On stability of lexicographical solution of vector minimax quadratic Boolean problem (Q2906894)

The paper deals with a lexicographical ordering by using \(k=1,2,\dots,s\) quadratic order functionals \(f_k(x,A_k)=\max_{y\in Y}y^TA_kx\) where \(Y,X\) are finite sets in finite dimensional real spaces and extensively estimates an introduced stability radius \(\rho\) by upper and lower bounds. The introduced stability do not seem to be a sustainable and elaborate concept. The finally introduced strict ordering yields (as can easily be shown) that the solution set \(S^s(A)\) is single valued or empty and it is defined only by using the first order functional \(f_1\). However the radius \(\rho\) is positive (equivalent to stability) if and only if \(S^s(A)\neq \emptyset\). Hence the paper essentially considers the very special case that already the first ordering in the \(s\)-time lexicographical order yields a single minimal element in \(X\) and thus further \(s-1\) orderings and finally the lexicographic ordering are unsubstantial in this concept.