A mathematical view of interior-point methods in convex - download pdf or read online

By James Renegar

ISBN-10: 0898715024

ISBN-13: 9780898715026

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

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

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3. Barrier Functionals 41 Proof. For brevity, let s := sym(x, Df). Assuming jc, y e Df, note that s)(x — y) € Df (the closure of £>/) since w, x, y are colinear and Since By(y, 1) c Df and D/ is convex, we thus have that is, By(x, y^) c Df. 3. 9 shows that /(*,) -> oo if / is a self-concordant functional and {*,} converges to a point in the boundary of Df. To close this subsection, we present a theorem that implies the rate at which /(*,•) goes to oo is "slow" if / is a barrier functional. 8. Assume f e SCB andx e Df.

Assume x' e Df, a point at which to initiate the barrier method. 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.

2. 8. If f e SC and the values of f are bounded from below, then f has a minimizer. ) Proof. 5 now implies / to have a minimizer. The conclusion of the next theorem is trivially verified for important self-concordant functionals like those obtained by adding linear functionals to logarithmic barrier functions. Whereas our definition of self-concordance plays a useful role in simplifying and unifying the analysis of Newton's method for many functionals important to ipm's, it certainly does not simplify the proof of the property established in the next theorem for those same functionals.

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

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