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

By James Renegar

I'm a working towards aerospace engineer and that i chanced on this e-book to be lifeless to me. It has nearly no examples. convinced, it has a whole lot mathematical derivations, proofs, theorms, and so forth. however it is dead for the kind of Interior-Point difficulties that i have to remedy each day.

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In the algorithm, we increase r}\ to a value 772 and then apply Newton's method to approximate 2(772), thus obtaining a point KI. ), we have the barrier method. One would like 772 to be much larger than 771. However, if 772 is "too" large relative to 771, Newton's method can fail to approximate 2(772); in fact, it can happen that x^ $. Z)/, bringing the algorithm to a halt. The main goal in analyzing the barrier method is to prove that 772 can be larger than 771 by a reasonable amount without the algorithm losing sight of the central path.

For an LP the most important self-concordant functionals are those of the form where r] > 0 is a fixed constant, / is the logarithmic barrier function for the nonnegative orthant, and L :— {x : Ax = b}. (Of course L is an affine space, a translation of a subspace; L is a subspace iff b = 0. 2. ) Another important self-concordant functional is the "logarithmic barrier function for the cone of psd matrices" in S"xn. , the pd matrices in S"xn). To prove self-concordance, it is natural to rely on the trace product, for which we know H(X)AX = X~l(AX)X~1.

Implies ||n^(jci)|| Xl < |, precisely the criterion we assumed xi to satisfy for the barrier method. i). 3 to /,,, yields The barrier method moves from x\ to x\ + nm(x\), where 772 = (1 + l/S^/^/)^!. The length of the barrier method step is thus In taking the step, the barrier method decreases the objective function value from {c, jci) to (c, x\ + «^ 2 (jci)); hence, the decrease is at most Assuming a > | for the predictor-corrector method, the predictor step is direction — cxi and has length at least ^, a consequence of Bx^ (x\, 1) c D/.

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