By Umberto Bertelé and Francesco Brioschi (Eds.)

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**Sample text**

Let the sequence of graphs resulting from the eliminations be G = GO,G1, G 2 , .. , G1 = G'. Assume that there exists a path connecting a and b through Y : a, yl', Y,', . . ,Y;, b. This is a path in GO. Let G i be the last graph in which this path is preserved. In the transformation from Gi to Gi+l, one vertex y' is eliminated and an edge is put between the adjacent vertices on the path, on both sides of y ' . Thus, in Gi++'there is still a path from a to b. This property is preserved after each elimination.

2 Consider the nonserial unconstrained problem min[fi(xl? x2) X + f2@2 9 x3) + f3(x3 9 XJ +fdxe 9 Xl) + fXX1 Y %)I where x = {XI, x2, x3,x4}; OXl= 3 2 ; OX%= UZs= ozp= is considered (see Fig. 2b). 2 3. 2. Elimination of Variables One by One: Procedure 50 is an optimal ordering belonging to 3, and that C ( Q ) = c = 4. 10 and C(w) = C ( 9 ) = 4. ~. ,x,} c X; and 1PI = p. The optimization procedure for the solution of this problem consists of fixing one ordering w = (yl,y z ,. . , . ,Y Z - ~ ( P )Thus .

Orderings. 1 The vector formed by the n nonnegative integers I y ( y j I y l , y,, . . , y+*) 1, ordered from the largest to the smallest, is denoted by 40) = (dl(O),d & J ) ,. . Y dll(w)) and called the vector dimension of the ordering w . 2 The scalar is called the number of functional evaluations of the ordering o. 3 The degree of B(w), which is a polynomial in and called the cost of the ordering w . Note that C(o) = d l ( o ) 0, is denoted by C ( w ) + 1. 4 The scalar B(,9) = min B(w) weS, is called the number of functional evaluations of the problem 9.