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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

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

Lead

Premium

Alaska 2

Unlead

Alaska Pool

Texas

Reg.

Alaska 1

input variables
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
Output Variables
  • Prem - Gals of Premium produced daily
  • Reg- Gals of Regular produced daily
  • NoL - Gals of No Lead produced daily
composition variables
Composition Variables
  • For convenience
  • AlaskaSulfur - sulfur content of Alaska pool
  • AlaskaOctane- octane of Alaska pool
constraints
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
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
Constrain Volumes
  • Prem  5000
  • Reg  5000
  • NoL  5000
  • Upper Limits
  • Texas  11000
  • Lead  6000
objective
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
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
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
Minimize Cost

Total Cost

Volume Purchased

slide18

Minimize Cost

Total Cost

Volume Purchased

easy problem
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
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))
slide21

Minimize Cost

Total Cost

Volume Purchased

maximize profit
Maximize Profit

Total Profit

Volume Sold

slide24

Maximize Profit

Total Profit

Volume Sold

easy problem1
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
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))
slide27

Maximize Profit

Total Profit

Volume Sold

what makes these easy
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
Tough Problems

Local Max

Local Max

Local Min

Local Min

convex sets
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 sets1
Convex Sets
  • Linear Programming Feasible regions
non convex sets
Non-convex Sets
  • Feasible Region of Integer Programs
easy problems
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 problems1
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
Non-convex Sets are Hard

Feasible

Profit

Volume Sold

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