l15 lp problems

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L15 LP Problems. Homework Review Why bother studying LP methods History N design variables, m equations Summary. H14 part 1. H14 Part 1. H14 Part 1. H14. Curve fitting. Curve Fitting. Need to find the parameters a i Another way? Especially for non-linear curve fits?. - PowerPoint PPT Presentation

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  • L15 LP Problems HomeworkReviewWhy bother studying LP methodsHistoryN design variables, m equationsSummary*

  • H14 part 1*

  • H14 Part 1*

  • H14 Part 1*

  • *

    LinearQuadraticPowerExponentiala04.0866.335a1-0.454-1.434a20.0775a3a45.908a5-1.113a68.965a7-0.463SSE9.3851.9143.5150.8304r20.7260.9440.8970.9758

  • H14*

  • Curve fitting*

  • Curve Fitting *Need to find the parameters ai Another way? Especially for non-linear curve fits?

  • Curve Fit example*

  • Goodness of fit?R2 = coefficient of determination 0 R2 1.R = correlation coefficien*

  • Curve Fit example*

  • Linear Programming Prob.s*

  • Why study LP methodsLP problems are convexIf there is a solutionits global optimumMany real problems are LPTransportation, petroleum refining, stock portfolio, airline crew scheduling, communication networks Some NL problems can be transformed into LPMost widely used method in industry*

  • Std Form LP Problem*Matrix formAll 0 i.e. non-neg. How do we transform an given LP problem into a Standard LP Prob.? All =

  • Recall LaGrange/KKT method*Add slack variableSubtractsurplus variable

  • Handling negative xi*When x is unrestricted in sign:

  • Transformation example*

  • Transformation example pg2*

  • Trans pg3*

  • Solving systems of linear equationsn equations in n unknownsProduces a unique solution, for example

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  • Elimination methods*Gaussian Elimination

  • Elimination methods contd*Gauss-Jordan Elimination

  • Can we find unique solutions forn unknowns with m equations?*5 unknowns and 2 equations!Whats the best you can do?MUST set 3 xi to zero! Solve for remaining 2. Just like us=0 in LaGrange Method!

  • m equations= m unknownsMost we can do is to solve for m unknowns,e.g. we can solve for 2 xi

    but which 2?*

  • Combinations?*m=2, n=5

  • Combinations from m=2, n=2*m=2, n=4

  • *Example 8.2Figure 8.1 Solution to the profit maximization problem. Optimum point = (4, 12). Optimum cost = -8800.

    5 unknowns, n=53 equations, m=310 combinations

  • Example 8.2 contd*Solutions are vertexes (i.e. extreme points, corners) of polyhedron formed by the constraints

  • Example 8.2 contdTen solutions created by setting (n-m) variables to zero, they are called basic solutionsSome of them were basic feasible solutionsAny solution in polygon is a feasible solutionVariables not set to zero are basic variablesVariables set to zero = non-basic variables*

  • Canonical form Ex 8.4 & TABLEAU*basis

  • Ex 8.4 contd*Pivot rowPivot column

  • Method?Set up LP prob in tableauSelect variable to leave basisSelect variable to enter basis (replace the one that is leaving)Use Gauss-Jordan elimination to form identity sub-matrix, (i.e. new basis, identity columns)Repeat steps 2-4 until opt soln is found!

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  • Can we be efficient?Do we need to calculate all the combinations?Is there a more efficient way to move from one vertex to another?How do we know if we have found the opt solution, or need to calculate another tableau?

    SIMPLEX METHOD! (Next class)

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  • SummaryCurve fit = min Sum Squared ErrorsMin SSE, check RMany important LP problemsLP probs are convex prog probsNeed to transform into Std LP format slack, surplus variables, non-negative b and xPolygon surrounds infinite # of solnsOpt solution is on a vertexMust find combinations of basic variables

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