A Matrix Stability Problem

PROBLEMS AND SOLUTIONS 295

and with the added condition that each of the variables xu and xi takes on onlythe values of 0 or 1.

If one ignores this last 0-1 condition and solves the problem as though it werea linear programming problem, then one will find that very often (but not always)an optimal extreme point solution to this linear programming problem will infact be a 0-1 extreme point [2]. suspect that this is due to the fact that most of theextreme points of the polyhedron determined by constraints (1) are in fact 0-1extreme points, but I cannot prove this. This in turn suggests the following prob-lems"

1. What are the smallest n and k for which there exists a linear programmingproblem of the above form which will have only non-0-1 optimal extremepoint solutions?

2. Can the non-0-1 extreme points of polyhedrons determined by the con-straints in (1) be characterized in any set theoretic way that would be usefulin developing more efficient algorithms for solving this facility locationproblem?

REFERENCES

[1] C. REVELLE AND R. SWAIN, Centralfacilities location, Geographical Analysis, 2(1970), Jan.[2] C. REVELLE, D. MARKS AND J. C. LIEBMAN, An analysis ofprivate andpublic sector location models,

Management Sci., 16 (1970), pp. 6292-6307.

Problem 76-8, A Matrix Inequality, by W. ANDERSON, JR. and G. TRAPP (WestVirginia University).

Let A and B be Hermitian positive definite (HD) matrices. Write A _>_ B ifA B is HD. Show that

A -1 + B -1 _>_ 4(A + B)-1.

Problem 76-9*, A Matrix Stability Problem, by S. VENIT (California State Univer-

sity at Los Angeles).

Let

be a real, square matrix of order 2n, partitioned into four n x n blocks. Assumethat I and 0 are the identity and null matrices (of order n), respectively, and thatthe only nonzero elements of B and C are given by bi 2rj/(1 + 2r) whenli -Jl 1, and cij (1 2r)/(1 + 2rj) when j (i,j 1, 2,..., n), where therj are arbitrary positive numbers.

Show either that the spectral radius of P is less than 1 for all positive integersn, or find a counterexample.

This problem arose in considering the matrix stability of a DuFort-Frankel-type difference scheme. See Numerical Methodsfor Partial Differerential Equations,by W. F. Ames, Barnes and Noble, New York, 1969. In this reference, the authorshows that the spectral radius is less than 1 in the special case when all rj areequal.

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