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...ets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the literature. Far from being complete, we simply mention references =-=[2, 3, 12, 14, 21]-=-. The application of this method to the (ETP) is demonstrated in Section 2. Although the ellipsoid method is known to converge eventually, the iterates (the centers) quite often are unfeasible, and th...

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...ets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the literature. Far from being complete, we simply mention references =-=[2, 3, 12, 14, 21]-=-. The application of this method to the (ETP) is demonstrated in Section 2. Although the ellipsoid method is known to converge eventually, the iterates (the centers) quite often are unfeasible, and th...

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...inner ellipsoids mentioned in Theorem 3.3, the analytic center serves as a good starting point for the so called Dikin's method. 3.4. Dikin's Method. Suppose D (c) is a feasible point. Dikin's method =-=[6]-=- amounts to approximating the (ETP) locally by the following subproblem (33) (34) Maximize e T d� subject to d 2 E(H(D (c) ) ;1 �d (c) ) where d (c) = diag(D (c) )ande := [1�:::�1] T . Optimizing line...

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...ets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the literature. Far from being complete, we simply mention references =-=[2, 3, 12, 14, 21]-=-. The application of this method to the (ETP) is demonstrated in Section 2. Although the ellipsoid method is known to converge eventually, the iterates (the centers) quite often are unfeasible, and th...

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...f a di erentiable matrix function are not themselves di erentiable at points where they coalesce. Furthermore, it has been observed quite so often that at an optimal solution the eigenvalues coalesce =-=[19]-=-. To overcome this di culty we can employ special techniques developed in, for example, [18, 19]. For convex programming problems, however, there are simple and e ective algorithms that do not require...

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...r how to express the positive semi-de nite constraints explicitly with m smooth and convex inequalities i(D) 0 where m is small. Discussion for this class of constraints can be found, for example, in =-=[1, 4, 8, 11, 19]-=-. One naive way of representing (2) is that all its principal minors are non-negative. Such an expression, however, is very expensive. Fletcher has tried to depict the normal cone [8, Formula (4.4)] b...

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...esce. Furthermore, it has been observed quite so often that at an optimal solution the eigenvalues coalesce [19]. To overcome this di culty we can employ special techniques developed in, for example, =-=[18, 19]-=-. For convex programming problems, however, there are simple and e ective algorithms that do not require smooth constraints or di erentiable objectives. For the above (ETP) in particular, the notion o...

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...3 , the boundary of the feasible domain 3 2 and the level curves of are plotted in Figure 1. We rst derive formulas for the gradient r (D) and the Hessian r2 (D). More general results can be found in =-=[4, 16]-=-. Lemma 3.1. The gradient vector of (D) is given by (28) r (D) =diag ; (S ; D) ;1 ; D ;1 : Proof. The derivatives of the second term in (27) is trivial. So the only concern is the partial derivative o...

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...ets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the literature. Far from being complete, we simply mention references =-=[2, 3, 12, 14, 21]-=-. The application of this method to the (ETP) is demonstrated in Section 2. Although the ellipsoid method is known to converge eventually, the iterates (the centers) quite often are unfeasible, and th...

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...ets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the literature. Far from being complete, we simply mention references =-=[2, 3, 12, 14, 21]-=-. The application of this method to the (ETP) is demonstrated in Section 2. Although the ellipsoid method is known to converge eventually, the iterates (the centers) quite often are unfeasible, and th...

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...much easier than that in [8]. The convergence property isnumerically demonstrated.s1. Introduction. The educational testing problem (ETP) is a nonlinear programming problem which arises in statistics =-=[7]-=-. The problem is to determine how much can be subtracted from the diagonal of a given symmetric and positive de nite matrix S such that the resulting matrix is positive semi-de nite. The (ETP) can be ...

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...e is that the solution to each subproblem is readily obtainable. It should be mentioned that recently Glunt has proposed another approach to the (ETP) on the basis of an alternating projection method =-=[11]-=-. A major component in Glunt's method is the use of Dkystra's algorithm [5] for computing projections onto the intersection of convex sets. It can be proved that Glunt's method converges globally at l...

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...r how to express the positive semi-de nite constraints explicitly with m smooth and convex inequalities i(D) 0 where m is small. Discussion for this class of constraints can be found, for example, in =-=[1, 4, 8, 11, 19]-=-. One naive way of representing (2) is that all its principal minors are non-negative. Such an expression, however, is very expensive. Fletcher has tried to depict the normal cone [8, Formula (4.4)] b...

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...e mentioned that recently Glunt has proposed another approach to the (ETP) on the basis of an alternating projection method [11]. A major component in Glunt's method is the use of Dkystra's algorithm =-=[5]-=- for computing projections onto the intersection of convex sets. It can be proved that Glunt's method converges globally at linear rate. Discussion on the ellipsoid method is fairly rich in the litera...

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...ble. Attention is paid to the Dikin's method where a special barrier function and interior ellipsoids for the feasible domain are explicitly formulated. The implementation is much easier than that in =-=[8]-=-. The convergence property isnumerically demonstrated.s1. Introduction. The educational testing problem (ETP) is a nonlinear programming problem which arises in statistics [7]. The problem is to deter...

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...mide nite yet trace(D) mayhave not reached its maximal value. Indeed, one di culty in implementing Dikin's method is that, in contrast to the ellipsoid methods, there is no general stopping criterion =-=[15]-=-. To reduce the risk of hitting boundaries of the feasible domain too soon, we nd it is a good idea to start out the Dikin's method from a point thatismostinterior to the feasible domain. Our numerica...

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...nter, for example, is always a good starting point. 4. Numerical Experiment. We have applied the algorithms discussed in this paper to solve the set of educational testing problems given by Woodhouse =-=[22]-=-. The Woodhouse data set is in general an N m matrix X =[xij] where xij gives the score of student i on subject j. Test problems are generated by selecting various subsets of columns for form the matr...