Checking robust nonsingularity is NP-hard

We consider the following problem: givenk+1 square matrices with rational entries,A0,A1,...,Ak, decide ifA0+r1A1+···+rkAk is nonsingular for all possible choices of real numbersr1, ...,rk in the interval [0, 1]. We show that this question, which is closely related to the robust stability problem, is NP-hard. The proof relies on the new concept ofradius of nonsingularity of a square matrix and on the relationship between computing this radius and a graph-theoretic problem.

[1]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[2]  J. Spencer,et al.  Minimization of ±1 matrices under line shifts , 1971 .

[3]  J. Spencer.  Probabilistic Methods in Combinatorics , 1974 .

[4]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[5]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[6]  Karl J. Lieberherr,et al.  Complexity of partial satisfaction , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[7]  John C. Doyle.  Analysis of Feedback Systems with Structured Uncertainty , 1982 .

[8]  Francisco Barahona,et al.  A solvable case of quadratic 0-1 programming , 1986, Discret. Appl. Math..

[9]  Svatopluk Poljak,et al.  A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound , 1986, Discret. Math..

[10]  A. Deif.  Sensitivity analysis in linear systems , 1986 .

[11]  Vojtech Rödl,et al.  Tournament Ranking with Expected Profit in Polynomial Time , 1988, SIAM J. Discret. Math..

[12]  J. Rohn.  Systems of linear interval equations , 1989 .

[13]  M. Mansour.  Robust stability of interval matrices , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[14]  B. Mohar,et al.  Eigenvalues and the max-cut problem , 1990 .

[15]  Michel Deza,et al.  Facets for the cut cone I , 1992, Math. Program..

[16]  Charles Delorme,et al.  Laplacian eigenvalues and the maximum cut problem , 1993, Math. Program..