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We are given a covariance matrix, thus it is positive definite.
We want to
maximize the sum of decrements to the diagonal of the matrix
such that the adjusted matrix is still positive semi-definite.
Application:
A comprehensive test consists of N subtests on different topics. The overall score of a student who takes the test is the sum of the scores on the N subtests.
If a large number of students take a test, one measure of the goodness of the test is how large the covariances among different subtests are relative to the variances (the diagonal) in scores on a test. The scores on the subtests of a well prepared test taker taking a well prepared test should be positively correlated. On the otherhand, if none of the test takers can understand the questions, then the variances in scores will be large, but the covariances between subtest scores will be close to zero. Thus, a measure of interest is how large the variances are relative to the covariances. One measure of this is to ask how much we can decrease the diagonal terms (the variances on each test) so the matrix remains positive semi-definite. For a good test, the diagonal (variances) should be small relative to the off-diagonal (covariances);
Make sure to turn on
Solver --> Options --> Nonlinear Solver --> Quadratic Recognition;
Ref.
Fletcher, R.(1981), "A nonlinear programming problem in statistics(Educational testing)", SIAM J. of Scientific and Statistical Computing, vol. 2, no. 3, pp. 257-267.;