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By James Renegar

I'm a training aerospace engineer and that i discovered this booklet to be lifeless 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 clear up every day.

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

Example text

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. In fact, x' = z'(v) for v = 1. Let n'v(;c) denote the Newton step for /„' at x. Rather than increasing the parameter v, we decrease it toward zero, following the central path to the analytic center z of /. From there, we switch to following the central path (2(77) : 77 > 0} as before. We showed rj can safely be increased by a factor of 1 + l/Sy/tf/.

Assuming t > 0 is in the domain of 0, and hence, for all s > 0 in the domain of 0, Thus, that is. Consequently, the domain of the convex functional (j> is contained in the open interval (—00, #//0'(0)). In particular, #//0'(0) is not in the domain. Since s = I is in the domain of 0 we thus have 1 < #//0'(0). 2 Analytic Centers The next theorem implies that for each x in the domain of a barrier functional, the ball Bx(x, 1) is, to within a factor of 4#/ + 1, the largest among all ellipsoids centered at x which are contained in the domain.

Adding / to the functional x \-+ -\n(R - \\x\\2) where R > \\x\\2, \\Xi\\2 (for all 0, one obtains a self-concordant functional / for which D? is bounded and for which lim, f ( x i ) = oo iff lim, /(*,•) = oo. Consequently, we may assume Df is bounded. Assuming Df is bounded, we will construct from {jc,} a sequence {v,} C Df which has a unique limit point, the limit point lying in dDf, and which satisfies 34 Chapter 2. Basic Interior-Point Method Theory Applying the same construction to the sequence {y,-}, and so on, we will thus conclude that if lim, /(*,•) 7^ oo, then / assumes arbitrarily small values, contradicting the lower boundedness of finite-valued convex functional having bounded domains.

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