By James Renegar
I'm a working towards aerospace engineer and that i stumbled on this publication to be dead to me. It has nearly no examples. definite, it has a whole lot mathematical derivations, proofs, theorms, and so on. however it is dead for the kind of Interior-Point difficulties that i have to remedy every day.
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Additional info for A mathematical view of interior-point methods in convex optimization
Relying on it, the algorithm is guaranteed to stay on track. It is a remarkable aspect of ipm's that safe values for quantities like ft depend only on the complexity value #/ of the underlying barrier functional /. Concerning LPs, if one relies on the logarithmic barrier function for the strictly nonnegative orthant M+ + , then y = (1 + g^) is safe regardless of A, b, and c. 5, II*—zOOL < |;mus, by the definition of self-concordance, II*—2(77)11^) < |. 4. 3). 14), given € > 0, iterations suffice to produce x satisfying (c, x) < val + € .
Restricting / to the line through x and y, we may assume / is univariate. , g(x) is a nonnegative number. *) + v\\x > |. Since 8x(x) > 0» we have || v \\x < |. 10, we find there exists u satisfying Note that \\u\\x < 1. 3. Barrier Functionals 39 where the last inequality makes use of gx(x) + v and y — x both being nonnegative. However, since H&tOt) + v\\x > | only if v = 0 (and hence only if u = 0), we Thus, from which the theorem is immediate. Minimizers of barrier functionals are called analytic centers.
Finally, we highlight an implicit assumption underlying our analysis, namely, the complexity value ftf is known. The value is used to safely increase the parameter r\. What is actually required is an upper bound & > #/. If one relies on an upper bound $ rather than the precise complexity value #/, then #/ in the theorem must be replaced by #. Except for #/, none of the quantities appearing in the theorem are assumed to be known or approximated. The quantities appear naturally in the analysis of the algorithm, but the algorithm itself does not rely on the quantities.
A mathematical view of interior-point methods in convex optimization by James Renegar