Homework 9 Solutions

Homework 9 Solutions - Math 171A: Mathematical Programming...

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Unformatted text preview: Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #9 Due Friday March 13, 2009 The final will be held Wednesday, March 18 at 3pm. Starred exercises require the use of Matlab . Exercise 9.1. * For the following linear program, construct an equivalent standard-form problem in which the objective function is minimized and the constraints have the form Ax = b , x ≥ 0. Starting at the vertex where all slack variables are basic, solve the problem using the standard-form simplex method (Algorithm 5.1 in the Class Text). At every iteration, record (i) the values of the basic variables; (ii) the indices of the basic and nonbasic variables; (iii) the multiplier π and the reduced cost vector z N ; (iv) the search direction p B ; and (v) the step to the nearest constraint. maximize- 10 x 1 + 32 x 2 + 48 x 3 + 54 x 4 subject to 2 x 1 + 3 x 2 + 5 x 3 + x 4 ≤ 24 5 x 1 + 2 x 2 + x 3 + 3 x 4 ≤ 32 8 x 1 + 5 x 2 + 6 x 3 + 10 x 4 ≤ 64 3 x 1 + 6 x 2 + 9 x 3 + 12 x 4 ≤ 81 x 1 , x 2 , x 3 , x 4 ≥ . >> A A = 2 3 5 1 1 5 2 1 3 1 8 5 6 10 1 3 6 9 12 1 >> b b = 24 32 64 81 >> c’ ans = 10-32-48-54 >> basic basic = 5 6 7 8 >> nonbasic nonbasic = 1 2 3 4 2 Mathematics 171A >> B=A(:,basic) B = 1 1 1 1 >> xB=solve(B,b) xB = 24 32 64 81 >> % Note that we can start with this xB since b>=0. >> N=A(:,nonbasic) N = 2 3 5 1 5 2 1 3 8 5 6 10 3 6 9 12 >> cB=c(basic) cB = >> c_N=c(nonbasic) c_N = 10-32-48-54 >> pi=solve(B’,cB) pi = >> z_N=c_N-N’*pi z_N = 10-32-48-54 >> s=4 s = 4 >> nu_s=nonbasic(s) nu_s = 4 >> pB=solve(B,-A(:,nu_s)) pB = Homework #9 3-1-3-10-12 >> sigma=xB./(-pB) sigma = 24.0000 10.6667 6.4000 6.7500 >> [alpha,t]=min(sigma) alpha = 6.4000 t = 3 >> xB=xB+alpha*pB xB = 17.6000 12.8000 4.2000 >> xB(t)=alpha xB = 17.6000 12.8000 6.4000 4.2000 >> beta_t=basic(t) beta_t = 7 >> basic(t)=nu_s basic = 5 6 4 8 >> nonbasic(s)=beta_t nonbasic = 1 2 3 7 >> % One iteration is complete. >> B=A(:,basic) B = 1 1 1 3 10 12 1 >> N=A(:,nonbasic) N = 2 3 5 5 2 1 8 5 6 1 3 6 9 >> cB=c(basic) cB = 4 Mathematics 171A-54 >> c_N=c(nonbasic) c_N = 10-32-48 >> pi=solve(B’,cB) pi =-5.4000 >> z_N=c_N-N’*pi z_N = 53.2000-5.0000-15.6000 5.4000 >> s=3 s = 3 >> nu_s=nonbasic(s) nu_s = 3 >> pB=solve(B,-A(:,nu_s)) pB =-4.4000 0.8000-0.6000-1.8000 >> decreasing=[1 3 4] decreasing = 1 3 4 >> sigma=zeros(4,1); >> sigma(2)=inf; >> sigma(decreasing)=xB(decreasing)./(-pB(decreasing)); >> sigma sigma = 4.0000 Inf 10.6667 2.3333 >> [alpha,t]=min(sigma) alpha = 2.3333 t = 4 Homework #9 5 >> beta_t=basic(t) beta_t = 8 >> xB=xB+alpha*pB xB = 7.3333 14.6667 5.0000 >> xB(t)=alpha xB = 7.3333 14.6667 5.0000 2.3333 >> basic(t)=nu_s basic = 5 6 4 3 >> nonbasic(s)=beta_t nonbasic = 1 2 8 7 >> % Two iterations are complete.>> % Two iterations are complete....
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Homework 9 Solutions - Math 171A: Mathematical Programming...

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