1.11 Miscellaneous Topics 37 in the form of a tableau, with the last column representing thebvector as
x1 x2 x3 x4 x5 x6 b x1
x2 x3
⎡
⎢⎣
1 0 0 1 1 −1 5
0 1 0 2 −3 1 3
0 0 1 −1 2 −1 −1
⎤
⎥⎦
Thex2– x5 switch results in the (2, 5) element being thepivot(= −3 here). If we now premultiply the tableau by a matrixTgiven by
⎡
⎢⎢
⎣
1 13 0 0 −13 0 0 23 1
⎤
⎥⎥
⎦
then we arrive at
x1 x2 x3 x4 x5 x6 b x1
x5
x3
⎡
⎢⎢
⎣
1 13 0 53 0 −23 6 0 −13 0 −23 1 −13 −1 0 23 1 13 0 −13 1
⎤
⎥⎥
⎦
from whichx2=x4=x6=0,x1=6,x5= −1,x3=1. The premultiplication byTis equivalent to ERO on the tableau. The structure of theTmatrix can be understood by considering a general situation, wherein the elements of the tableau are denoted by ai j, and we wish to interchange a basic variablexpwith a nonbasic variablexq. Then,apqis the pivot andTis formed with 1’s along the diagonal, 0’s elsewhere, and with theqth column defined as
−a1q
apq • • −a(p−1)q apq
1 apq
−a(p+1)q
apq • • −amq
apq
T
In view of the above, we may express the coefficients of the new tableaua in terms of the old tableauaas
ai j =ai j− ai q
apq ap j and bi =bi− ai q
apq bp, i = p,i =1 tom,j =1 ton (1.17) ap j =ap j
apq and bp= bp
apq, j =1 ton
A FEW LEADS TO OPTIM IZATION SOFTWARE
Book by J.J. More and S.J. Wright, Optimization Software Guide, 1993; see:
http://www.siam.org/catalog/mcc12/more.htm (Series: Frontiers in Applied Mathe- matics 14, ISBN 0-89871-322-6)
www.mcs.anl.gov/ (Argonne National Labs Web site)
www.gams.com (GAMS code)
www.uni-bayreuth.de/departments/
math/org
(by K. Schittkowski)
www.altair.com (Altair corp. – structural optimization) www.mscsoftware.com (MSC/Nastran code for structural
optimization)
www.altair.com (includes topology optimization)
www.vrand.com (by G.N. Vanderplaats – DOC and
GENESIS, the latter code for structural optimization)
SOM E OPTIM IZATION CONFERENCES
INFORMS (ORSA-TIMS) Institute for operations research and the management sciences
SIAM Conferences Society for industrial and applied math NSF Grantees Conferences
WCSMO World Congress on Structural and
Multidisciplinary Optimization
ASMO UK/ISSMO Engineering Design Optimization
AIAA/USAF/NASA Symposium on Multidisciplinary Analysis and Optimization
ASME Design Automation Conference (American Society of Mechanical Engineers)
EngOpt (2008) International Conference on Engineering
Optimization. Rio de Janeiro, Brazil Optimization In Industry (I and II), 1997, 1999
(Industry applications)
Problems 1.1–1.2 39 SOM E OPTIM IZATION SPECIFIC JOURNALS
Structural and Multidisciplinary Optimization (official Journal for ISSMO) AIAA Journal
Journal of Global Optimization Optimization and Engineering Evolutionary Optimization (Online) Engineering Optimization
Interfaces
Journal of Optimization Theory and Applications (SIAM) Journal of Mathematical Programming (SIAM)
COM PUTER PROGRAM S
1) Program GRADIENT – computes the gradient of a function supplied by user in “Subroutine Getfun” using forward, backward, and central differences, and compares this with analytical gradient provided by the user in “Subroutine Gradient”
2) Program HESSIAN – similar to GRADIENT, except a Hessian matrix is output instead of a gradient vector
PROBLEM S P1.1. Consider the problem
minimize f =(x1−3)2+(x2−3)2 subject to x1≥0, x2≥0, x1+2x2≤1
Let the optimum value of the objective to the problem bef∗. Now if the third con- straint is changed tox1+2x2≤1.3, will the optimum objective decrease or increase in value?
P1.2. Consider the problem
minimize f =x1+x2+x3
subject to x1x2−x3≥1
xi ≥0.1, i =1,2,3
Quickly, determine an upper boundMto the optimum solutionf∗.
P1.3. Are the conditions in the Weirstrass theorem necessary or sufficient for a minimum to exist?
P1.4. For the cantilever problem in Example 1.14 in the text, redraw the feasible region (Figure E1.14b) with the additional constraint:x2≤x1/3. Does the problem now have a solution? Are Wierstraas conditions satisfied?
P1.5. Using formulae in Example 1.2, obtainx∗and the shortest distancedfrom the point (1, 2)Tto the liney=x– 1. Provide a plot.
P1.6. In Example 3, beam on two supports, develop a test based on physical insight to verify an optimum solution.
P1.7. Formulate two optimization problems from your everyday activities, and state how the solution to the problems will be beneficial to you.
P1.8. Using the definition, determine if the following vectors are linearly indepen- dent:
⎛
⎜⎝ 1
−5 2
⎞
⎟⎠ ,
⎛
⎜⎝ 0 6
−1
⎞
⎟⎠ ,
⎛
⎜⎝ 3
−3 4
⎞
⎟⎠
P1.9. Determine whether the following quadratic form is positive definite:
f =2x12+5x22+3x32−2x1x2−4x2x3, x∈R3
P1.10. Givenf=[max(0,x– 3)]4, determine the highest value ofnwhereinf∈Cn P1.11. Iff=gTh, wheref: Rn→R1,g: Rn→Rm,h: Rn→Rm, show that
∇f =[∇h]g+[∇g]h
P1.12. Referring to the following graph, isfa function ofxin domain [1, 3]?
3 1
Figure P1.12
P1.13. Show why the diagonal elements of a positive definite matrix must be strictly positive.Hint:use the definition given in Section 1.7 and choose appropriatey’s.
Problems 1.14–1.20 41 P1.14. Ifnfunction evaluations are involved in obtaining the gradient∇fvia for- ward differences, where n equals the number of variables, how many function evaluations are involved in (i) obtaining∇f via central differences as in Eq. (1.8) and (ii) obtaining the Hessian as per Eq. (1.9)?
P1.15. Compare analytical, forward difference, backward difference and central difference derivatives of f with respect to x1 and x2, using program GRADI- ENT for the function f = 10000 (x4x22−2xx13x2
1−x14) at the point x0 = (0.5, 1.5)T. Note: pro- vide the function in Subroutine Getfun and the analytical gradient in Subroutine Gradient.
P1.16. Givenf=100(x2−x12)2+(1−x1)2,x0=(2, 1)T,s=(−1, 0)T.
(i) Write the expression for f(α)≡ f(x(α)) along the direction s,from the pointx0.
(ii) Use forward differences to estimate the directional derivative offalongs atx0; do not evaluate∇f.
P1.17. Plot the derivative ddxλ2 versusεusing a forward difference scheme. Do not consider analytical expression. Takex0=0. For what range ofεdo you see stable results?
1 −1
−1 1
yi =λi
c x x c
yi, i =1,2
P1.18. Repeat P1.17 with central differences, and plot forward and central differ- ences on same figure. Compare your results.
P1.19. Given the inequality constraintg≡100 –x1 x22≤0, and a point on the bound- aryx0=(4, 5)T, develop an expression for:
(i) a linear approximation togatx0and (ii) a quadratic approximation togatx0.
Plot the originalgand the approximations and indicate the feasible side of the curves.
P1.20. Given the inequality constraintg ≡100 –x1 x2 ≤ 0 develop an expression for a linear approximation tog at x0. Plot the original g and the approximation and indicate the feasible side of the curves. Do the problem including plots for two different choices ofx0:
(i) a point on the boundaryx0=(4, 25)Tand (ii) a point not on the boundaryx0=(5, 25)T.
P1.21. Consider the forward difference approximation in Eq. (1.7) and Taylor’s the- orem in Eq. (1.10). Assume that the second derivative offis bounded byMonR1, or∂2f
∂x2≤M.
(i) Provide an expression for the “truncation” error in the forward difference formula.
(ii) What is the dependence of the truncation error on the divided difference parameterh.
P1.22. Using Taylor’s theorem, show why the central difference formula, Eq. (1.8), gives smaller error compared with the forward difference formula in Eq. (1.7).
P1.23. Consider the following structure. Determine the maximum loadPsubject to cable forces being within limits, when:
x
P
Cable Strength
=F = known
massless, rigid platform 1
Figure P1.23
(i) the position of the loadPcan be anywhere between 0≤x≤1 and (ii) the loadPis fixed at the locationx=0.5 m.
(Solve this by any approach.)
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