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G. e. a m  m orthogonal matrix U, a m  n block diagonal matrix S, a n  n orthogonal matrix V, such that A = USVT. In particular, it results S S¼ 0 0 0 ! 31) with S ¼ diagðs1 ; :::; sn Þ. The diagonal entries of S are the singular values of A. 32) with S À1 SÀ1 ¼ 0 0 0 ! 33) À Á À1 T À1 and SÀ1 ¼ diag sÀ1 1 ; :::; sn . Also matrix VS U is a pseudo-inverse of A. 1). For this class of integral equations, in fact, the solution f(y) does not depend continuously on the given function g(x). 2). 1) the unknown f(y) can be discretized by means of a finite-difference grid composed of n nodes, while the known term g(x) can be discretized on another grid of m > n nodes.

Y be a vector of nf objectives, with X 

It provides some metrics to identify non-dominated solutions, which in turn are given in terms of the design space X. Following this line, a preliminary information about how the objective space is bounded is provided by some characteristic points, which can be determined prior to solving the multiobjective problem. Referring to a min-min problem, as each scalar objective function is bounded and so it is supposed to have a minimum, there exists a vector in the Y space, called ideal objective vector, whose coordinates are the minima of the corresponding single objectives.

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