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Amity University Haryana
Optimization TechniqueIntroduction: Historical
Developments, Engineering applications of Optimization
Classical Optimization Techniques: Introduction, Review of
single and multivariable optimization
methods with and without constraints, Non-linear one-dimensional
minimization problems, Eamples!
Constrained Optimization Techniques: Introduction, Direct
methods - "utting plane method and #ethod
of $easible directions, Indirect methods - "onve programming
problems, Eterior penalt% functionmethod, Eamples and problems
Unconstrained Optimization Techniques: Introduction, Direct
search method - Random, &nivariate and
'attern search methods, Rosenbroc()s method of rotating
co-ordinates, Descent methods - *teepest Decent
methods-+uasi-Newton)s and ariable metric method, Eamples!
Geometric Programming: Introduction, &nconstrained
minimization problems, solution of unconstrained
problem from arithmetic-geometric ineualit% point of view,
"onstrained minimization problems,
.eneralized pol%nomial optimization, /pplications of geometric
problems, Introduction to stochastic
optimization!
ovel methods !or Optimization: Introduction to simulated
annealing, selection of simulated annealing
parameters, simulated annealing algorithm0 .enetic
/lgorithm 1./2, Design of ./, 3e% concepts of ./,
Neural Networ(s, / frame wor( for Neural Networ( models,
"onstruction of Neural Networ( algorithm,
Eamples of simulated algorithm, genetic annealing and Neural
Networ( method!
+4 .ive an% ten engineering application of optimization
techniues!+5! Describe Stochastic Programming in brief.+6! Describe
the *7' !+8! .ive the classification of optimization techniues!
+9! $ind etreme point of function
+:! Describe the univariate method!+;.ive a review of historical
development of optimization techniue!+ design vector +?! .ive
the necessar% > sufficient conditions for a point to maima
1minima2!formultivariable optimization problem!+4@! Discuss the
ph%sical meaning of 7agrange multiplier!+44! *tate the 3uhn-Auc(er
conditions+45! Describe the penalt% function method!
+46! / manufacturing firm producing small refrigerators has
entered into a contract tosuppl% 9@ refrigerators at the end of the
first month, 9@ at the end of the second month,
and 9@ at the end of the third! Ahe cost of producing x
refrigerators in an% month is given b% B1 Ahe firm can produce
more refrigerators in an% month and carr% them to
a subseuent month! However, it costs B5@ per unit for an%
refrigerator carried over fromone month to the net! /ssuming that
there is no initial inventor%, determine the number of
refrigerators to be produced in each month to minimize the total
cost!+48! $ind the dimensions of a bo of largest volume that can be
inscribed in a sphere of unitradius with i! Direct *ubstitution
#ethod ,ii! "onstrained ariation #ethod , iii! 7agrangemultiplier
#ethod
+49! #inimize starting from the point ! with
4! &nivaraient #ethod5! 'owel #ethod
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6! Newtons #etod8! Hoo(e Ceev #ethod
9! $letcher-Reevs #ethod:! "auch% #ethod
+4:! / manufacturing firm produces two products, A
and B, using two limited resources!Ahe maimum amounts of
resources 4 and 5 available per da% are 4@@@ and 59@
units,respectivel%! Ahe production of 4 unit of product A
reuires 4 unit of resource 4 and @!5unit of resource 5, and the
production of 4 unit of product B reuires @!9 unit of
resource4 and @!9 unit of resource 5! Ahe unit costs of resources 4
and 5 are given b% the relations1@!6;9 - @!@@@@9&42 and 1@!;9 -
@!@@@4&52, respectivel%, where &, denotes thenumber of
units of resource i used 1 i 4 , 5 2 ! Ahe selling prices per unit
of products Aand B, P A and P
B , are given b%' A = 5!@@ - @!@@@9A- @!@@@49F
, P B = 6!9@ - @!@@@5/ -
@!@@49F where A and B indicate,
respectivel%, the number of units of products A and B
sold!$ormulate the problem of maimizing the profit assuming that
the firm can sell all theunits it manufactures!+4;! $our identical
helical springs are used to support a milling machine weighing
9@@@Ib! $ormulate the problem of finding the wire diameter (d),
coil diameter 1D2, and thenumber of turns (N) of each spring 1*hown
in $ig below2 for minimum weight b%
limiting the deflection to @!4 in! and the shear stress to
4@,@@@ psi in the spring! Inaddition, the natural freuenc% of
vibration of the spring is to be greater than 4@@ Hz! Ahestiffness
of the spring (k), the shear stress in the spring 1G2, and the
natural freuenc% of vibration of the spring 1f n2 are
given b%
where G is the shear modulus, $ the compressive load on the
spring, w the weight of the spring, pthe weight densit% of the
spring, and K s the shear stress correction factor!
/ssume that the material
is spring steel with . 45 4@: psi and p @!6 lbin6, and the
shear stress correction factor is K s4!@9!
+4
+4?! rite short note an% two
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i2 'attern search methodsii2 "onstrained Optimization
AechniuesJiii2 +uasi Newton #ethodiv2 Neural Networ(s,
+5@! rite short note an% twoi2 "onve programming problemsii2
.enetic /lgorithm 1./2,
iii2 +uasi Newton #ethodiv2 *imulated /nnealingQ21. Enlist the
methods used to solve the optimization problem with equality
constrained & explain two of them.
+55! Eplain the geometric programming problem!+56! Eplain the
Newton method > modif% Newton method > their limitation!+!58
#inimize the function f(x) = @!:9 - K@!;914
L x2 )M- @!:9 x tan-414 x2 using the
goldensection method with n = :!
+!59 *how that the Newtons method finds the minimum of a
uadratic function in one iteration!!+!5: $ind the minimum of
x(x — 4!92 in the interval 1@!@,4!@@2 to within 4@ of the
eactvalue!
+! 5; $ind the maimum of the function12
2 x1 L x2 L 4@ sub=ect to g(X)
= x1 L 5 x2 2 = 6 usingthe 7agrange
multiplier method!
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