L13 Optimization using Excel

• See revised scheduleread 8(1-4) + Excel “help” for Mar 12

• Test Answers• Review: Convex Prog. Prob.• Worksheet modifications• Excel optimization• Summary

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Trendline in Excel

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Excel help

“trendline”

for Wed

Theorem 4.9

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}to1,0)(g; to1for,0)(|{

SetConstraint

j mjpihS i

xxx

Given:

S is convex if:1. hi are linear2. gj are convex i.e. Hg PD or PSD

When f(x) and S are convex= “convex programming problem”

“Sufficient” Theorem 4.10, pg 165

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The first-order KKT conditions are Necessary and Sufficient for a GLOBAL minimum….if:

1. f(x) is convexHf(x) Positive definite

2. x is defined as a convex feasible set SEquality constraints must be linearInequality constraints must be convex

HINT: linear functions are convex!

Worksheet Modifications

• Naming cells• Inserting shapes• Inserting MS Equation “object”• Recording macros• Attaching a macro to a shape• Creating a SOLVER hot button• Visual basic, tools/references/solver

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Figure 6.1 Excel worksheet for finding roots of 2x/3 – sin x : (a) worksheet; (b) worksheet with formulas showing.

Excel Applications

Solver parameters

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Figure 6.2 A Solver Parameters dialog box to define the problem.

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Figure 6.3 A Solver Results dialog box and the final worksheet.

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Figure 6.4 A Solver Answer Report for roots of 2x/3 – sin x = 0.

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Figure 6.5 Worksheet and Solver Parameters dialog box for KKT conditions for Example 4.31.

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Figure 6.6 Solver Results for KKT conditions for Example 4.31.

KKT system of NL EQNs

Prob 4.59 and 4.122

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Figure 6.7 Excel worksheet and Solver Parameters dialog box for unconstrained problem.

Constrained Optimization

Prob. 4.69 and 4.122

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4

13subject to

)3()3(),(

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21

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2121

xxg

xxh

xxxxfMin

Graphical Solution

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125.500

075.025.175.025.3

2

1

fsu

xx

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Figure 6.8 Excel worksheet for the linear programming problem.

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Figure 6.9 Solver Parameters dialog box for the linear programming problem.

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Figure 6.10 Solver Results dialog box for the linear programming problem.

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Figure 6.11 Answer Report from Solver for linear programming problem.

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Figure 6.12 Sensitivity Report from Solver for the linear programming problem.

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Figure 6.13 Excel worksheet for the spring design problem.

Summary• KKT pt from a Convex Prog. Prob. Is a

global min!• Use modifications for “ease of use”• Pay attention to layout

– Design variables– Parameters– Analysis/Performance “Variables”– Objective function– Constraints

• May need multiple starting points22