Post on 09-Aug-2020
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