By James Renegar
I'm a training aerospace engineer and that i stumbled on this booklet to be lifeless to me. It has almost no examples. convinced, it has lots of mathematical derivations, proofs, theorms, and so on. however it is dead for the kind of Interior-Point difficulties that i have to resolve each day.
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Additional info for A mathematical view of interior-point methods in convex optimization
Assuming \\Y - X\\x < 1, all of the values A,y are thus positive, and hence X~l/2YX~1/2 is pd, which is easily seen to be equivalent to Y being pd. Consequently, if ||y - X\\x < 1, then Y <= Df (= §++"), as required by the definition of self-concordance. i. In establishing the bounds for || V||y/|| V\\x in the definition of self-concordance, we rely on the inequality To verify this inequality, let Q be an orthogonal matrix diagonalizing Si so that QS\! QT is a diagonal matrix A~ 1/2 with diagonal entries l/>An" < • • • < 1/V^T- Since the Frobenius norm of a symmetric matrix is determined solely by the eigenvalues, it is immediate that However, as the Frobenius norm of a matrix M satisfies ||M|| — (^ w,27)1/2, it is easily seen that where the last equality once again relies on the Frobenius norm of a symmetric matrix being determined solely by the eigenvalues.
Consider the univariate barrier functional // obtained by restricting / to the line through e in the direction ej. t. t. the restricted inner product. Since Df. is unbounded, and hence // does not have 40 Chapter 2. 3. In light of the preceding paragraphs, we see that with regards to the complexity value, the logarithmic barrier function for the nonnegative orthant R++ is the optimal barrier functional having domain M++. Likewise, viewing R" as a subspace of S nxn , we can conclude that the logarithmic barrier function for the cone of pd matrices is the optimal barrier functional having that cone as its domain.
IfO<€ < 1, then within iterations of the algorithm, all points x computed thereafter satisfy Consider the following modification to the algorithm. Choose V > (c, x'}. 4. Primal Algorithms 49 and whose complexity value does not exceed #/ + 1. In the theorem, the quantity V — val is then replaced by the potentially much smaller quantity V — val. 20) about x'. Finally, we highlight an implicit assumption underlying our analysis, namely, the complexity value ftf is known. The value is used to safely increase the parameter r\.