# The Shape of the Symmetric Solution Set

• Götz Alefeld
• Günter Mayer
Chapter
Part of the Applied Optimization book series (APOP, volume 3)

## Abstract

We give a new deduction of the set of inequalities which characterize the solution set S of real linear systems Ax = b with the n × n coefficient matrix A varying between a lower bound $$\underline A$$ and an upper bound $$\overline A$$, and with b similarly varying between $$\underline b$$ and $$\overline b$$. The idea of this deduction can also be used to construct a set of inequalities which describe the so-called symmetric solution set S sym, i.e., the solution set of Ax = b with A = A T varying between the bounds $$\underline A = {\underline A ^T}$$ and $$\overline A = {\overline A ^T}.$$ This is the main result of our paper. We show that in each orthant S sym is the intersection of S with sets of which the boundaries are quadrics.

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• Götz Alefeld
• 1