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Non-Linear Programming

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Non-Linear Programming - PowerPoint PPT Presentation

Non-Linear Programming. Same structure variables objective function constraints No restrictions Except typically variables must be continuous. Examples. How to model binary variables x is 0 or 1 Equivalent continuous formulation x(1-x) = 0 NOT LINEAR!. Location Problem. Variables

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Non-Linear Programming
• Same structure
• variables
• objective function
• constraints
• No restrictions
• Except typically variables must be continuous
Examples
• How to model binary variables
• x is 0 or 1
• Equivalent continuous formulation
• x(1-x) = 0 NOT LINEAR!
Location Problem

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

Formulation
• minimize

200*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)

Pooling Problem
• Blend crudes in pools
• Blend Alaska 1 and Alaska 2
• Make 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
Diagram

Lead

Premium

Alaska 2

Unlead

Alaska Pool

Texas

Reg.

Alaska 1

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
Output Variables
• Prem - Gals of Premium produced daily
• Reg- Gals of Regular produced daily
• NoL - Gals of No Lead produced daily
Composition Variables
• For convenience
• AlaskaSulfur - sulfur content of Alaska pool
• AlaskaOctane- octane of Alaska pool
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
Constraints Cont’d
• Define 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
Constrain Volumes
• Prem  5000
• Reg  5000
• NoL  5000
• Upper Limits
• Texas  11000
• Lead  6000
Objective
• Maximize Profit
• Revenues from Products
• 0.86*Reg + 0.93*NoL + 1.06*Prem
• Costs of Raw Materials
• 0.78*Alaska 1 + 0.88*Alaska 2 + 0.75*Texas + 1.30*Lead
Formulating NLPs
• As in the book
• No need for abstraction
• Some off the shelf software (MINOS)
• Requires more sophistication to use
• Does not typically provide guarantees
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
Minimize Cost

Total Cost

Volume Purchased

Minimize Cost

Total Cost

Volume Purchased

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
Convex Function
• Lies below the line
• Technically: 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))

Minimize Cost

Total Cost

Volume Purchased

Maximize Profit

Total Profit

Volume Sold

Maximize Profit

Total Profit

Volume Sold

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
Concave Function
• Lies above the line
• Technically: 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))

Maximize Profit

Total Profit

Volume Sold

What makes these easy
• With no constraints
• Local 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.
Tough Problems

Local Max

Local Max

Local Min

Local Min

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
Convex Sets
• Linear Programming Feasible regions
Non-convex Sets
• Feasible Region of Integer Programs
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
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
Non-convex Sets are Hard

Feasible

Profit

Volume Sold