Abstract.
In this paper we consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space, with a fixed and arbitrary (possibly infinite) index set. This parameter space is endowed with the topology of the uniform convergence of the coefficient vectors by means of an extended distance. Some authors, in a different context in which the index set is finite and, accordingly, the coefficients are bounded, consider the boundary of the set of consistent systems as the set of ill-posed systems. The distance from the nominal system to this boundary (‘distance to ill-posedness’), which constitutes itself a measure of the stability of the system, plays a decisive role in the complexity analysis of certain algorithms for finding a solution of the system. In our context, the presence of infinitely many constraints would lead us to consider separately two subsets of inconsistent systems, the so-called strongly inconsistent systems and the weakly inconsistent systems. Moreover, the possible unboundedness of the coefficient vectors of a system gives rise to a special subset formed by those systems whose distance to ill-posedness is infinite. Attending to these two facts, and according to the idea that a system is ill-posed when small changes in the system’s data yield different types of systems, now the boundary of the set of strongly inconsistent systems arises as the ‘generalized ill-posedness’ set. The paper characterizes this generalized ill-posedness of a system in terms of the so-called associated hypographical set, leading to an explicit formula for the ‘distance to generalized ill-posedness’. On the other hand, the consistency value of a system, also introduced in the paper, provides an alternative way to determine its distance to ill-posedness (in the original sense), and additionally allows us to distinguish the consistent well-posed systems from the inconsistent well-posed ones. The finite case is shown to be a meeting point of our linear semi-infinite approach to the distance to ill-posedness with certain results derived for conic linear systems. Applications to the analysis of the Lipschitz properties of the feasible set mapping, as well as to the complexity analysis of the ellipsoid algorithm, are also provided.
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References
Anderson, E.J., Nash, P.: Linear Programming in Infinite Dimensional Spaces: Theory and Applications, Wiley, Chichester (UK), 1987
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, 1993
Cánovas, M.J., López, M.A., Parra, J., Todorov, M.I.: Stability and well-posedness in linear semi-infinite programming. SIAM J. Optim. 10, 82–98 (1999)
Cánovas, M.J., López, M.A., Parra, J., Todorov, M.I.: Solving strategies and well-posedness in linear semi-infinite programming. Ann. Oper. Res. 101, 171–190 (2001)
Cánovas, M.J., López, M.A., Parra, J: Upper semicontinuity of the feasible set mapping for linear inequality systems. Set-Val. Anal., 10, 361–378 (2002)
Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to insolvability for linear optimization problems. Technical Report, Operations Research Center, Miguel Hernández University of Elche, 2004
Dantzig, G.B., Folkman, J., Shapiro, N.: On the continuity of the minimum set of a continuous function, J. Math. Anal. Appl. 17, 519–548 (1967)
Dontchev, A.L., Lewis, A.S., Rockafellar R.T.: The radius of metric regularity, Trans. Amer. Math. Soc. 355(2), 493–517 (2002)
Duffin, R.J.: Infinite programs. In: H.W. Kuhn, A.W. Tucker, (eds.), Linear Equalities and Related Systems, Princeton University Press, Princeton, 1956, pp. 157–170
Epelman, M., Freund, R.M.: Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system. Math. Program. 88 (3), 451–485 (2000)
Freund, R.M., Vera, J.R.: Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system. Math. Program. 86 (2), 225–260 (1999)
Freund, R.M., Vera, J.R.: Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm, SIAM J. Optim. 10 (1), 155–176 (1999)
Goberna, M.A., López, M.A.: Topological stability of linear semi-infinite inequality systems. J. Optimization Theory Appl. 89, 227–236 (1996)
Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization, John Wiley and Sons, Chichester (UK), 1998
Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17, 730–743 (1996)
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)
Hiriart-Urruty, J.B., Lemarechal, C. (1993): Convex Analysis and Minimization Algorithms I, Springer-Verlag, New York.
Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. National Bureau of Standards 49, 263–265 (1952)
Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Transactions of the American Mathematical Society 266, 1–56 (1981)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications, Kluwer Academic Publishers, Dordrecht (NL), 2002
Luenberger, D.G.: Optimization by Vector Space Methods, John Wiley and Sons, New York (USA), 1969
Nunez, M.A.: A characterization of ill-posed data instances for convex programming. Math. Program. 91 (2), 375–390 (2002)
Nunez, M.A., Freund, R.M.: Condition measures and properties of the central trajectory of a linear program. Math. Program. 83 (1), 1–28 (1998)
Peña, J.: Understanding the geometry of infeasible perturbations of a conic linear system. SIAM J. Optim. 10(2), 534–550 (2000)
Renegar, J.: Some perturbation theory for linear programming. Math. Program. 65A, 73–91 (1994)
Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Program. 70, 279–351 (1995)
Robinson, S.M.: Stability theory for systems of inequalities. Part I: linear systems. SIAM J. Numer. Anal. 12, 754–769 (1975)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ, 1970
Rockafellar, R.T., Wets R.J.-B.: Variational Analysis. Springer-Verlag, Berlín, 1998
Tuy, H.: Stability property of a system of inequalities. Math. Oper. Statist. Series Opt. 8, 27–39 (1977)
Vera, J.R.: Ill-posedness and the complexity of deciding existence of solutions to linear programs. SIAM J. Optim. 6 (3), 549–569 (1996)
Yong-Jin, Z.: Generalizations of some fundamental theorems on linear inequalities. Acta Math. Sin. 16, 25–40 (1966)
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This research has been partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), and Bancaja-UMH (Spain).
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Cánovas, M., López, M., Parra, J. et al. Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math. Program. 103, 95–126 (2005). https://doi.org/10.1007/s10107-004-0519-6
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DOI: https://doi.org/10.1007/s10107-004-0519-6