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73 MUTTIDIMENSIONAT OPTIMIZATION
Aside from one-dimensionalfunctions, optimization also deals with multidimensional functionsRecallfrom Fig7 3a that our visual imageof a one-dimensional search was like a roller coasterFor two-dimensionalcases,the image becomesthat of mountainsand valleys(Fie13b)As in the lbllowing example, MAILAB's graphiccapabilities provide a handy meansto visualizesuchfunctions
EXAMPLE 4 7
OPTIMIZATION
F V i s u o l i z i n gq T w o - D i m e n s i o n q lu n c t i o n Problem Stqtemeni Use MATLAB's graphicalcapabilities display the following to function and visually estimateits minimum in the range-2 < xr < 0 and 0 < r2 < 3: 2 f(xt'x) :2 i xt - xz -'l- xl +2xrrz + xi Solution The fbllowing scriptgenerates contourand meshplotsof the function: - 2 , C) 4 A ) ; i, = l i n s p a c e ( 0 , 3 , 4 0 ) ; x - l i n sp ac e ( , ( t]:' Yl = meshqtrid x'Y) ; z = 2 + x - '+ 2 * x ^ 2 + 2* x * Y + Y ' ' 2 ; { subploL i'7,2,7);
c ! ; = c o n t o l l r ( X , Y , Z ' ) ; ( -l : r b e l ( c s ) ; xlabel ('r-1' ) ;\'l;rbel (':<-2' ) ; (a) Contour ploL' ) ;,Jrid; trtle(' subplol \I,2,2 t; cs=surf c lX,'{,Z); zmin=f l,ror(min (Z)); z m a x - r - - e iI ( m r x( Z ) ) ; x l a b c l { ' y _ 1 ' ) ; : r I a b e l ( ' : ' , _ ), '; z l a b e l title('(b) M es l - r t r ' l o t ' ) ;
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As displayed in lrig 79, both plots indicatc that function has a minimurn value of about -l andr, : l-5 f t r ' , " r 2 ) : 0 t o I l o c a t e da t a b o u t r l :
FIGURE 79
(o) Confourond (b) rneshplc,ls o Mo-dimensionol functron of
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74 CASE STUDY
Techniques multidimensional for unconstrained optimizationcan be classifiedin a we numberof waysFor purposes the present of discussion, will divide them depending on whetherthey require derivative evaluationThose that require derivativesare called (or gradient, descent ascent), methods The approaches do not requirederivativeevalor that uationare callednongradient, or direct, methodsAs describednext the built-in MATLAB functionfniinsearch is a directmethod 731 IVIATLAB Functiohs fminsearch the Standard MMLAB has a function fminsearch that can be usedto determine minifunction It is basedon the Nelder-Meadmethod,which is a lnum of a rnultidimensional direct-search method that uses <lnly function values (does not require derivatives)and A handles non-snrooth obiective functions simpleexpression its syntaxis of fxnrr, fvall = fminsearch( function,xl,x2)
andvalueof theminimum,fttnction is thenameof where xmrn and frzal arethe location the tunctionbeing evaluated, and xl and x2 are the boundsof the intervalbeing searched that usesf rninsearch to determine rninimumfor Hereis a simpleMATLAB session in thefunctionwe just graphed Example74:
: > f = @( x ) 2+x\1 ) -x(2) +2*xlI) ^ 2 + 2 * x ( l * x ( 2 ) + x ( t : , )^ 2 ; )
f v e ri
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15001)
ENERGY EAUILIBRIUM AND MINIMUM POTENTIAL
to Bockground As in Fig 770a,an unloadedspring can be attached a wall mount is The displacement relatedto the When a horizontalforce is applied,the spring stretches force by Hookes law, F : kx The potential energyof the deformed stateconsistsof the difference betweenthe strainenergyofthe springand the work doneby the force: PE(i :05kx2 * Fx (711)
7IO FIGURE force An spring oitoched o wollmount Applicotiono horizoniol siretches to of {b} {o) unlooded isdescribed Hookes thespring the force by low where relotionship beiween onddisplocement
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