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16] P. Wolfe, M. Held and H. Crowder, "Validation of subgradient optimization", Mathematical Programming 6 (1974) 62-88. Mathematical Programming Study 3 (1975) 35-55. E. DONATH and P. WOLFE IBM Thomas J. A. Received Revised manuscript received 28 April 1975 Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived. Such a sum is convex but not necessarily everywhere differentiable. A convergent procedure is presented for determining a minimizingpoint of any such sum subject to the condition that the trace of the matrix is held constant.
5] J. E. Donath and P. Y. (November 1973). F. Demyanov, "On the solution of certain minimax problems", Kibernetica 2 (1966). F. M. Rubinov, Approximate methods in optimization problems (American Elsevier, New York, 1970). V. P. McCormick, Nonlinear programming : sequential unconstrained minimization techniques (Wiley, New York, 1968). A. Goldstein, Constructive real analysis (Harper & Row, New York, 1967). [10] M. Held, P. P. Crowder, "Validation of subgradient optimization", Mathematical Programmin0 6 (1) (1974) 62-88.
For any e > 0, define G(d, 8) = {u 9 u = T(Yt(d, e)) + T(Y2(d, e)H) for some H e H,(~)+ rt~)+s(,) a + * }. 1) The sets Yj(d, e ) j = l, 2 as well as the t-multiplicities r(e) and s(e) were defined in Section 3. Clearly, G(d, e) ~_ G(d). Let S(d, e) = c o n v G(d, e), and P S(d, e) and P G(d, e) denote the corresponding projections of these sets onto the constraint
