A mathematical view of interior-point methods in convex by James Renegar

By James Renegar

I'm a working towards aerospace engineer and that i discovered this e-book to be lifeless to me. It has almost no examples. sure, it has hundreds mathematical derivations, proofs, theorms, and so forth. however it is dead for the kind of Interior-Point difficulties that i have to remedy each day.

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Example text

2. ) Another important self-concordant functional is the "logarithmic barrier function for the cone of psd matrices" in S"xn. , the pd matrices in S"xn). To prove self-concordance, it is natural to rely on the trace product, for which we know H(X)AX = X~l(AX)X~1. For arbitrary 7 e S"x", keeping in mind that the trace of a matrix depends only on the eigenvalues, we have where A-i < • • • < A,n are the eigenvalues of X~l/2YX~l/2. 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.

The operator norms induced by the local norms. 5. Indeed, the original definition of self-concordance in [15] is phrased as a bound on the third differential. 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.

A barrier functional / : K° —> R is said to be logarithmically homogeneous if for all # e K° and ? > 0, It is easily established that the logarithmic barrier functions for the nonnegative orthant and the cone of psd matrices are logarithmically homogeneous, as are barrier functionals of the form x i—> f(Ax) where / is logarithmically homogeneous. Another important example of a logarithmically homogeneous barrier functional is the domain of this functional being the interior of the second-order cone It has complexity value #/ = 2.

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