l15 lp problems
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DESCRIPTIONL15 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
L15 LP Problems HomeworkReviewWhy bother studying LP methodsHistoryN design variables, m equationsSummary*
H14 part 1*
H14 Part 1*
H14 Part 1*
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 pg2*
Solving systems of linear equationsn equations in n unknownsProduces a unique solution, for example
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 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!
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)
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