This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: braceleftBigg 6.079/6.975, Fall 2009–10 S. Boyd & P. Parrilo Homework 5 additional problems 1. Heuristic suboptimal solution for Boolean LP. This exercise builds on exercises 4.15 and 5.13 in Convex Optimization , which involve the Boolean LP minimize c T x subject to Ax precedesequal b x i ∈ { , 1 } , i = 1 , . . . , n, with optimal value p ⋆ . Let x rlx be a solution of the LP relaxation minimize c T x subject to Ax precedesequal b 0 precedesequal x precedesequal 1 , so L = c T x rlx is a lower bound on p ⋆ . The relaxed solution x rlx can also be used to guess a Boolean point ˆ x , by rounding its entries, based on a threshold t ∈ [0 , 1]: rlx 1 x i ≥ t x ˆ i = 0 otherwise , for i = 1 , . . ., n . Evidently ˆ x is Boolean ( i.e. , has entries in { , 1 } ). If it is feasible for the Boolean LP, i.e. , if Ax ˆ precedesequal b , then it can be considered a guess at a good, if not optimal, point for the Boolean LP. Its objective value, U = c T x ˆ, is an upper bound on p ⋆ . If U and L are close, then ˆ x is nearly optimal; specifically, ˆ x cannot be more than ( U − L )suboptimal for the Boolean LP. This rounding need not work; indeed, it can happen that for all threshold values, ˆ x is infeasible. But for some problem instances, it can work well. Of course, there are many variations on this simple scheme for (possibly) constructing a feasible, good point from x rlx . Finally, we get to the problem. Generate problem data using rand(’state’,0); n=100; m=300; A=rand(m,n); b=A*ones(n,1)/2; c=rand(n,1); You can think of x i as a job we either accept or decline, and − c i as the (positive) revenue we generate if we accept job i . We can think of Ax precedesequal b as a set of limits on 1 summationdisplay...
View
Full
Document
This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.
 Fall '06
 boyde

Click to edit the document details