By James Renegar

I'm a practising aerospace engineer and that i chanced on this booklet to be lifeless to me. It has nearly no examples. convinced, it has a whole bunch mathematical derivations, proofs, theorms, and so forth. however it is dead for the kind of Interior-Point difficulties that i must resolve every day.

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**Extra resources for A mathematical view of interior-point methods in convex optimization**

**Sample text**

Implies ||n^(jci)|| Xl < |, precisely the criterion we assumed xi to satisfy for the barrier method. i). 3 to /,,, yields The barrier method moves from x\ to x\ + nm(x\), where 772 = (1 + l/S^/^/)^!. The length of the barrier method step is thus In taking the step, the barrier method decreases the objective function value from {c, jci) to (c, x\ + «^ 2 (jci)); hence, the decrease is at most Assuming a > | for the predictor-corrector method, the predictor step is direction — cxi and has length at least ^, a consequence of Bx^ (x\, 1) c D/.

The insight gained from the simplicity of the nonnegative orthant justifies the redundancy. 5 we show that the self-concordance of each of these two logarithmic barrier functions is a simple consequence of the original definition of self-concordance due to Nesterov and Nemirovskii [15]. ) To apply the definition of self-concordance in developing the theory, it is useful to rephrase it in terms of Hessians. 1. Assume the functional f has the property that Bx(x, 1) C Df for all x e Df. 2. Self-Concordant Functionals 27 Proof.

1. Assume f is such that for all x e Df we have Bx (x, 1) C Df and is such that whenever y e Bx(x, I) we have Then Proof.