Linear Programming: Chapter 16
Interior-Point Methods
Robert J. Vanderbei
November 6, 2007
Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
http://www.princeton.edu/∼rvdb
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Linear Programming Interior Point Methods, Lecture Notes - Mathematics, Study notes of Linear Programming
Linear Programming Interior Point Methods, Lecture Notes - Mathematics - Prof. J Vanderbei.pdf, Prof. J Vanderbei, Mathematics, Linear Programming, Interior Point Methods, AT
Typology: Study notes
2010/2011
1 / 16
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Linear Programming Interior Point Methods, Lecture Notes - Mathematics and more Study notes Linear Programming in PDF only on Docsity!
Linear Programming: Chapter 16
Interior-Point Methods
Robert J. Vanderbei
November 6, 2007
Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb
Interior-Point Methods—The Breakthrough
AT&T Patents the Algorithm, Announces
KORBX
What Makes LP Hard?
Primal
maximize cT^ x subject to Ax + w = b x, w ≥ 0
Dual
minimize bT^ y subject to AT^ y − z = c y, z ≥ 0 Complementarity Conditions
xj zj = 0 j = 1, 2 ,... , n wiyi = 0 i = 1, 2 ,... , m
Optimality Conditions
Ax + w = b AT^ y − z = c ZXe = 0 W Y e = 0 w, x, y, z ≥ 0
Ignore (temporarily) the nonnegativities.
2 n + 2m equations in 2 n + 2m unknowns.
Solve’em.
Hold on. Not all equations are linear.
It is the nonlinearity of the complementarity conditions that makes LP fundamentally harder than solving systems of equations.
The Interior-Point Paradigm
Since we’re ignoring nonnegativities, it’s best to replace complementarity with μ-complementarity:
Ax + w = b AT^ y − z = c ZXe = μe W Y e = μe
Start with an arbitrary (positive) initial guess: x, y, w, z.
Introduce step directions: ∆x, ∆y, ∆w, ∆z.
Write the above equations for x + ∆x, y + ∆y, w + ∆w, and z + ∆z:
A(x + ∆x) + (w + ∆w) = b AT^ (y + ∆y) − (z + ∆z) = c (Z + ∆Z)(X + ∆X)e = μe (W + ∆W )(Y + ∆Y )e = μe
Paradigm Continued
Pick a smaller value of μ for the next iteration.
Repeat from beginning until current solution satisfies, within a tolerance, optimality conditions:
primal feasibility b − Ax − w = 0. dual feasibility c − AT^ y + z = 0. duality gap bT^ y − cT^ x = 0.
Theorem.
- Primal infeasibility gets smaller by a factor of 1 − θ at every iteration.
- Dual infeasibility gets smaller by a factor of 1 − θ at every iteration.
- If primal and dual are feasible, then duality gap decreases by a factor of 1 − θ at every iteration (if μ = 0, slightly slower convergence if μ > 0 ).
loqo
Hard/impossible to “do” an interior-point method by hand.
Yet, easy to program on a computer (solving large systems of equations is routine).
LOQO implements an interior-point method.
Setting option loqo options ’verbose=2’ in AMPL produces the fol- lowing “typical” output:
A Generalizable Framework
Start with an optimization problem—in this case LP: maximize^ c
T (^) x subject to Ax ≤ b x ≥ 0 Use slack variables to make all inequality constraints into non- negativities:
maximize cT^ x subject to Ax + w = b x, w ≥ 0 Replace nonnegativity constraints with logarithmic barrier terms in the objective:
maximize cT^ x + μ
j log^ xj^ +^ μ^
i log^ wi subject to Ax + w = b
Incorporate the equality constraints into the objective using Lagrange multipliers:
L(x, w, y) = cT^ x + μ
j
log xj + μ
i
log wi + yT^ (b − Ax − w)
Set derivatives to zero:
c + μX−^1 e − AT^ y = 0 (deriv wrt x) μW −^1 e − y = 0 (deriv wrt w) b − Ax − w = 0 (deriv wrt y)
Introduce dual complementary variables:
z = μX−^1 e
Rewrite system:
c + z − AT^ y = 0 XZe = μe W Y e = μe b − Ax − w = 0
_x g=_* ^0 ∆^ f
maximize f (x) subject to g 1 (x) = 0 g 2 (x) = 0
_x_*
g 2 = 0
∆ g 2 ∆ g 1 ∆ f
∆^ f