1 / 35

# 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

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

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

• Same structure

• variables

• objective function

• constraints

• No restrictions

• Except typically variables must be continuous

• How to model binary variables

• x is 0 or 1

• Equivalent continuous formulation

• x(1-x) = 0 NOT LINEAR!

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

• 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)

• Blend crudes in pools

• Make products from the pools

• Regular

• Composition constraints on final products

• Premium  2.8% Sulfur  90 Octane

• Sells for \$0.86/gal, minimum 5000 gals

Texas

Reg.

• 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

• Prem - Gals of Premium produced daily

• Reg- Gals of Regular produced daily

• NoL - Gals of No Lead produced daily

• For convenience

• Define Products

• Constrain Composition

• Prem  5000

• Reg  5000

• NoL  5000

• Upper Limits

• Texas  11000

• Maximize Profit

• Revenues from Products

• 0.86*Reg + 0.93*NoL + 1.06*Prem

• Costs of Raw Materials

• As in the book

• No need for abstraction

• Some off the shelf software (MINOS)

• Requires more sophistication to use

• Does not typically provide 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

Total Cost

Volume Purchased

Total Cost

Volume Purchased

• The Cost Function lies below the linear approximation

• No incentive to use any weights other than consecutive ones

• Don’t need Integer Programming

• 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))

Total Cost

Volume Purchased

Total Profit

Volume Sold

Total Profit

Volume Sold

• The Profit Function lies above the linear approximation

• No incentive to use any weights other than consecutive ones

• Don’t need Integer Programming

• 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))

Total Profit

Volume Sold

• 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.

Local Max

Local Max

Local Min

Local Min

• 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

• Linear Programming Feasible regions

• Feasible Region of Integer Programs

• 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

• 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

Feasible

Profit

Volume Sold