LP relaxations for a class of linear semi-infinite programming problems

References

• Pardalos PM, Vavasis SA. Quadratic programming with one negative eigenvalue is NP-hard. J Global Optim. 1991;1(1):15–22.
   Google Scholar
• Goberna MA. Linear semi-infinite optimization: recent advances. In: Jeyakumar V, Rubinov A, editors. Continuous optimization. New York (NY): Springer; 2005. p. 3–22. (Applied optimization; vol. 99).
   Google Scholar
• Goberna MA, López MA. A comprehensive survey of linear semi-infinite optimization theory. In: Reemtsen R, Rückmann J-J, editors. Semi-infinite programming. Dordrecht: Kluwer Academic Publishers; 1998. p. 3–27. (Nonconvex optimization and its applications; vol. 25).
   Google Scholar
• Goberna MA, López MA. Linear semi-infinite optimization. Chichester: John Wiley & Sons; 1998.
   Google Scholar
• Hettich R, Kortanek KO. Semi-infinite programming: theory, methods, and applications. SIAM Rev. 1993;35(3):380–429.
   Web of Science ®Google Scholar
• López M, Still G. Semi-infinite programming. Eur J Oper Res. 2007;180(2):491–518.
   Web of Science ®Google Scholar
• Reemtsen R, Görner S. Numerical methods for semi-infinite programming: a survey. In: Reemtsen R, Rückmann J-J, editors. Semi-infinite programming. Dordrecht: Kluwer Academic Publishers; 1998. p. 195–275. (Nonconvex optimization and its applications; vol. 25).
   Google Scholar
• Shapiro A. Semi-infinite programming, duality, discretization and optimality conditions. Optimization. 2009;58(2):133–161.
   Web of Science ®Google Scholar
• Parpas P, Rustem B. An algorithm for the global optimization of a class of continuous minimax problems. J Optim Theory Appl. 2009;141(2):461–473.
   Web of Science ®Google Scholar
• Lasserre JB. An algorithm for semi-infinite polynomial optimization. TOP. 2012;20(1):119–129.
   Web of Science ®Google Scholar
• Wang L, Guo F. Semidefinite relaxations for semi-infinite polynomial programming. Comput Optim Appl. 2013;58(1):133–159.
   Web of Science ®Google Scholar
• Xu Y, Sun W, Qi L. On solving a class of linear semi-infinite programming by SDP method. Optimization. 2015;64(3):603–616.
   Web of Science ®Google Scholar
• Lasserre JB. Moments, positive polynomials and their applications. London: Imperial College Press; 2009.
   Google Scholar
• Laurent M. Sums of squares, moment matrices and optimization over polynomials. In: Putinar M, Sullivant S. editors. Emerging applications of algebraic geometry. New York (NY): Springer; 2009. p. 157–270. (The IMA volumes in mathematics and its applications; vol. 149).
   Google Scholar
• Marshall M. Positive polynomials and sums of squares. Providence (RI): American Mathematical Society; 2008. (Mathematical surveys and monographs; vol. 146).
   Google Scholar
• Parrilo PA, Sturmfels B. Minimizing polynomial functions. In: Basu S, Gonzalez-Vega L. editors. Algorithmic and quantitative real algebraic geometry. Providence (RI)American Mathematical Society; 2003. p. 83–99. (DIMACS series in discrete mathematics and theoretical computer science; vol. 60).
   Google Scholar
• Guo F. Semidefinite programming relaxations for linear semi-infinite polynomial programming. Available from: http://arxiv.org/abs/1509.06394,2015
   Google Scholar
• Putinar M. Positive polynomials on compact semi-algebraic sets. Indiana Univ Math J. 1993;42(3):969–984.
   Web of Science ®Google Scholar
• Lasserre JB. Semidefinite programming vs. LP relaxations for polynomial programming. Math Oper Res. 2002;27(2):347–360.
   Web of Science ®Google Scholar
• Lasserre JB. Polynomial programming: LP-relaxations also converge. SIAM J Optim. 2005;15(2):383–393.
   Web of Science ®Google Scholar
• Sherali HD, Adams WP. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J Discrete Math. 1990;3(3):411–430.
   Web of Science ®Google Scholar
• Cassier G. Problèm des moments sur un compact de ℝ et décomposition de polynômes a plusieurs variables. J Funct Anal. 1984;58(3):254–266.
   Web of Science ®Google Scholar
• Handelman D. Representing polynomials by positive linear functions on compact convex polyhedra. Pac J Math. 1988;132(1):35–62.
   Web of Science ®Google Scholar
• Lasserre JB. A new look at nonnegativity on closed sets and polynomial optimization. SIAM J Optim. 2011;21(3):864–885.
   Web of Science ®Google Scholar
• Haviland EK. On the momentum problem for distribution functions in more than one dimension. Am J Math. 1935;57(3):562–568.
   Google Scholar
• Baldoni V, Berline N, De Loera JA, et al. A User’s Guide for LattE integrale v1.7.2, 2013, software package LattE. Available from: http://www.math.ucdavis.edu/latte/
   Google Scholar
• Grimm D, Netzer T, Schweighofer M. A note on the representation of positive polynomials with structured sparsity. Archiv der Mathematik. 2007;89(5):399–403.
   Web of Science ®Google Scholar
• Lasserre JB. Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J Optim. 2006;17(3):822–843.
   Web of Science ®Google Scholar
• Waki H, Kim S, Kojima M, et al. Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J Optim. 2006;17(1):218–242.
   Web of Science ®Google Scholar
• Schrijver A. Theory of linear and integer programming. England: Wiley; 1998.
   Google Scholar
• Guo F, Wang L, Zhou G. Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities. J Global Optim. 2014;58:261–284.
   Web of Science ®Google Scholar
• Bertsekas DP. Convex optimization theory. Belmont (MA): Athena Scientific; 2009.
   Google Scholar
• Charnes A, Cooper WW, Kortanek K. Duality in semi-infinite programs and some works of Haar and Carathéodory. Manage Sci. 1963;9(2):209–228.
   Web of Science ®Google Scholar
• Rogosinski WW. Moments of non-negative mass. Proc Roy Soc London Ser A, Math Phys Sci. 1958;245(1240):1–27.
   Web of Science ®Google Scholar
• Cánovas MJ, López MA, Parra J, et al. Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math Program. 2005;103(1):95–126.
   Web of Science ®Google Scholar
• Cánovas MJ, López MA, Parra J, et al. Distance to ill-posedness in linear optimization via the fenchel-legendre conjugate. J Optim Theory Appl. 2006;130(2):173–183.
   Web of Science ®Google Scholar
• Cánovas MJ, López MA, Parra J, et al. Distance to solvability/unsolvability in linear optimization. SIAM J Optim. 2006;16(3):629–649.
   Web of Science ®Google Scholar
• Cánovas MJ, López MA, Parra J, et al. Ill-posedness with respect to the solvability in linear optimization. Linear Algebra Appl. 2006;416(2):520–540.
   Web of Science ®Google Scholar
• Cánovas MJ, López MA, Parra J, et al. Distance to ill-posedness for linear inequality systems under block perturbations: convex and infinite-dimensional cases. Optimization. 2011;60(7):925–946.
   Web of Science ®Google Scholar
• OPTimization Interface (OPTI) Toolbox. Available from: http://www.i2c2.aut.ac.nz/Wiki/OPTI/index.php/Main/HomePage
   Google Scholar
• CLP solver. Available from: https://projects.coin-or.org/Clp
   Google Scholar
• Ferris MC, Philpott AB. An interior point algorithm for semi-infinite linear programming. Math Program. 1989;43(1):257–276.
   Web of Science ®Google Scholar