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## PowerPoint Slideshow about ' Non-Linear Programming' - cassia

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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

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

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))

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))

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.

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

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