non-linear programming zsame structure yvariables yobjective function yconstraints zno restrictions...
TRANSCRIPT
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Non-Linear Programming
Same structure variables objective function constraints
No restrictions Except typically variables must be
continuous
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Examples
How to model binary variables x is 0 or 1 Equivalent continuous formulation
x(1-x) = 0 NOT LINEAR!
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Location ProblemCustomer X Coordinate Y Coordinate Number of Shipments
1 5 10 2002 10 5 1503 0 12 2004 12 0 300
Variables
x is the X coordinate of the facility
y is the Y coordinate of the facility
Objective
Minimize Distance traveled to deliver goods
Constraints - None
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Formulation
minimize200*sqrt((x-5)2 + (y-10)2) +150*sqrt((x-10)2 + (y-5)2) +200*sqrt((x-0)2 + (y-12)2) +300*sqrt((x-12)2 + (y-0)2)
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Pooling ProblemBlend crudes in pools
Blend Alaska 1 and Alaska 2Make products from the pools
Regular Unleaded Premium
Composition constraints on final products Premium 2.8% Sulfur 90 Octane Sells for $0.86/gal, minimum 5000 gals
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Diagram
Alaska 1
Alaska 2
Alaska Pool
Premium
Unlead
Reg.
Texas
Lead
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Input Variables Lead - gallons daily
LeadPrem - gallons of lead used in premium daily LeadReg - gallons of lead used in regular daily
Alaska - gallons of Alaska pool daily Alaska1 - gallons of Alaska 1 used in pool daily Alaska2 - gallons of Alaska 2 used in pool daily
AlaskaPrem - gals of Alaska pool used in prem. daily
AlaskaReg - gals of Alaska pool used in reg. daily
AlaskaNoL - gals of Alaska pool used in No lead daily
Texas - gallons of Texas used daily TexasPrem - gals of Texas used in prem. daily
TexasReg - gals of Texas used in reg. daily
TexasNoL - gals of Texas used in No lead daily
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Output Variables
Prem - Gals of Premium produced daily
Reg- Gals of Regular produced dailyNoL - Gals of No Lead produced daily
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Composition Variables
For convenienceAlaskaSulfur - sulfur content of Alaska poolAlaskaOctane- octane of Alaska pool
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Constraints
Define Alaska Pool Alaska = Alaska 1 + Alaska 2 AlaskaSulfur = (4%*Alaska 1 + 1% * Alaska
2)/Alaska AlaskaOctane=(91*Alaska 1 + 97*Alaska
2)/Alaska
Use Alaska Pool Alaska = AlaskaPrem + AlaskaReg +
AlaskaNoL
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Constraints Cont’dDefine Products Prem = AlaskaPrem+ TexasPrem + LeadPrem Reg = AlaskaReg+ TexasReg + LeadReg NoL = AlaskaNoL+ TexasNoL
Constrain Composition AlaskaSulfur*AlaskaPrem + .02*TexasPrem .028*Prem AlaskaSulfur*AlaskaNoL + .02*TexasNoL .03*NoL AlaskaSulfur*AlaskaReg + .02*TexasReg .03*Reg AlaskaOctane*AlaskaPrem + 83*TexasPrem 94*Prem AlaskaOctane*AlaskaNoL + 83*TexasNoL 88*NoL AlaskaOctane*AlaskaReg + 83*TexasReg 90*Reg
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Constrain Volumes
Prem 5000Reg 5000NoL 5000Upper LimitsTexas 11000Lead 6000
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Objective
Maximize ProfitRevenues from Products0.86*Reg + 0.93*NoL + 1.06*Prem Costs of Raw Materials0.78*Alaska 1 + 0.88*Alaska 2 + 0.75*Texas +
1.30*Lead
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Formulating NLPs
As in the book No need for abstractionSome off the shelf software (MINOS)Requires more sophistication to useDoes not typically provide
guarantees
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Getting Guarantees
When we can use an LP formulation with a non-linear objective
Minimize Cost and things get more expensive as we get more
Maximize Profit and profits decrease as we sell more
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Minimize Cost
Volume Purchased
Tot
al C
ost
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Minimize Cost
Volume Purchased
Tot
al C
ost
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Easy Problem
The Cost Function lies below the linear approximation
No incentive to use any weights other than consecutive ones
Don’t need Integer Programming
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Convex Function
Lies below the lineTechnically: A convex function has the
property that for each pair of points x and y and weight w between 0 and 1 the function evaluated at wx + (1-w)y (a fraction w of the way from y towards x) is wf(x) + (1-w)f(y) (the same fraction of the way from f(y) towards f(x))
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Minimize Cost
Volume Purchased
Tot
al C
ost
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Convex in 2 dimensions
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Maximize Profit
Volume Sold
Tot
al P
rofi
t
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Maximize Profit
Volume Sold
Tot
al P
rofi
t
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Easy Problem
The Profit Function lies above the linear approximation
No incentive to use any weights other than consecutive ones
Don’t need Integer Programming
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Concave Function
Lies above the lineTechnically: A concave function has the
property that for each pair of points x and y and weight w between 0 and 1 the function evaluated at wx + (1-w)y (a fraction w of the way from y towards x) is wf(x) + (1-w)f(y) (the same fraction of the way from f(y) towards f(x))
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Maximize Profit
Volume Sold
Tot
al P
rofi
t
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What makes these easy
With no constraintsLocal Optimum is best in a small
neighborhood, e.g., as good as every point within epsilon of it.
Convex minimization: A local optimum is a global optimum, e.g., a best answer
Concave maximization: A local optimum is a global optimum.
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Tough Problems
Local Max
Local Max
Local Min Local Min
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Convex Sets
A set with the property that for every pair of points in the set, the line joining the points is in the set as well is a CONVEX SET
Points in a convex set can see each other
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Convex Sets
Linear Programming Feasible regions
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Non-convex Sets
Feasible Region of Integer Programs
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Easy Problems
Convex Minimization over a convex set Objective is a convex function Constraints define a feasible region that
is a convex set
Any Local minimum is a global minimum
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Easy Problems
Concave Maximization over a convex set Objective is a concave function Constraints define a feasible region that
is a convex set
Any Local maximum is a global maximum
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Non-convex Sets are HardP
rofi
t
Volume Sold
Feasible