By James Renegar
I'm a working towards aerospace engineer and that i stumbled on this e-book to be dead to me. It has nearly no examples. definite, it has lots of mathematical derivations, proofs, theorms, and so on. however it is lifeless for the kind of Interior-Point difficulties that i have to resolve every day.
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Extra info for A mathematical view of interior-point methods in convex optimization
We have been assuming that an initial point x\ near the central path is available. What if, instead, we know only some arbitrary point x' e D/? How might we use the barrier method to solve the optimization problem efficiently? We now describe a simple approach, assuming Df is bounded and hence / has an analytic center. Consider the optimization problem obtained by replacing the objective vector c with —#(*'). The central path then consists of the minimizers z'(v) of the self-concordant functional The point x' is on the central path for this optimization problem.
2 Self-Concordancy and Newton's Method The following theorems display the simplifying role the conditions of self-concordance play in analysis. The first theorem bounds the error of the quadratic approximation and the second guarantees progress made by Newton's method. 2. Recall that qx is the quadratic approximation of / at x, that is, where n(x) := —H(x) l g(x) is the Newton step for / at x. 28 Chapter 2. 2. Iff e SC, x e Df and y e Bx(x, 1), tfzen Proof. 3. Assume f e SC andx e Df. Ifz minimizes f and z € Bx(x, I), then satisfies Proof.
A bound on ||jc+ — z\\x+ rather than a bound on ||*+ — z\\x. Given that the definition of self-concordance restricts the norms to vary nicely, it is no surprise that the theorem can easily be transformed into a statement ready for induction. 2. Self-Concordant Functionals 29 and as are immediate from the definition of self-concordance when x e Bz(z, 1), we find as a corollary to the theorem that if ||jc — z\\z < \, then Consequently, if one assumes ||jc — z\\z < \, then and, inductively, where x\ — x+, X 2 , .
A mathematical view of interior-point methods in convex optimization by James Renegar