m.tech. assignment
TRANSCRIPT
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8/18/2019 M.tech. Assignment
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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!
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+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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