Abstract
This chapter deals with linear systems with an arbitrary (possibly infinite) number of weak and/or strict inequalities and their solution sets, the so-called evenly convex sets, which can be seen as the two faces of a same coin. Section 1.1 provides different characterizations of evenly convex sets and shows that this class of sets enjoys most of the well-known properties of the subclass of closed convex sets. Since the intersection of evenly convex sets belongs to the same family, any set has an evenly convex hull. Section 1.2 is focussed on the operations with evenly convex hulls and their relationships with other hulls. Section 1.3 reviews different types of separation theorems involving evenly convex sets. Section 1.4 provides existence theorems for linear systems with strict inequalities and characterizations of the linear inequalities which are consequence of consistent systems (those systems with nonempty solution set), which allows us to tackle set containment problems involving evenly convex sets. Section 1.5 is aimed to study the so-called evenly linear semi-infinite programming problems (i.e., linear semi-infinite programming problems with strict inequalities). Finally, Sect. 1.6 describes applications to polarity (treated in a detailed way as it was the problem which inspired the concept of evenly convex set), semi-infinite games, approximate reasoning, optimality conditions in mathematical programming, and strict separation of families of sets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abrams R, Kerzner L (1978) A simplified test for optimality. J Optim Theory Appl 25:161–170
Anderson EJ, Goberna MA, López MA (2001) Simplex-like trajectories on quasi-polyhedral convex sets. Math Oper Res 26:147–162
Anderson EJ, Lewis AS (1989) An extension of the simplex algorithm for semi-infinite linear programming. Math Program 44:247–269
Bair J (1980) Strict separation of several convex sets. Bull Soc Math Belg Ser B 32:135–148
Bateman PT, Rådström H, Hanner O et al. (1954) Seminar on convex sets: 1949–1950. Institute for Advanced Study, Princeton
Carver WB (1922) Systems of linear inequalities. Ann Math 23:212–220
Charnes A, Cooper WW, Kortanek K (1965) On representations of semi-infinite programs which have no duality gaps. Manag Sci 12:113–121
Cozman FG (2018) Evenly convex credal sets. Int J Approx Reason 103:124–138
Daniilidis A, Martínez-Legaz JE (2002) Characterizations of evenly convex sets and evenly quasiconvex functions. J Math Anal Appl 273:58–66
Doagooei AR, Mohebi H (2009) Dual characterizations of the set containments with strict cone-convex inequalities in Banach spaces. J Global Optim 43:577–591
Dür M, Jargalsaikhan B, Still G (2016) Genericity results in linear conic programming—a tour d’horizon. Math Oper Res 42:77–94
Eremin II (1995) A finite iterative method for finding an interior point of an algebraic polyhedron, and estimation of the number of steps. Russ Math 39:16–22
Fan K (1968) On infinite systems of linear inequalities. J Math Anal Appl 21:475–478
Fenchel W (1951) Convex cones, sets, and functions. Lecture notes. Princeton University, Princeton
Fenchel W (1952) A remark on convex sets and polarity. Comm Sèm Math Univ Lund Tome Supplémentaire 82–89
Frittelli M, Maggis M (2014) Conditionally evenly convex sets and evenly quasi-convex maps. J Math Anal Appl 413:169–184
Fung GM, Mangasarian OL, Shavlik JW (2002 Knowledge-based support vector machine classifiers. In: Becker S, Thrun S, Obermayer K (eds) Advances in neural information processing systems. NIPS proceedings, pp 537–544
Gale D (1960) The theory of linear economic models. McGraw-Hill, New York
Gale D, Klee V (1959) Continuous convex sets. Math Scand 7:379–391
Ghosh PK (1990) A solution of polygon containment, spatial planning, and other related problems using Minkowski operations. Comput Vis Graph Image Process 49:1–35
Giannessi F, Maugeri A (2013) Variational inequalities and network equilibrium problems. Springer, New York
Goberna MA, Jeyakumar V, Dinh N (2006) Dual characterizations of set containments with strict convex inequalities. J Global Optim 34:33–54
Goberna MA, Jornet V, Puente R, Todorov MI (1999) Analytical linear inequality systems and optimization. J Optim Theory Appl 103:95–119
Goberna MA, Jornet V, Rodríguez MML (2003) On linear systems containing strict inequalities. Linear Algebra Appl 360:151–171
Goberna MA, López MA (1998) Linear semi-infinite optimization. Wiley, Chichester
Goberna MA, López MA (2018) Recent contributions to linear semi-infinite optimization: an update. Ann Oper Res 271:237–278
Goberna MA, López MA, Pastor J, Vercher E (1984) Alternative theorems for infinite systems with applications to semi-infinite games. Nieuw Arch Wisk (4) 2:218–234
Goberna MA, López MA, Volle M (2014) Primal attainment in convex infinite optimization duality. J Convex Anal 21:1043–1064
Goberna MA, López MA, Volle M (2015) New glimpses on convex infinite optimization duality. RACSAM, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales 109:431–450
Goberna MA, Rodríguez MML (2006) Analyzing linear systems containing strict inequalities via evenly convex hulls. Eur J Oper Res 169:1079–1095
Goberna MA, Rodríguez MML, Vicente-Pérez J (2020) Evenly convex sets, and evenly quasiconvex functions, revisited. J Nonlinear Var Anal 4:189–206
Gordan P (1873) Über die Auflösungen linearer Gleichungen mit reelen Coefficienten (German). Math Ann 6:23–28
Gruber G, Kruk S, Rendl F, Wolkowicz H (1997) Presolving for semidefinite program without constraints qualifications. In: Gruber G et al. (eds) Proceedings of second workshop on high performance optimization techniques (HPOP97). Rotterdam
Han D, Rizaldi A, El-Guindy A, Althoff M (2016) On enlarging backward reachable sets via Zonotopic set membership. In: 2016 IEEE international symposium on intelligent control, pp 1–8
Hiriart-Urruty JB, Lemaréchal C (2001) Fundamentals of convex analysis. Springer, Berlin
Jeyakumar V (2003) Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J Optim 13:947–959
Jeyakumar V, Lee GM, Lee JH (2016) Sums of squares characterizations of containment of convex semialgebraic sets. Pac J Optim 12:29–42
Klee V (1968) Maximal separation theorems for convex sets. Trans Am Math Soc 134:133–147
Klee V (1969) Separation and support properties of convex sets—a survey. In: Balakrishnan AV (ed) Control theory and the calculus of variations. Academic Press, New York, pp 235–303
Klee V, Maluta E, Zanco C (2007) Basic properties of evenly convex sets. J Convex Anal 14:137–148
Kortanek KO, Medvedev VG (2001) Semi-infinite programming and applications in finance. In: Floudas CA, Pardalos PM (eds) Encyclopedia of optimization. Springer, Boston
Kostyukova OI (1988) Investigation of the linear extremal problems with continuum constraints (in Russian). Preprint no. 26 (336), Institute of Mathematics, Academy of Sciences of BSSR, Minsk
Kostyukova OI, Tchemisova TV (2012) Optimality criteria without constraint qualifications for linear semidefinite problems. J Math Sci 182:126–143
Kostyukova OI, Tchemisova TV (2014) On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets. Optimization 63:67–91
Kostyukova OI, Tchemisova TV (2017) Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets. J Optim Theory Appl 175:76–103
Kuhn HW (1956) Solvability and consistency for linear equations and inequalities. Am Math Mon 63:217–232
Llorca N, Tijs S, Timmer J (2003) Semi-infinite assignment problems and related games. Math Methods Oper Res 57:67–78
López MA, Vercher E (1986) Convex semi-infinite games. J Optim Theory Appl 50:289–312
Mangasarian OL (2000) Generalized support vector machines. In: Smola AJ, Bartlett PL, Schölkopf B, Schuurmans D (eds) Advances in large margin classifiers. MIT Press, Cambridge, pp 135–146
Mangasarian OL (2002) Set containment characterization. J Global Optim 24:473–480
Martínez-Legaz JE (1983) A generalized concept of conjugation. In: Hiriart-Urruty JB, Oettli W, Stoer J (eds) Optimization, theory and algorithms (Lecture notes in pure and applied mathematics), vol 86. Marcel Dekker, New York, pp 45–59
Martínez-Legaz (1988) Quasiconvex duality theory by generalized conjugation methods. Optimization 19:603–652
Martínez-Legaz JE (1991) Duality between direct and indirect utility functions under minimal hypotheses. J Math Econ 20:199–209
Minkowski H (1896) Geometrie der Zahlen, Erste Lieferung (German). Teubner, Leipzig
Minkowski H (1911) Theorie der Konvexen Körper (German). Insbesondere Begründung ihres Oberflächenbegriffs, Gesammelte Abhandlungen II, Leipzig
Motzkin TS (1936) Beiträge zur theorie der linearen ungleichungen (German). Azriel [Reprinted in: Cantor D, Gordon B, Rothschild B (eds) Theodore S. Motzkin: selected papers. Birkhäuser, Boston, 1983, pp 1–80]
Naraghirad E (2009) On linear systems containing strict inequalities in reflexive Banach spaces. Appl Math Sci 3:2119–2132
Passy U, Prisman EZ (1984) Conjugacy in quasiconvex programming. Math Program 30:121–146
Passy U, Prisman EZ (1985) A convexlike duality scheme for quasiconvex programs. Math Program 32:278–300
Penot JP, Volle M (1990) On quasiconvex duality. Math Oper Res 15:597–625
Połowczuk W, Radzik T, Wiȩcek P (2012) Simple equilibria in semi-infinite games. Int Game Theory Rev 14:1–19
Puente R, Vera de Serio VN (1999) Locally Farkas–Minkowski linear semi-infinite systems. TOP 7:103–121
Rockafellar RT. Polyhedral convex sets with some closed faces missing. Private communication to Victor Klee
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Rockafellar RT (1984) Network flows and monotropic optimization. Wiley-Interscience, New York
Rodríguez MML, Vicente-Pérez J (2011) On evenly convex functions. J Convex Anal 18:721–736
Rubinstein GS (1981) A comment on Voigt’s paper “A duality theorem for linear semi-infinite programming”. Optimization 12:31–32
Sadraddini S, Tedrake R (2019) Linear encodings for polytope containment problems. In: 2019 IEEE 58th conference on decision and control (CDC), Nice, France, pp 4367–4372
Sánchez-Soriano J, Llorca N, Tijs S, Timmer J (2001) Semi-infinite assignment and transportation games. In: Goberna MA, López MA (eds) Semi-infinite programming: recent advances. Kluwer, Dordrecht, pp 349–363
Sánchez-Soriano J, Llorca N, Tijs S, Timmer J (2002) On the core of semi-infinite transportation games with divisible goods. Eur J Oper Res 142:463–475
Schröder J (1970) Range-domain implications for linear operators. SIAM J Appl Math 19:235–242
Schröder J (1972) Duality in linear range-domain implications. In: Shisha O (ed) Inequalities III. Academic Press, New York, pp 321–332
Slater ML (1950) Lagrange multipliers revisited. Cowles commission discussion paper no. 403 (report) [Reprinted in: Giorgi G, Kjeldsen TH (eds) Traces and emergence of nonlinear programming. Birkhäuser, Basel, 2014, pp 293–306]
Soyster AL (1975) A semi-infinite game. Manag Sci 21:806–812
Steinitz E (1914) Bedingt konvergente Reihen und konvexe Systeme II (German). J Reine Angew Math 144:1–40
Stoer J, Witzgall C (1970) Convexity and optimization in finite dimensions I. Springer, Berlin
Suzuki S (2010) Set containment characterization with strict and weak quasiconvex inequalities. J Global Optim 47:273–285
Suzuki S, Kuroiwa D (2009) Set containment characterization for quasiconvex programming. J Global Optim 45:551–563
Suzuki S, Kuroiwa D (2011) On set containment characterization and constraint qualification for quasiconvex programming. J Optim Theory Appl 149:554–563
Szilágyi P (1999) Nonhomogeneous linear theorems of the alternative. Pure Math Appl 10:141–159
Tijs SH (1979) Semi-infinite linear programs and semi-infinite matrix games. Nieuw Arch Wisk (3) 27:197–214
Walley P (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall, London
Wolkowicz H (1983) Method of reduction in convex programming. J Optim Theory Appl 40:349–378
Yang XQ, Yen ND (2010) Structure and weak sharp minimum of the Pareto solution set for piecewise linear multiobjective optimization. J Optim Theory Appl 147:113–124
Zhu YJ (1966) Generalizations of some fundamental theorems on linear inequalities. Acta Math Sin 16:25–39
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Fajardo, M.D., Goberna, M.A., Rodríguez, M.M.L., Vicente-Pérez, J. (2020). Evenly Convex Sets: Linear Systems Containing Strict Inequalities. In: Even Convexity and Optimization. EURO Advanced Tutorials on Operational Research. Springer, Cham. https://doi.org/10.1007/978-3-030-53456-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-53456-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-53455-4
Online ISBN: 978-3-030-53456-1
eBook Packages: Business and ManagementBusiness and Management (R0)