The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients
Götz E. Alefeld, Владик Крейнович, Günter Mayer · Mathematische Nachrichten · 1998
Abstract A standard system of interval linear equations is defined by Ax = b, where A is an m × n coefficient matrix with (compact) intervals as entries, and b is an m‐dimensional vector whose components are compact intervals. It is known that for systems of interval linear equations the solution set, i. e., the set of all vectors x for which Ax = b for some A ϵ A and b ϵ b, is a polyhedron. In some cases, it makes sense to consider not all possible A ϵ A and b ϵ b, but only those A and b that satisfy certain linear conditions describing dependencies between the coefficients. For example, if we allow only symmetric matrices A (aij = aji), then the corresponding solution set becomes (in general) piecewise‐quadratic. In this paper, we show that for general dependencies, we can have arbitrary (semi)algebraic sets as projections of solution sets.