Todorov, MI, Sensitivity analysis in linear semi-infinite programming via partitions, Mathematics of Operations Research, 35, 14-25, (2010) · Zbl 1230.90190
[116] Goberna, MA; Todorov, MI, Primal-dual stability in continuous linear optimization, Mathematical Programming, 116B, 129-146, (2009) · Zbl 1176.90592
[117] González-Gutiérrez, E.; Rebollar, LA; Todorov, MI, Relaxation methods for solving linear inequality systems: Converging results, Top, 20, 426-436, (2012) · Zbl 1258.49043
[118] González-Gutiérrez, E.; Todorov, MI, A relaxation method for solving systems with infinitely many linear inequalities, Optimization Letters, 6, 291-298, (2012) · Zbl 1258.90092
[119] Gui, Zeying; Li, Mo; Guo, Ping, Simulation-Based Inexact Fuzzy Semi-Infinite Programming Method for Agricultural Cultivated Area Planning in the Shiyang River Basin, Journal of Irrigation and Drainage Engineering, 143, 05016011, (2017)
[120] Guo, F. Semidefinite programming relaxations for linear semi-infinite polynomial programming. ArXiv:1509.06394v3, 14 Nov 2017.
[121] Guo, F.; Sun, X., LP relaxations for a class of linear semi-infinite programming problems, Optimization, 66, 657-673, (2017) · Zbl 1401.90245
[122] Guo, P.; Huang, GH; He, L., ISMISIP: An inexact stochastic mixed integer linear semi-infinite programming approach for solid waste management and planning under uncertainty, Stochastic Environmental Research and Risk Assessment, 22, 759-775, (2008)
[123] Gustafson, SA, On the computational solution of a class of generalized moment problems, SIAM Journal on Numerical Analysis, 7, 343-357, (1970) · Zbl 0217.21401
[124] Gustafson, SA; Kortanek, KO, Numerical treatment of a class of semi-infinite programming problems, Naval Research Logistics Quarterly, 20, 477-504, (1973) · Zbl 0272.90073
[125] Haar, A., Über lineare ungleichungen (in German), Acta Scientiarum Mathematicarum, 2, 1-14, (1924) · JFM 50.0699.02
[126] Hayashi, S.; Okuno, T.; Ito, Y., Simplex-type algorithm for second-order cone programmes via semi-infinite programming reformulation, Optimization Methods and Software, 31, 1272-1297, (2016) · Zbl 1386.90076
[127] He, L.; Huang, GH, Optimization of regional waste management systems based on inexact semi-infinite programming, Canadian Journal of Civil Engineering, 35, 987-998, (2008)
[128] He, L.; Huang, GH; Lu, H., Bivariate interval semi-infinite programming with an application to environmental decision-making analysis, European Journal of Operational Research, 211, 452-465, (2011) · Zbl 1237.90239
[129] Henrion, R., & Roemisch, W. (2017). Optimal scenario generation and reduction in stochastic programming. Preprint (http://www.optimization-online.org/DB_HTML/2017/03/5919.html).
[130] Hu, H.; Du, D-Z (ed.); Sun, J. (ed.), A projection method for solving infinite systems of linear inequalities, 186-194, (1994), Dordrecht · Zbl 0829.90127
[131] Huang, GH; He, L.; Zeng, GM; Lu, HW, Identification of optimal urban solid waste flow schemes under impacts of energy prices, Environmental Engineering Science, 25, 685-695, (2008)
[132] Huynh, DBP; Rozza, G.; Sen, S.; Patera, AT, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, Comptes Rendus de l’Académie des Sciences de Paris, 345, 473-478, (2007) · Zbl 1127.65086
[133] Ioffe, AD, Necessary and sufficient conditions for a local minimum. I. A reduction theorem and first order conditions, SIAM Journal on Control and Optimization, 17, 245-250, (1979) · Zbl 0417.49027
[134] Jeyakumar, V.; Li, GY; Lee, GM, A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty, Operations Research Letters, 39, 109-114, (2011) · Zbl 1218.91006
[135] Jeroslow, RG, Some relaxation methods for linear inequalities, Cahiers du Cero, 21, 43-53, (1979) · Zbl 0405.65036
[136] Ji, Y., A worst-case regret approach to multiperiod portfolio optimization, Technical Bulletin, 55, 398-406, (2017)
[137] Ji, Y.; Wang, T.; Goh, M.; Zhou, Y.; Zou, B., The worst-case discounted regret portfolio optimization problem, Applied Mathematics and Computation, 239, 310-319, (2014) · Zbl 1335.91063
[138] Jinglai, S., Positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification, IEEE Transactions on Automatic Control, 57, 3-18, (2012) · Zbl 1369.93450
[139] Karimi, A.; Galdos, G., Fixed-order \(H_{\infty }\) controller design for nonparametric models by convex optimization, Automatica, 46, 1388-1394, (2010) · Zbl 1204.93042
[140] Kashyap, H.; Ahmed, HA; Hoque, N.; Roy, S.; Bhattacharyya, DK, Big data analytics in bioinformatics: Architectures, techniques, tools and issues, Network Modeling Analysis in Health Informatics and Bioinformatics, 5, 28, (2016)
[141] Klabjan, D.; Adelman, D., An infinite-dimensional linear programming algorithm for deterministic semi-Markov decision processes on Borel spaces, Mathematics of Operations Research, 32, 528-550, (2007) · Zbl 1279.90111
[142] Klatte, D., & Kummer, B. (2002). Nonsmooth equations in optimization: Regularity, calculus, methods and applications. Dordrecht: Kluwer. · Zbl 1173.49300
[143] Klatte, D.; Kummer, B., Optimization methods and stability of inclusions in Banach spaces, Mathematical Programming, 117B, 305-330, (2009) · Zbl 1158.49007
[144] Kortanek, KO, Classifying convex extremum problems over linear topologies having separation properties, Journal of Mathematical Analysis and Applications, 46, 725-755, (1974) · Zbl 0283.90043
[145] Kortanek, KO; Goberna, MA (ed.); López, MA (ed.), On the 1962-1972 decade of semi-infinite programming: A subjective view, 3-34, (2001), Dordrecht · Zbl 1055.90078
[146] Kortanek, KO; Zhang, Q., Extending the mixed algebraic-analysis Fourier-Motzkin elimination method for classifying linear semi-infinite programmes, Optimization, 65, 707-727, (2016) · Zbl 1384.90111
[147] Kruger, AY; Ngai, H.; Théra, M., Stability of error bounds for convex constraint systems in Banach spaces, SIAM Journal on Optimization, 20, 3280-3296, (2010) · Zbl 1208.49030
[148] Larriqueta, M.; Vera de Serio, VN, On metric regularity and the boundary of the feasible set in linear optimization, Set-Valued and Variational Analysis, 22, 1-17, (2014) · Zbl 1297.90165
[149] Lasserre, JB, An algorithm for semi-infinite polynomial optimization, Top, 20, 119-129, (2012) · Zbl 1285.90076
[150] Leibfritz, F.; Maruhn, JH, A successive SDP-NSDP approach to a robust optimization problem in finance, Computational Optimization and Applications, 44, 443-466, (2009) · Zbl 1181.90217
[151] Li, C.; Ng, KF; Pong, TK, Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM Journal on Optimization, 19, 163-187, (2008) · Zbl 1170.90009
[152] Li, M. H.; Meng, K. W.; Yang, X. Q., On error bound moduli for locally Lipschitz and regular functions, Mathematical Programming, 171, 463-487, (2017) · Zbl 1397.65095
[153] Li, X.; Lu, H.; He, L.; Shi, B., An inexact stochastic optimization model for agricultural irrigation management with a case study in China, Stochastic Environmental Research and Risk, 28A, 281-295, (2014)
[154] Li, X.; Mao, W.; Jiang, W., Multiple-kernel-learning-based extreme learning machine for classification design, Neural Computing and Applications, 27, 175-184, (2016)
[155] Liu, Y., New constraint qualification and optimality for linear semi-infinite programing, Pac J Optim, 12, 223-232, (2016) · Zbl 1343.90100
[156] Liu, Y., Generalized corner optimal solution for LSIP: Existence and numerical computation, TOP, 24, 19-43, (2016) · Zbl 1336.90093
[157] Liu, Y.; Ding, MF, A ladder method for semi-infinite programming, Journal of Industrial and Management Optimization, 10, 397-412, (2014) · Zbl 1281.90076
[158] Liu, Y.; Goberna, MA, Asymptotic optimality conditions for linear semi-infinite programming, Optimization, 65, 387-414, (2016) · Zbl 1332.90321
[159] López, MA, Stability in linear optimization and related topics, A personal tour. Top, 20, 217-244, (2012) · Zbl 1257.90096
[160] López, MA; Still, G., Semi-infinite programming, European Journal of Operational Research, 180, 491-518, (2007) · Zbl 1124.90042
[161] Lou, Y.; Yin, Y.; Lawphongpanich, S., Robust congestion pricing under boundedly rational user equilibrium, Transportation Research Part B: Methodological, 44, 15-28, (2010)
[162] Luo, Z-Q; Roos, C.; Terlaky, T., Complexity analysis of a logarithmic barrier decomposition method for semi-infinite linear programming, Applied Numerical Mathematics, 29, 379-394, (1999) · Zbl 0948.90137
[163] Mangasarian, OL, Knowledge-based linear programming, SIAM Journal on Optimization, 12, 375-382, (2004) · Zbl 1114.90070
[164] Mangasarian, OL; Wild, EW, Nonlinear knowledge in kernel approximation, IEEE Transactions on Neural Networks and Learning System, 18, 300-306, (2007)
[165] Mangasarian, OL; Wild, EW, Nonlinear knowledge-based classification, IEEE Transactions on Neural Networks and Learning System, 19, 1826-1832, (2008)
[166] Martínez-Legaz, JE; Todorov, MI; Zetina, C., \(\gamma \) -Active constraints in convex semi-infinite programming, Numerical Functional Analysis and Optimization, 35, 1078-1094, (2014) · Zbl 1295.90096
[167] Maruhn, J. H. (2009). Robust static super-replication of barrier options. Berlin: De Gruyter. · Zbl 1196.91007
[168] Mehrizi, S.; Khosravi, S.; Ahmadian, M., An efficient procedure for bilayer-expurgated LDPC codes design in cooperative relay channels, IEEE Communications Letters, 21, 2114-2117, (2017)
[169] Mejia, C., Linear secret sharing and the automatic search of linear rank inequalities, Applied Mathematical Sciences, 9, 5305-5324, (2015)
[170] Miao, DY; Li, YP; Huang, GH; Yang, ZF, Optimization model for planning regional water resource systems under ucertainty, Journal of Water Resources Planning and Management, 140, 238-249, (2014)
[171] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation, I: Basic theory. Berlin: Springer.
[172] Motzkin, TS; Schoenberg, IJ, The relaxation method for linear inequalities, Canadian Journal of Mathematics, 6, 393-404, (1954) · Zbl 0055.35002
[173] Ochoa, PD; Vera de Serio, VN, Stability of the primal-dual partition in linear semi-infinite programming, Optimization, 61, 1449-1465, (2012) · Zbl 1282.90205
[174] Oskoorouchi, MR; Ghaffari, HR; Terlaky, T.; Aleman, DM, An interior point constraint generation algorithm for semi-infinite optimization with health-care application, Operations Research, 59, 1184-1197, (2011) · Zbl 1235.90167
[175] Ozogur, S.; Ustunkar, G.; Weber, G-W, Adapted infinite kernel learning by multi-local algorithm, International Journal of Pattern Recognition and Artificial Intelligence, 30, 1651004, (2016)
[176] Ozogur, S.; Weber, G-W, On numerical optimization theory of infinite kernel learning, Journal of Global Optimization, 48, 215-239, (2010) · Zbl 1198.90377
[177] Ozogur, S.; Weber, G-W, Infinite kernel learning via infinite and semi-infinite programming, Optimization Methods and Software, 25, 937-970, (2010) · Zbl 1225.90141
[178] Papp, D., Semi-infinite programming using high-degree polynomial interpolants and semidefinite programming, SIAM Journal on Optimization., 27, 1858-1879, (2017) · Zbl 1370.90279
[179] Patera, AT; Yano, M., An LP empirical quadrature procedure for parametrized functions, Comptes Rendus Mathematique, 355, 1161-1167, (2017) · Zbl 1377.65074
[180] Peña, J.; Vera, JC; Zuluaga, LF, Static-arbitrage lower bounds on the prices of basket options via linear programming, Quantitative Finance, 10, 819-827, (2010) · Zbl 1204.91132
[181] Powell, MJD, Karmarkar’s algorithm: A view from nonlinear programming, Bulletin Institute of Mathematics and its Applications, 26, 165-181, (1990) · Zbl 0711.90052
[182] Prékopa, A., Inequalities for discrete higher order convex functions, Journal of Mathematical Inequalities, 3, 485-498, (2009) · Zbl 1264.90130
[183] Prékopa, A.; Ninh, A.; Alexe, G., On the relationship between the discrete and continuous bounding moment problems and their numerical solutions, Annals of Operations Research, 238, 521-575, (2016) · Zbl 1334.90184
[184] Priyadarsini, PI; Devarakonda, N.; Babu, IR, A chock-full survey on support vector machines, International Journal of Advanced Research in Computer Science and Software Engineering, 3, 780-799, (2013)
[185] Puente, R.; Vera de Serio, VN, Locally Farkas-Minkowski linear semi-infinite systems, Top, 7, 103-121, (1999) · Zbl 0936.15012
[186] Remez, E., Sur la détermination des polynômes d’approximation de degré donné (in French), Commun Soc Math Kharkoff and Inst Sci Math et Mecan, 10, 41-63, (1934)
[187] Robinson, SM, Some continuity properties of polyhedral multifunctions. Mathematical programming at Oberwolfach (Proc. Conf., Math. Forschungsinstitut, Oberwolfach, 1979), Mathematical Programming Studies, 14, 206-214, (1981)
[188] Rockafellar, R. T., & Wets, R. J. B. (1998). Variational analysis. Berlin: Springer.
[189] Rozza, G.; Huynh, DBP; Patera, AT, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Archives of Computational Methods in Engineering, 15, 229-275, (2008) · Zbl 1304.65251
[190] Rubinstein, GS, A comment on Voigt’s paper a duality theorem for linear semi-infinite programming (in Russian), Optimization, 12, 31-32, (1981)
[191] Rudolph, H., Some applications of the semi-infinite simplex algorithm, Optimization, 64, 1739-1757, (2015) · Zbl 1337.90041
[192] Shani, B.; Solan, E., Strong approachability, Journal of Dynamics & Games, 71, 507-535, (2014) · Zbl 1403.91040
[193] Singh, C.; Sarkar, S.; Aram, A.; Kumar, A., Cooperative profit sharing in coalition-based resource allocation in wireless networks, IEEE/ACM Transactions on Networking, 20B, 69-83, (2012)
[194] Sommer, B.; Dingersen, T.; Gamroth, C.; Schneider, SE; Rubert, S.; Krüger, J.; etal., CELLmicrocosmos 2.2 MembraneEditor: A modular interactive shape-based software approach to solve heterogenous membrane packing problems, Journal of Chemical Information and Modeling, 5, 1165-1182, (2009)
[195] Sonnenburg, S.; Rätsch, G.; Schäfer, C.; Schölkopf, B., Large scale multiple kernel learning, Journal of Machine Learning Research, 7, 1531-1565, (2006) · Zbl 1222.90072
[196] Stein, O., How to solve a semi-infinite optimization problem, European Journal of Operational Research, 223, 312-320, (2012) · Zbl 1292.90300
[197] Suakkaphong, N.; Dror, M., Managing decentralized inventory and transshipment, Top, 19, 480-506, (2011) · Zbl 1257.90007
[198] Summerfield, NS; Dror, M., Stochastic pogramming for decentralized newsvendor with transshipment, International Journal of Production Economics, 137, 292-303, (2012)
[199] Tan, M.; Tsang, IW; Wang, L., Towards ultrahigh dimensional feature selection for big data, Journal of Machine Learning Research, 15, 1371-1429, (2014) · Zbl 1318.68156
[200] Thibault, L., Sequential convex subdifferential calculus and sequential Lagrange multipliers, SIAM Journal on Control and Optimization, 35, 1434-1444, (1997) · Zbl 0891.90138
[201] Tian, Y.; Chen, Y. (ed.); Immorlica, N. (ed.), Strategy-proof and efficient ofline interval scheduling and cake, 436-437, (2013), New York · Zbl 06385637
[202] Tian, Y.; Shi, Y.; Liu, X., Recent advances on support vector machines research, Technological and Economic Development of Economy, 18, 5-33, (2012)
[203] Todd, MJ, Interior-point algorithms for semi-infinite programming, Math Programing, 65A, 217-245, (1994) · Zbl 0831.90114
[204] Todorov, M. I. (1985/86). Generic existence and uniqueness of the solution set to linear semi-infinite optimization problems. Numerical Functional Analysis and Optimization, \(8\), 27-39.
[205] Tong, X.; Wu, S-Y; Zhou, R., New approach for the nonlinear programming with transient stability constraints arising from power systems, Computational Optimization and Applications, 45, 495-520, (2010) · Zbl 1198.90361
[206] Tong, X.; Ling, Ch; Qi, L., A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints, Journal of Computational and Applied Mathematics, 217, 432-447, (2008) · Zbl 1160.65029
[207] Toriello, A.; Uhan, NA, Dynamic linear programming games with risk-averse players, Mathematical Programming, 163A, 25-56, (2017) · Zbl 1366.91029
[208] Trofimov, S.; Ivanov, A.; Fettser, Y., An infinitesimal approach for analysis of convex optimization problem with duality gap, CEUR Workshop Proceedings, 1987, 570-577, (2017)
[209] Tunçel, L.; Todd, MJ, Asymptotic behavior of interior-point methods: A view from semi-infinite programming, Mathematics of Operations Research, 21, 354-381, (1996) · Zbl 0857.90085
[210] Uhan, NA, Stochastic linear programming games with concave preferences, European Journal of Operational Research, 243, 637-646, (2015) · Zbl 1346.90649
[211] Vanderbei, RJ, Affine-scaling trajectories associated with a semi-infinite linear program, Mathematics of Operations Research, 20, 163-174, (1995) · Zbl 0835.90108
[212] Vaz, A.; Fernandes, E.; Gomes, M., A sequential quadratic programming with a dual parametrization approach to nonlinear semiinfinite programming, Top, 11, 109-130, (2003) · Zbl 1069.90101
[213] Vaz, A.; Fernandes, E.; Gomes, M., SIPAMPL: Semi-infinite programming with AMPL, ACM Transactions on Mathematical Software, 30, 47-61, (2004) · Zbl 1068.90001
[214] Vercher, E., Portfolios with fuzzy returns: Selection strategies based on semi-infinite programming, Journal of Computational and Applied Mathematics, 217, 381-393, (2008) · Zbl 1136.91458
[215] Vinh, NT; Kim, DS; Tam, NN; Yen, ND, Duality gap function in infinite dimensional linear programming, Journal of Mathematical Analysis and Applications, 437, 1-15, (2016)
[216] Wang, Y.; Ni, H., Multivariate convex support vector regression with semi-definite programming, Knowledge-Based Systems, 30, 87-94, (2012)
[217] Wu, S-Y; Li, DH; Qi, LQ; Zhou, GL, An iterative method for solving KKT system of the semi-infinite programming, Optimization Methods and Software, 20, 629-643, (2005) · Zbl 1127.90410
[218] Xu, Y.; Sun, W.; Qi, LQ, On solving a class of linear semi-infinite programming by SDP method, Optimization, 64, 603-616, (2015) · Zbl 1311.90160
[219] Yamangil, E.; Altinel, IK; Çekyay, B.; Feyzioğlu, O.; Özekici, S., Design of optimum component test plans in the demonstration of diverse system performance measures, IIE Transactions, 43, 535-546, (2011)
[220] Yiu, KFC; Gao, MJ; Shiu, TJ; Wu, SY; Tran, T.; Claesson, I., A fast algorithm for the optimal design of high accuracy windows in signal processing, Optimization Methods and Software, 28, 900-916, (2013) · Zbl 1307.90187
[221] Yu, G., & Yang, Y. (2017). Dynamic routing with real-time traffic information. Operational Research. https://doi.org/10.1007/s12351-017-0314-9.
[222] Zălinescu, C. (2002). Convex analysis in general vector spaces. Singapore: World Scientific. · Zbl 1023.46003
[223] Zhang, L.; Wu, S-Y; López, MA, A new exchange method for convex semi-infinite programming, SIAM Journal on Optimization, 20, 2959-2977, (2010) · Zbl 1229.90247
[224] Zhang, Q., Strong duality and dual pricing properties in semi-infinite linear programming: A non-Fourier-Motzkin elimination approach, Journal of Optimization Theory and Applications, 175, 702-17, (2017) · Zbl 1387.90250
[225] Zheng, XY; Ng, KF, Metric regularity and constraint qualifications for convex inequalities on Banach spaces, SIAM Journal on Optimization, 14, 757-772, (2003)
[226] Zhu, Y.; Huang, GH; Li, YP; He, L.; Zhang, XX, An interval full-infinite mixed-integer programming method for planning municipal energy systems: A case study of Beijing, Applied Energy, 88, 2846-2862, (2011)
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