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

By James Renegar

I'm a working towards aerospace engineer and that i discovered this booklet to be dead to me. It has almost no examples. certain, it has a whole bunch mathematical derivations, proofs, theorms, and so on. however it is dead for the kind of Interior-Point difficulties that i have to resolve each day.

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Choose rj2 larger than 771, perhaps significantly larger. i which approximates 2(772), the algorithm will generate a finite sequence of points and then let KI := yK. At each point y^, the algorithm will determine if the point is close to z(r]2) by, say, checking whether ||n,, 2 (yjt)IU < \- (We choose the specific value \ becau 50 Chapter 2. ) The point yK will be the first point that is determined to satisfy this inequality. To compute yk+i from y*, the algorithm minimizes the univariate functional This is the place in the algorithm to which the phrase "exact line search" alludes.

3. 5 Logarithmic Homogeneity In Chapter 3, when we tie ipm's to duality theory, attention will often focus on a particular type of barrier functional / whose domain is the interior K° of a closed, convex cone K (if 42 Chapter 2. Basic Interior-Point Method Theory jti, X2 € K and t\,ti > 0, then t\x\ + hxi e £). A barrier functional / : K° —> R is said to be logarithmically homogeneous if for all # e K° and ? > 0, It is easily established that the logarithmic barrier functions for the nonnegative orthant and the cone of psd matrices are logarithmically homogeneous, as are barrier functionals of the form x i—> f(Ax) where / is logarithmically homogeneous.

H*. Let Bx(y, r) denote the closed ball. A functional / is said to be (strongly nondegenerate) self-concordant if for all x e Df we have Bx(x, 1) c Df, and if whenever y e Bx(x, 1) we have Let SC denote the family of functionals thus defined. Self-concordant functionals play a central role in the general theory of ipm's, as was developed in the pioneering work of Nesterov and Nemirovskii [15]. Although our definition of strongly nondegenerate self-concordant functionals is on the surface quite different from the original definition given in [15], it is in fact equivalent except in assuming / € C2 as opposed to the ever-so-slightly stronger assumption in [15] that / is thrice continuously differentiate.

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