By James Renegar
This compact booklet, throughout the simplifying point of view it provides, will take a reader who understands little of interior-point the right way to close by of the learn frontier, constructing key rules that have been over a decade within the making by way of various interior-point strategy researchers. It goals at constructing a radical realizing of the main common thought for interior-point equipment, a category of algorithms for convex optimization difficulties. The examine of those algorithms has ruled the continual optimization literature for almost 15 years. In that point, the idea has matured enormously, yet a lot of the literature is tough to appreciate, even for experts. by means of focusing merely on crucial components of the speculation and emphasizing the underlying geometry, A Mathematical View of Interior-Point tools in Convex Optimization makes the speculation obtainable to a large viewers, letting them speedy improve a primary figuring out of the cloth.
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Extra resources for A Mathematical View of Interior-Point Methods in Convex Optimization (MPS-SIAM Series on Optimization)
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.
However, by choosing x so that in the line L through x and z, the distance from x to the boundary of Df D L is smaller than the distance from x to z, the containment Bx(x, 1) c Df implies \\z — x\\x > 1, a contradiction. Hence &f > ± for all / e SCB. 5, if / e SCB and Df is unbounded—hence / has no minimizer—then \\gx(x)\\x > I := ^ for all x e Df. It is worth noting that any universal lower bound t as in the preceding paragraph implies a lower bound nt < #/ for each barrier functional / whose domain is the nonnegative orthant K+ + .
The number of exact line searches) in increasing the parameter from an initial value r}\ to some value 77 > 771 is In the case of linear programming where #/ = n, such a bound was first established by Gonzaga  (see also den Hertog, Roos, and Vial ). 17) for the short-step method by a factor y#/. It is one of the ironies of the ipm literature that algorithms which are more efficient in practice often have somewhat worse complexity bounds. 4 Chapter 2. Basic Interior-Point Method Theory A Predicto^Corrector Method The Newton step nn (x) := — r]cx — gx (x) for the barrier method can be viewed as the sum of two steps, one of which predicts the tangential direction of the central path and the other of which corrects for the discrepancy between the tangential direction and the actual position of the (curving) path.