Abstract
Let (X, ) be a Hilbert space, M/sup T/ = M : X/spl rarr/ X linear self-adjoint map, u/sub 0/, u/sub 1/,......u/sub m//spl isin/ X, a /spl isin/ X, Z be a closed vector subspace in X. Consider the following optimization problem: f(x) = ((( )/2)+( )+((1/2)/spl Sigma/(/sub i,j=1/,/sup m/)l/sub ij/ )) /spl rarr/ min (1) x /spl isin/ a + Z (2). We assume that the matrix L = (l/sub ij/) is symmetric and f is convex. Suppose that we can solve (numerically or analytically) the problem of the form f/sub v/(x) = ((( )/2) + ( )) (3) x/spl isin/ a + Z (4) for any v /spl isin/ X. The question we address in this paper is whether it is possible to obtain the optimal solution to (1), (2) based on the information obtained by solving problems (3), (4).
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