(20-21)-search method.pdf

8/18/2019 (20-21)-search method.pdf

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Optimization

MEL 806

Thermal System Simulation (2-0-2)

Dr. Prabal Talukdar Associate Professor

Department of Mechanical Engineering

e


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Introduction• In the preceding lectures, we focused our attention on obtaining a

workable, feasible, or acceptable design of a system. Such a design

satisfies the requirements for the given application, without violating

any imposed constraints. A system fabricated or assembled

ecause o s es gn s expec e o per orm e appropr a e as s

for which the effort was undertaken.

• However, the design would generally not be the best design, where

e e n on o es s ase on cos , per ormance, e c ency or

some other such measure.

• In actual practice, we are usually interested in obtaining the best

qua y or per ormance per un cos , w accep a e env ronmen aeffects. This brings in the concept of optimization, which minimizes

or maximizes quantities and characteristics of particular interest to a

.


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UNCONSTRAINED SEARCH WITH MULTIPLE

VARIABLES

Let us now consider the search for an optimal

design when the system is governed by two ormore independent variables.

However, the complexity of the problem rises

,

therefore, attention is generally directed at the most

important variables, usually restricting these to two

or three.


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• ,

be well characterized in terms of two or threepredominant variables.

• Examples of this include the length and diameter

of a heat exchanger, fluid flow rate andevaporator temperature in a refrigeration

system, and so on.


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•

approach to the optimum design, a

or lines of constant values of the objective

.


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Lattice Search Method

Lattice search method in a two-variable space.


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Univariate Search•

objective function with respect to one variable ata time. Therefore, the multivariable problem is

reduced to a series of single-variable

optimization problems, with the processconverg ng o e op mum as e var a es are

alternated


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Graphical presentation

Various steps in the univariate search method.


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The method A starting point is chosen based on available information on the system

.

First, one of the variables, say x, is held constant and the function is

optimized with respect to the other variable y.

Point A represents the optimum thus obtained. Then y is held constant

at the value at point A and the function is optimized with respect to x to

obtain the optimum given by point B.

Again, x is held constant at the value at point B and y is varied to obtain

the optimum, given by point C.

This process is continued, alternating the variable, which is changedwhile keeping the others constant, until the optimum is attained.

This is indicated b the chan e in the ob ective function from one ste

to the next, becoming less than a chosen convergence criterion or

tolerance


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If y is kept constant, the value of x at the optimum is given by

Similarly, if x is held constant, the value of y at the optimum is given by

let us choose x = y = 0.5 as the starting point.

First x is held constant and y is varied to obtain an optimum value

of U. Then y is held constant and x is varied to obtain an optimum

value of U. In both cases, the recedin e uations are used.


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Calculations

x y u

0.5 1.632993 9.839626

1.944161 0.828139 5.598794

2.437957 0.739531 5.427791

. . .

2.547644 0.723436 5.422363

2.550314 0.723057 5.422359

2.55076 0.722994 5.422359

2.550834 0.722983 5.422359

2.550847 0.722982 5.422359


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Steepest Ascent/Descent Method• The stee est ascent/descent method is a ver efficient

search method for multivariable optimization and iswidely used for a variety of applications, including

.

• It is a hill-climbing technique in that it attempts to move

toward the peak, for maximizing the objective function, ortoward the valley, for minimizing the objective function,

over the shortest possible path.

case and steepest descent in the latter.


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• At each step, starting with the initial trial point, the

rec on n w c e o ec ve unc on c anges a e

greatest rate is chosen for moving the location of theoint, which re resents the desi n on the multivariable

space.

Steepest ascent method, shown in terms of (a) the climb toward the peakof a hill and (b) in terms of constant U contours.


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• It was shown that the radient vector is normal toU∇

the constant U contour line in a two-variable space, tothe constant U surface in a three-variable space, and so

.

• Since the normal direction represents the shortest

distance between two contour lines, the direction of thegradient vector is the direction in which U changes at

the greatest rate.U∇

,written as


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• At each trial oint the radient vector is determined and

the search is moved along this vector, the direction beingchosen so that U increases if a maximum is sought, or U

.

• The direction represented by the gradient vector is given

by the relationship between the changes in theindependent variables. Denoting these by Δx1, Δx2 , ---

Δxn , we have


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First approach• Choose a startin oint. Select Δx. Calculate the

derivatives.• Decide the direction of movement, i.e., whether Δx is

pos ve or nega ve. a cu a e y. a n e new va ues

of x, y, and U.

• Calculate the derivatives a ain at this oint. Re eat

previous steps to attain new point.

• This procedure is continued until the change in the

var a es e ween wo consecu ve era ons s w n adesired convergence criterion.


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Example Problem•

before and apply the two approaches just

method to obtain the minimum cost U.


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• The startin oint is taken as x = = 0.5. The results

obtained for different values ofΔ

x are


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Multivariable Constrained

•

optimization, which is much more involved thanthe various unconstrained optimization cases

considered thus far.

• The number of independent variables must belarger than the number of equality constraints;

otherwise, these constraints may simply be used

possible


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Penalt Function Method• The basic a roach of this method is to convert the

constrained problem into an unconstrained one byconstructing a composite function using the objective

• Let us consider the optimization problem given by the

equations

• The composite function, also known as the penaltyfunction, may be formulated in many different ways.


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• If a maximum in U is being sought, a new objective

•

function is defined as

• Here the r’s are scalar quantities that vary the

importance given to the various constraints and are

.

• They may all be taken as equal or different.

• Higher values may be taken for the constraints that are

critical and smaller values for those that are not as

important.


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• If the penalty parameters are all taken as zero, the constraints

have no effect on the solution and therefore the constraints

are not satisfied.

• On the other hand, if these parameters are taken as large, the

constraints are satisfied but the conver ence to the o timum is

slow.

• Therefore, by varying the penalty parameters we can vary the

rate of conver ence and the effect of the different constraints

on the solution.

• The general approach is to start with small values of the

enalt arameters and raduall increase these as the G’s

which represent the constraints, become small.• This implies going gradually and systematically from an

unconstrained roblem to a constrained one.


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different values of the penalty parameter r.


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Example Problem

In a two-component system, the cost is the objective function given by the

,

where x and y represent the specifications of the two components. Thesevariables are also linked by mass conservation to yield the constraint

G(x, y) = xy -12 = 0

Solve this problem by the penalty function method to obtain minimum cost.

The new objective function V(x, y), consisting of the objective function ande cons ra n , s e ne as

the optimum. An exhaustive search can be used because of the simplicityof the method and the given functions.

, ,

large, the constraints are satisfied, but the convergence is slow.


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We may also derive x and y in terms of the penalty parameter r, by

,

expressions to zero, as

See the spreadsheet

(20-21)-search method.pdf

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