Boundedness Theorems for the Relaxation Method

References

• Agmon S. The relaxation method for linear inequalities. Canadian J. Math. (1954) 6:382–392Crossref, Google Scholar
• Aharoni R., Duchet P., Wajnryb B. Successive projections on hyperplanes. J. Math. Anal. Appl. (1984) 103:134–138Crossref, Google Scholar
• Amaldi E. From finding maximum feasible subsystems of linear systems to feedforward neural network design. (1994) . Ph.D. thesis, Department of Mathematics, Swiss Federal Institute of Technology, Lausanne, SwitzerlandGoogle Scholar
• Amaldi E., Agnetis A., Di Pillo G. The maximum feasible subsystem problem and some applications. Modelli e Algoritmi per l’Ottimizzazione di Sistemi Complessi (2003) (Pitagora Editrice Bologna, Italy) 31–69Google Scholar
• Amaldi E., Belotti P., Hauser R., Jünger M., Kaibel V. Randomized relaxation methods for the maximum feasible subsystem problem. Integer Programming and Combinatorial Optimization, Proc. 11th Internat. IPCO Conf., Lecture Notes in Computer Science (2005) 3509(Springer, Berlin, Germany) 249–264Crossref, Google Scholar
• Bauschke H. H., Borwein J. M. Legendre functions and the method of random Bregman projections. J. Convex Anal. (1997) 4:27–67Google Scholar
• Bauschke H. H., Borwein J. M., Lewis A. The method of cyclic projections for closed convex sets in Hilbert space. Contemporary Math. (1997) 204:1–38Crossref, Google Scholar
• Block H. D., Levin S. A. On the boundedness of an iterative procedure for solving a system of linear inequalities. Proc. Amer. Math. Soc. (1970) 26:229–235Crossref, Google Scholar
• Censor Y., Zenios S. A.Parallel Optimization: Theory, Algorithms and Application (1997) (Oxford University Press, New York) Google Scholar
• Chinneck J. Fast heuristics for the maximum feasible subsystem problem. INFORMS J. Comput. (2001) 13:210–213Link, Google Scholar
• Cimmino G. Calcolo approssimato per soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica XVI., Series II, Anno IX (1938) 1:326–333Google Scholar
• Duda R. O., Hart P. E.Pattern Classification and Scene Analysis (1973) (John Wiley & Sons, New York) . (Second edition, 2001.)Google Scholar
• Dunagan J., Vempala S. A simple polynomial-time rescaling algorithm for solving linear programs. Proc. 36th Annual ACM Sympos. Theory Comput. (2004) (ACM Press, New York) 315–320Crossref, Google Scholar
• Effron B. The perceptron correction procedure in nonseparable situations. (1964) . Rome Air Development Center, Technical Report RADC-TDR-63-533Google Scholar
• Gallant S. I. Perceptron-based learning algorithms. IEEE Trans. Neural Networks (1990) 1:179–191Crossref, Google Scholar
• Goffin J.-L. The relaxation method for solving systems of linear inequalities. Math. Oper. Res. (1980) 5:388–414Link, Google Scholar
• Goffin J.-L. On the non-polynomiality of the relaxation method for systems of linear inequalities. Math. Programming (1982) 22:93–103Crossref, Google Scholar
• Greenberg H. J., Murphy F. H. Approaches to diagnosing infeasible linear programs. ORSA J. Comput. (1991) 3:253–261Link, Google Scholar
• Hiriart-Urruty J.-B., Lemaréchal C.Convex Analysis and Minimization Algorithms. Grundlehren der Mathematischen Wissenschaften (1993) (Springer-Verlag, Berlin, Germany) . 305Google Scholar
• Khachiyan L. G. A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR (1979) 244:1093–1096(In Russian.) (English translation. 1979. Soviet Math. Dokl. 20 191–194.)Google Scholar
• Khachiyan L. G. On the complexity of approximating extremal determinants in matrices. J. Complexity (1995) 11:138–153Crossref, Google Scholar
• Lee E. K., Gallagher R. J., Zaider M. Planning implants of radionuclides for the treatment of prostate cancer: An application of MIP. Optima (1999) 61:1–7Google Scholar
• Mangasarian O., Fischer H., Riedmueller B., Schaeffler S. Machine learning via polyhedral concave minimization. Applied Mathematics and Parallel Computing—Festschrift for Klaus Ritter (1996) (Physica-Verlag, Heidelberg, Germany) 175–188Crossref, Google Scholar
• Meller J., Wagner M., Elber R. Solving huge linear programming problems for the design of protein folding potentials. Math. Programming B (2004) 101:301–318Google Scholar
• Minsky M. L., Papert S. A.Perceptrons: An Introduction to Computational Geometry, expanded edition (1988) (MIT Press, Cambridge, MA) Google Scholar
• Motzkin T., Schoenberg I. Y. The relaxation method for linear inequalities. Canadian J. Math. (1954) 6:393–404Crossref, Google Scholar
• Robertson A. P., Robertson W. J.Topological Vector Spaces (1973) (Cambridge University Press, London, UK, and New York) Google Scholar
• Rosenblatt R.Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms (1962) (Spartan Books, Washington, D.C.) Google Scholar
• Rossi F., Sassano A., Smriglio S. Models and algorithms for terrestrial digital broadcasting. Ann. Oper. Res. (2001) 107:267–283Crossref, Google Scholar
• Schrijver A.Theory of Linear and Integer Programming (1987) revised ed.(John Wiley & Sons, Chichester, UK) . Wiley-Interscience Series in Discrete Mathematics and OptimizationGoogle Scholar
• Shor N. Z. On the structure of algorithms for the numerical solution of optimal planning and design problems. (1964) . (In Russian.) Ph.D. dissertation, Cybernetics Institute, Academy of Sciences of the Ukrainian SSR, Kiev, RussiaGoogle Scholar
• Shor N. Z. Utilization of the operation of space dilatation in the minimization of convex functions. Kibernetika (Kiev) (1970) 1:6–12(In Russian.)Google Scholar
• Telgen J. On relaxation methods for systems of linear inequalities. Eur. J. Oper. Res. (1982) 9:184–189Crossref, Google Scholar
• Yudin D. B., Nemirovskii A. S. Informational complexity and efficient methods for the solution of convex extremal problems. Èkonomika i Matematicheskie Metody (1976) 12:357–369(In Russian.)Google Scholar