Please Check the Solution is right or not 5.30 Derive the KKT conditions for the

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Please Check the Solution is right or not

5.30 Derive the KKT conditions for the problem minimize trX log det X subject to Xs=y Rn are given, with 8TV with variable X e S" and domain S.. y Verify that the optimal solution is given by R', and 8 1 Solution. We introduce a Lagrange multiplier z E R" for the equality constraint. The KKT optimality conditions are: (5.30.A) from the condition Xs = y. Multiplying the gradient equation on (5.30.B) yTy We first determine the right with y gives By taking the inner product with y on both sides and simplifying, we getTy Substituting in (5.30.B) we get 1 and substituting this expression for z in (5.30.A) gives Finally we verify that this is the inverse of the matrix X* given above: To complete the solution, we prove that X* 0. An easy way to see this is to note that

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