PROF. DR. VEDAT CEYHAN

NON-LINEAR PROGRAMMING (NLP)

Similarities & differences

Characteristics of NLP

NLP models

One Variable NLP

Multi Variable NLP

Unlimited NLP

Limited NLP

Basic concept

Case studies

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Similarities & differences

• Linear programming(LP)– Linear target function+ linear constraints

– Continuous variable

• Integer programming (IP)– Linear target function+ linear constraints

– Discrete variable

• Non-linear programming (NLP)– The objective is linear but constraints are non-linear

– Non-linear objectives and linear constraints

– Non-linear objectives and constraints

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Characteristics of NLP

Solution is difficult

Solution may tie initial point.

Initial point is subjective.

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• NLP forms:– Non-linear objectives

– Non-linear constraints

– Non-linear objectives and constraints

The main problem is algorithm selection

2min ( -3)

. . x< 3

x

s t

1 2

2 21 2

min

. . 4

x x

s t x x

21 2

2 21 2

min ( 2)

. . 4

x x

s t x x

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• Constructing model is required for solving optimizationproblem. Mathematical model is based on determination ofvariables and defining their functional relationship

• 3 components of mathematical model

– Objective function,

– Constraints

– Non-negativity restriction

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• Linear programming, integer programming and goalprogramming assume that objective function and constraintsis linear. Not include non-linear expressions such as 1/X2, logX3

• NLP procedures does not produce optimum solution everytime in contrary with linear programming (Render ve ark,2012).

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Objective function in NLP

• gi(x) ≤ bi constraints

• z = f(x) objective function

We find the vector of x= (x1, x2,…, xn), which produce optimum solution for objective

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NLP models

• Associated with the number of variable

–One variable or multivariate,

Associated with the presence of restrictions

– restricted or unrestricted

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NLP models

Constraints: equality

Limited Model

One variable models Constraints: inequality

Unrestricted Model

Constraints: equality

Limited Model

Multivariate models Constraints:inequality

Unresticted Model

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Basic concepts

• Increasing and decreasing function

y=f(x)

x1 and x2 is a random figüre

If the function is f(x1)<f(x2) when inequality is x1<x2,

then function is called increasing.

If the function is f(x1)>f(x2) when inequality is x1<x2,

then function is called decreasing.

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Increasing and decreasing function

Increasing function Decreasing function

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– Local Maximum and Minimum

f(x)’’> f(x)’

f(x)’’< f(x)’ local min. or max

–Gradiant of function

Gradiant function of Z=f(x1,x2,…,xn) is vector offirst derivatives.

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Hessian Matrix

• Hessian matrix is a nxn matrix of second partial derivative of f(x1,x2,…,xn) function

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In LP, solution region is convex set andoptimum solution is one of the corner point

In NLP, it is not necessary that optimumsolution is one of the corner point

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Concave function Convex function

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Example:

1) Is f(x)= √x function whether convex or concave at [S= 0, ∞)?

f(x)=√x concave function

Since line between two point take place under the curve, the

function is concave.17

2) Is f(x)=x3 function whether convex or concave at S = R1=(-∞, ∞)?

f(x)=x³ Neither convex nor concave function

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3) Is f(x)=3x-3 function whether convex or concave at S = R1=(-∞, ∞)?

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Both convex and concave function

-3

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One variable unrstricted NLP

• Deal with finding maximum or minimum pointof one variable function with no restriction

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Multivariate unrestricted NLP

• Deal with finding optimum point(x1*,x2*,…,xn*) that is maximum or minimumpoint of multivariate function of f(x1,x2,…,xn)with no restriction

• Demand function explained by price,preference, income, etc. İs a example formultivariate unrestricted NLP.

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Restricted NLP

• Examining the colineairty among x1,x2,……,xn that is variablesof multivariate function of f(x1,x2,……,xn

• f(x1,x2,……,xn) function,

• g1(x1,x2,……,xn) = b1

• g2(x1,x2,……,xn) =b2 restrictions

• gm(x1,x2,……,xn) =bm

We are looking for point that is maximum or minimum underupper restrictions.

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Solution

• Lagrange function

L(x1,x2,…………….xn,λ1,λ2,……….λm)= f(x1,x2,……..xn) + i[ bi- gi(x1,x2,………xn)]

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Example

1 1

2

2 2

1 2

1 2

1 2

1 2

2 2

1 2 1 1 2 2 1 2

1 1 2 1

2

Minimize ( 4) ( 4)

subject to 2 3 6

3 2 12

and , 0

The Lagrangian is:

( 4) ( 4) (6 2 3 ) ( 12 3 2 )

Kuhn Tucker Conditions:

2( 4) 2 3 0 0 and 0

2( 4)

x x

x

C x x

x x

x x

x x

Z x x x x x x

Z x x x Z

Z x

2

1 1

2 2

1 2 2

1 2 1 1

1 2 2 2

28 36 161 2 1 213 13 13

3 2 0 0 and 0

6 2 3 0 0 and 0

12 3 2 0 0 and 0

Solution: , , 0,

xy x Z

Z x x Z

Z x x Z

x x

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Non-linear objective function and linear constraints

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Case of non-linear objective function and linear constraints

Great Western Appliance firm sell toast machine ofMikrotoaster (X1) and Self-Clean Toaster Oven (X2). Firm gainnet profit by $28 per toaster. Profit function is 21X2+0,25

Objective function is non-linear:

Maximum profit =28X1+21X2+0,25

There are two linear constraints

X1+X2 ≤ 1,000 (production capacity)

0.5X1+0.4X2 ≤ 500 (term of sale)

X1, X2 ≥ 0

(Source: Render ve ark., 2012)

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2. Quadratic programming

Objective function includes terms such as 0,25 and when the restrictions is linear

Quadratic programming problems can be solved by using adjusted simplex algorithms.

(Source: Render and ark., 2012)

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EXAMPLES

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WIN QSB for NLP

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WİN QSB interface

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WİN QSB EKRANINI TANIYALIM

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Unrestricted NLP example

• minimum x(sin(3.14159x))

• 0 <= x <= 6

• We have one non-linear expression such as sin(3.14159x) for minimizing objective function

•

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Forunrestrictedcase, enter

“0”

Unrestricted NLP example

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Attention!

Interval of X1

Graphicalsolution

x=3.5287 Objective function= -3.5144.

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Example 2:

Objective function:

maximum 2x1 + x2 - 5loge(x1)sin(x2)

Constraintsx1x2 <= 10 | x1 - x2 | <= 20.1 <= x1 <= 5 0.1 <= x2 <= 3

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x1=3.3340 ve x2=2.9997 Objective function=8.8166

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Non-linear function and constraints

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• In medium size hospital having 200-400 patient bed,Hospicare Corporation, annual profit depend onnumber of patient(X1) ve number of patientsurgeon(X2) bağlıdır.

• Non-linear objective function for Hospicare :$13X1 + $6X1X2 + $5X2 + $1/X2

Constraints;

• 2X12+4X2≤90 (nursery capacity)

• X1+X23≤75 (X-ray capacity)

• 8X1-2X2≤61 (marketing budget)

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Example:Pickens Memorial Hospital

Patient demand exceeds capacity of hospital

Objective: Maximum profit

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Decision variables

M =number of served patient

S = number of served patient for surgery

P = number of served child patient

Profit functionWhen increasing patient, profit increasing non-

linearly.

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Constraints

• Hospital capacity: Total 200 patient

• X-ray capacity: 560 x-rays per week

• Marketing budget: $1000 per week

• Laboratory capacity : 140 hours per week

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Objective function

Max 45M + 2M2 + 70S + 3S2 + 2MS + 60P + 3P2

Constraints:

M + S + P < 200 (patient capacity)

M + 3S + P < 560 (x-ray capacity)

3M + 5S + 3.5P < 1000 (budget $)

(0.2+0.001M)x(3M+3S+3P) < 140 (lab hour)

M, S, P > 0

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Objective functionMax 45M + 2M2 + 70S + 3S2 + 2MS + 60P + 3P2

Constraints:

M + S + P < 200 (patient capacity)

M + 3S + P < 560 (x-ray capacity)

3M + 5S + 3.5P < 1000 (budget $)

(0.2+0.001M)x(3M+3S+3P) < 140 (lab hour)

M, S, P > 0