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Step 2 : [Further improve the model] Let C(,&) = S n7(02),where and If p [ q lies on the boundary of I ( p 2 ) , set = p and stop. (If 11 . ) Otherwise, recompute plF] so that (41) is satisfied and either p[3-1 lies strictly interior to C(p2) with or plF-)lies on the boundary of C(,&). Reset x p ] to x [ q +pp). Step 3 : [Test for convergence] If pi71 lies strictly interior to C(p2) and (45) is satisfied or if it is decided that sufficient passes have been made, set p(”j) = p and stop. lies another pass by returning to Step 2.

4 IMPACT OF PARTIAL SEPARABILITY ON LARGE-SCALE OPTIMIZATION 37 The ratios in Tables 3 and 4 are below the corresponding ratios in Table 2. This result can be explained by noting that the ratios in Tables 3 and 4 can be expressed as Tad + Ta1o Thc + Talg ’ where Tad,Talg,and ThCare the computing times for the function and AD-generated gradient evaluation, the vmlm algorithm, and the function and hand-coded gradient evaluation, respectively. Since Tad > The, we have which is the desired result. If Tad and Thcare the dominant costs, the ratio (1 1) should be close to Tad/Thc.

Reset x p ] to x [ q +pp). Step 3 : [Test for convergence] If pi71 lies strictly interior to C(p2) and (45) is satisfied or if it is decided that sufficient passes have been made, set p(”j) = p and stop. lies another pass by returning to Step 2. End of Algorithm Figure 3. Model-reduction Algorithm 51 52 CONN. 4) Assume that AS1 and AS2 hold, that x* is a limit point of the sequence { x ( ~})generated by the Outer-iteration Algorithm and that for i = 1, - - - , m. Then x* is a Kuhn-Tucker ('rst order stationary) point for ( I ) , ( 2 ) and (9)and the corresponding subsequences of { i ( k } and ) { V, \I,( I c ) } converge to a set of Lagrange multipliers, A*, and the gradient of the Lagrangian, ge(x*, A*), for the problem, respectively.

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