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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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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