m step I wlth the addlflonal con$ramt tnat tne varue or oojecrrve z ls aty. Varying v (over values of v preferred to v2) will give you other points oncurve.

step 3 Inbest valueoff cuwe.

step 1 we obtained one endpoint of the trade-off curve. If weof objective 2 that can be attained, we obtain the other endpoint oi

REI,ITlllGroup A

PROBLI[l/l$

I Show that/("r) : e-' is a convex firnction on -R1.

2 Five ofa store's major customeN are located as in Figure55. Determine \ltlere fte store should be located to minimizethe sum of the squarcs of the distances that each customerwould have to travel to the storc. Can you generalize this resultto the case of, customers located at po {s.x1,x2,...,x,?

3 A company uses a ruw material to produce two t,?es ofproducts. When processed, each unit of raw material yields2 t'mits of Foduct I and I unit of product 2. Ifrl units ofproduct I are produce4 then each unit can be sold for$49 - xr, ifxz units of product 2 are produced, then eachunit can be sold for $30 - 2r2. It costs $5 to purchase andprocess each unit of raw material.

a Use the Kuhn-Tircker conditions to determine howthe company can maximize profits.

b Use LINGO or 'Wolfe's method to determine howthe company can maximize profts.c What is the most that the company would be will-ing to pay for an extra unit of mw material?

- 4 Show that/(.t) : lrl is a convex function on R1.

5 Use Golden Section Search to locate, withi[ 0.5, theoptimal solution to

nax 3x - *s.t. 0<x<5

I Perform two iterations of the method of steepest ascentin an attempt to maximize

f(xu xr) = (\ + xL)e (\+',\ - xlBegin at the point (0,1).

7 The cost of producing .x units of a product during amoqth is x2 dollarc. Find the minimum cost method ofproducitrg 60 units dudng the next thrce months. Can yougeneralize this result to the case where the cost ofproducingr units during a month is an increasing convex function?

8 Solve the following NLP:max z : xyws.t. 2x + 3y + 4w:36

FIGURE 553456

744 cirrrri I 2 ll0nlinear Pr00rdlnlni[[

I Solve the following NLP:mirz = ! + ! + xys.t. *> l,y> 1

10 If a company charges a price p for a pro{uct an6$a on advenising, it can sell 10.000 - 5Vc - l0(of the product. If the product costs $ I 0 per unit to nthen how can the compaly maximize profits?

11 With Z labor hours and M machiqe hours. acar, prodtre Lttl iu?/1 computer disk &ives. Each dislsells for $150. Iflabor can be purchased at $50 per hq1machine hous can be puchased at $100 per hour,how the company can maximize profits.

Group B

,2 Irt time ,, a tree can grow to a size F(t), where ]"0 and F'(D < 0. Assume that for large r,F,(r) is near 0.the tree is cut at time ,, then a revenue F(r) is receivrAssume that revenues are discdunted continuously at a/, so $ I rcceived at time , is equivalent to $e-' received it I

time 0. The goal is to cut the tree at the time r* thaimaximizes discount€d revenue. Show that the tree should be

cut at the time l* satisrying the equation

f '(d)F(C)

h the answer explain *trv tiffifl > /) this equation has a

unique solutioo. Also show that the answer is a maximum"

not a minimum. [Hir,r. Why is it sumcient to choose ,* tomaximize h(e-'rF(r)?l

'13 Suppose we arc hiring a weather forccaster to predict

the probability that next summer will be rainy or surmy. The

following suggests a metltod that can be used to ensue that

the forecaster is accurate. Suppose that the achral probability

ofrain next summer is 4. For simplicity, we assume that the

summer can only be rainy or sunny. If the forccasterarnounces a probability p that the summcr v.ill be raioy'

rheo she receives a palmint of 1 - t I - p)2 if rhe surnmer

is rainy and a payment of I - p' if the sumrner is sumy'

Show ihat the iorecaster will maximize expected profits by

announcing that the probability of a rainy summer is q

14 Show that if D > a -- e, tltel, ab ) 0". Use this result

to show that e' > rr'. (,Fl,r,r.- Show that m*$1 ov"r., >a occurs for r = a.]

RI

Then

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Lueqin

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;4ft[#,'**rl$Ji,.r, iff]"j":ft:tl

#H#-et+;difff#ffidtr#17 Consider the problem

max z = .f(x)s.t. a=x=h

fi ,,iiffi *;T,#iliffi 'dTr;H*,,f;iff:.:",,",J

.1.. -::eeo.s:,fg) is a convex tuncrion that has de va_uves tor all values of.y. Show thatu" oprma ro, Ge l,rri i#J*Ti[;;.".' = ,.^,3^. Yrf*;.71*, O u.onvex fiuction for whi ch f ,k)

mav

^ trJ ffi il &"y#xH,=,:f :i*[f ,*."## fiI

ffi*ffiffi,'*ffi$f,tj.rfi Hl###x:I'',jflIiffi :'#

nu^mber ofunits of raw material that are purcbased and

il::ffi t,l;"3'ff '#;",',li'J:,[ffi t:,,"l;frff i;max w =xgt1(x) * x2p2(t2) _ czs.t. xt s krz

xz s kzz

H.,##Hiillr^*#:*l;:$1;x$s;l^*_9:i:i&, a modified version of the probtem. The

*e*,'+f*f.*,n-##,[-*:^-yl]" T rnrerpretarion oftr, and L, rhar mighr beuserut to the company.s accountant.

Zo. fle-al-&lnalels_lritb sides of tenglb a. b, and (

l"r;j,;;:,1,;,^?lq _.^c)i wlgre s i, r,aitr,. p".#"i".

__" --, aw w, a uunqle wlth sides oi1". yr(".- oX, - bXs _ c.), where s is

i.l,*:,fl :'l::y" & uo ? ;i[ii i,1fl,i#,1"H,""jjfenced area.[:a:crular-shaeedu,*.o"",",ii"'i'"f ;"Hlili::?:

can be sold "tpr1r; a.ri_"!"?il;;:,r;Hii:

tFtRiltcts;Jll1l#i.U*Hf;llasize the theoreticar aspects or

ffi,fir1;:#::),;tr:;;,:y.y#;:iii:ff#;

*iH'.1il:ff::%ir*n r"''& cambridge, Mass. :

n"*3;?n #fl"? !i;; g.P roerunn i n s \ead-

i:," 1l"fi'Pe,'J,Il, ;i&ffi T."XitrJ F ff;:ff r,ol l+" l** li_,|[V' Vo' VE ')

illffitt oo* to ,,*"ize the energy used in compressing

22 Prove Lemma I (use Lagrarge multipliers).

McCormick. C., Nonlinear programming: Theory, Algo_

:i,.,*y\#ji:ri,xffi ,Y*";t?J')"iff :,lJ;i:?i#i%

progromming. Ensrewood crifs.

Tbe fo-llowing.book emphasizes rarious nor inear program_mrng algorithrns:

"'"*ir"i'#J!?i^l,T;7r."0'oo'*.*zs.NewDerhi:

O, Nonlinear piiainizg New york Mc_