# Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems

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## 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.

### Keywords

Stability Well-posedness Linear inequality systems Distance to ill-posedness Regularity## Preview

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### References

- 1.Anderson, E.J., Nash, P.: Linear Programming in Infinite Dimensional Spaces: Theory and Applications, Wiley, Chichester (UK), 1987Google Scholar
- 2.Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, 1993Google Scholar
- 3.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)CrossRefGoogle Scholar - 4.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)CrossRefGoogle Scholar - 5.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)Google Scholar - 6.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, 2004Google Scholar
- 7.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)CrossRefGoogle Scholar - 8.Dontchev, A.L., Lewis, A.S., Rockafellar R.T.: The radius of metric regularity, Trans. Amer. Math. Soc.
**355**(2), 493–517 (2002)Google Scholar - 9.Duffin, R.J.: Infinite programs. In: H.W. Kuhn, A.W. Tucker, (eds.), Linear Equalities and Related Systems, Princeton University Press, Princeton, 1956, pp. 157–170Google Scholar
- 10.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)Google Scholar - 11.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)CrossRefGoogle Scholar - 12.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)CrossRefGoogle Scholar - 13.Goberna, M.A., López, M.A.: Topological stability of linear semi-infinite inequality systems. J. Optimization Theory Appl.
**89**, 227–236 (1996)Google Scholar - 14.Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization, John Wiley and Sons, Chichester (UK), 1998Google Scholar
- 15.Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl.
**17**, 730–743 (1996)Google Scholar - 16.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)CrossRefGoogle Scholar - 17.Hiriart-Urruty, J.B., Lemarechal, C. (1993): Convex Analysis and Minimization Algorithms I, Springer-Verlag, New York.Google Scholar
- 18.Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. National Bureau of Standards
**49**, 263–265 (1952)Google Scholar - 19.Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Transactions of the American Mathematical Society
**266**, 1–56 (1981)Google Scholar - 20.Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications, Kluwer Academic Publishers, Dordrecht (NL), 2002Google Scholar
- 21.Luenberger, D.G.: Optimization by Vector Space Methods, John Wiley and Sons, New York (USA), 1969Google Scholar
- 22.Nunez, M.A.: A characterization of ill-posed data instances for convex programming. Math. Program.
**91**(2), 375–390 (2002)CrossRefGoogle Scholar - 23.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)Google Scholar - 24.Peña, J.: Understanding the geometry of infeasible perturbations of a conic linear system. SIAM J. Optim.
**10**(2), 534–550 (2000)Google Scholar - 25.Renegar, J.: Some perturbation theory for linear programming. Math. Program.
**65**A, 73–91 (1994)Google Scholar - 26.Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Program.
**70**, 279–351 (1995)Google Scholar - 27.Robinson, S.M.: Stability theory for systems of inequalities. Part I: linear systems. SIAM J. Numer. Anal.
**12**, 754–769 (1975)Google Scholar - 28.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ, 1970Google Scholar
- 29.Rockafellar, R.T., Wets R.J.-B.: Variational Analysis. Springer-Verlag, Berlín, 1998Google Scholar
- 30.Tuy, H.: Stability property of a system of inequalities. Math. Oper. Statist. Series Opt.
**8**, 27–39 (1977)Google Scholar - 31.Vera, J.R.: Ill-posedness and the complexity of deciding existence of solutions to linear programs. SIAM J. Optim.
**6**(3), 549–569 (1996)Google Scholar - 32.Yong-Jin, Z.: Generalizations of some fundamental theorems on linear inequalities. Acta Math. Sin.
**16**, 25–40 (1966)Google Scholar