classification of optimization problems

8/18/2019 Classification of Optimization Problems

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D Nagesh Kumar, IISc Optimization Methods: M1L31

Introduction and Basic Concepts

(iii) Classificationof Optimization

Problems


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D Nagesh Kumar, I Optimization Metho s! M"%

Introduction

Optimization problems can be classifie base on

the t&pe of constraints , nature of esign 'ariables ,ph&sical structure of the problem , nature of thee uations in'ol'e , eterministic nature of the'ariables , permissible 'alue of the esign 'ariables ,

separabilit& of the functions an number of ob ecti'efunctions * +hese classifications are briefl& iscussein this lecture*


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D Nagesh Kumar, I Optimization Metho s! M"$

Classification based on existence of constraints.

Constrained optimization problems: hich aresub ect to one or more constraints*

Unconstrained optimization problems ! in hich

no constraints e-ist*


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D Nagesh Kumar, I Optimization Metho s! M".

Classification based on the nature of the designvariables

+here are t o broa categories of classificationithin this classification

/irst categor& ! the ob ecti'e is to fin a set of esignparameters that ma0e a prescribe function of theseparameters minimum or ma-imum sub ect to certainconstraints * 1 /or e-ample to fin the minimum eight esign of a strip

footing ith t o loa s sho n in the figure, sub ect to alimitation on the ma-imum settlement of the structure *


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+he problem can be efine as follo s

sub ect to the constraints

+he length of the footing (l) the loa s P" an P% , the istance bet een the loa s areassume to be constant an the re uire optimization is achie'e b& 'ar&ing b an *

Such problems are calle parameter or static optimization problems*


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D Nagesh Kumar, I Optimization Metho s! M"2

Classification based on the nature of the designvariables (contd.)

Secon categor&! the ob ecti'e is to fin a set ofesign parameters, hich are all continuous

functions of some other parameter, that minimizesan ob ecti'e function sub ect to a set of constraints*

1/or e-ample, if the cross sectional imensions of therectangular footing is allo e to 'ar& along its length assho n in the follo ing figure*


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+he problem can be efine as follo s

sub ect to the constraints

l

+he length of the footing (l) the loa s P " an P % , the istance bet een the loa s areassume to be constant an the re uire optimization is achie'e b& 'ar&ing b an *

Such problems are calle trajectory or dynamic optimization problems*


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D Nagesh Kumar, I Optimization Metho s! M"3

Classification based on the physical structure of theproblem

Based on the physical structure !e can classifyoptimization problems are classified as optimal control

and non"optimal control problems *(i) 4n optimal control (#C) problem is a mathematical

programming problem in'ol'ing a number of stages, hereeach stage e'ol'es from the prece ing stage in aprescribe manner *

It is efine b& t o t&pes of 'ariables! the control or esign'ariables an state 'ariables*


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+he problem is to fin a set of control or esign 'ariables such that thetotal ob ecti'e function (also 0no n as the performance in e-, PI) o'erall stages is minimize sub ect to a set of constraints on the control anstate 'ariables* 4n OC problem can be state as follo s!

5here x i is the i th control 'ariable, y i is the i th state 'ariable, an f i is thecontribution of the i th stage to the total ob ecti'e function* g $ h%, an q i are the functions of xj, yj ; x k, y k an x i an y i , respecti'el&, an l is thetotal number of states* +he control an state 'ariables x i an y i can be'ectors in some cases*

(ii) &he problems !hich are not optimal control problems are called non-optimal control problems.

sub ect to the constraints!


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D Nagesh Kumar, I Optimization Metho s! M""6

Classification based on the nature of the e'uationsinvolved

Based on the nature of expressions for the ob$ectivefunction and the constraints optimization problems canbe classified as linear nonlinear geometric and 'uadraticprogramming problems.


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D Nagesh Kumar, I Optimization Metho s! M"""

Classification based on the nature of the e'uationsinvolved (contd.)

(i) Linear programming problemIf the ob ecti'e function an all the constraints are linear functions ofthe esign 'ariables, the mathematical programming problem is callea linear programming (#P) problem*

often state in the stan ar form ! sub ect to


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D Nagesh Kumar, I Optimization Metho s! M""$


(iii) Geometric programming problem 1 4 geometric programming (7MP) problem is one in hich the

ob ecti'e function an constraints are e-presse as pol&nomials in 8* 4 pol&nomial ith N terms can be e-presse as

1 +hus 7MP problems can be e-presse as follo s! /in hich

minimizes ! sub ect to!


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D Nagesh Kumar, I Optimization Metho s! M"".


here N 0 an N k enote the number of terms in the ob ecti'e an k th constraint function, respecti'el&*

(iv) Quadratic programming problem 4 ua ratic programming problem is the best beha'e nonlinearprogramming problem ith a ua ratic ob ecti'e function an linearconstraints an is conca'e (for ma-imization problems)* It is usuall&formulate as follo s!

Sub ect to!

here c, q i , Q ij , a ij , an b j are constants*


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Classification based on the permissible values ofthe decision variables

Under this classification problems can be classified as integer and real"valued programming problems

(i) Integer programming problemIf some or all of the esign 'ariables of an optimization problem arerestricte to ta0e onl& integer (or iscrete) 'alues, the problem is callean integer programming problem*

(ii) Real-valued programming problem

4 real:'alue problem is that in hich it is sought to minimize orma-imize a real function b& s&stematicall& choosing the 'alues of real'ariables from ithin an allo e set* 5hen the allo e set containsonl& real 'alues, it is calle a real:'alue programming problem*


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Classification based on deterministic nature of thevariables

Under this classification optimization problemscan be classified as deterministic and stochasticprogramming problems

(i) Deterministic programming problem; In this t&pe of problems all the esign 'ariables are

eterministic*

(ii) Stochastic programming problemIn this t&pe of an optimization problem some or all theparameters ( esign 'ariables an


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D Nagesh Kumar, I Optimization Metho s! M""=

Classification based on separability of the functions

Based on the separability of the ob$ective and constraintfunctions optimization problems can be classified as

separable and non"separable programming problems(i) Separable programming problems

In this t&pe of a problem the ob ecti'e function an the constraints areseparable* 4 function is sai to be separable if it can be e-presse asthe sum of n single:'ariable functions an separable programmingproblem can be e-presse in stan ar form as !

sub ect to !

here b j is a constant*


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Classification based on the number of ob$ectivefunctions

Under this classification ob$ective functions can beclassified as single and multiob$ective programming

problems.(i) Single-objective programming problem in hich there is onl& asingle ob ecti'e*

(ii) Multi-objective programming problem 4 multiob ecti'e programming problem can be state as follo s!

here f 1 , f 2 , . . . f k enote the ob ecti'e functions to be minimizesimultaneousl&*

sub ect to !


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D Nagesh Kumar, I Optimization Metho s! M"">

Thank You

classification of optimization problems

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