Transcript
Ho Chi Minh City University of Technology
Faculty of Electrical and Electronics Engineering
Department of Telecommunications
Lectured by Ha Hoang Kha, Ph.D.
Ho Chi Minh City University of Technology
Email: hahoangkha@gmail.com
Constrained Optimization
Content
Constrained Optimization 2 H. H. Kha
References
Slides are adapted from the following resources: Lagrange Multipliers and the Karush-Kuhn-Tucker conditions, Sildes at
http://www.csc.kth.se/utbildning/kth/kurser/DD3364/Lectures/KKT.pdf
Constrained Optimization 3 H. H. Kha
Optimization
Constrained Optimization 4 H. H. Kha
Optimization
Constrained Optimization 5 H. H. Kha
Unconstrained Optimization
Constrained Optimization 6 H. H. Kha
Unconstrained Minimization
Constrained Optimization 7 H. H. Kha
Unconstrained Maximization
Constrained Optimization 8 H. H. Kha
Unconstrained Maximization
Constrained Optimization 9 H. H. Kha
Constrained Optimization:
Equality Constraints
Constrained Optimization 10 H. H. Kha
Tutorial Example
Constrained Optimization 11 H. H. Kha
Tutorial Example-Cost function
Constrained Optimization 12 H. H. Kha
Tutorial Example-Feasible Region
Constrained Optimization 13 H. H. Kha
Feasible region
Constrained Optimization 14 H. H. Kha
Given a pint xF on the constraint surface
Feasible region
Constrained Optimization 15 H. H. Kha
Given a pint xF on the constraint surface
Condition to decrease the cost function
Constrained Optimization 16 H. H. Kha
Condition to decrease the cost function
Constrained Optimization 17 H. H. Kha
Condition to remain on
the constraint surface
Constrained Optimization 18 H. H. Kha
Condition to remain on
the constraint surface
Constrained Optimization 19 H. H. Kha
Condition to remain on
the constraint surface
Constrained Optimization 20 H. H. Kha
To minimize function
with a equality constraint
Constrained Optimization 21 H. H. Kha
Condition for a local optimum
Constrained Optimization 22 H. H. Kha
Condition for a local optimum
Constrained Optimization 23 H. H. Kha
From this fact Lagrange Multipliers
make sense
Constrained Optimization 24 H. H. Kha
From this fact Lagrange Multipliers
make sense
Constrained Optimization 25 H. H. Kha
Multiple Equality Constraints
Constrained Optimization 26 H. H. Kha
Constrained Optimization:
Inequality Constraints
Constrained Optimization 27 H. H. Kha
Tutorial Example-Case 1
Constrained Optimization 28 H. H. Kha
Tutorial Example-Cost function
Constrained Optimization 29 H. H. Kha
Tutorial Example-Feasible Region
Constrained Optimization 30 H. H. Kha
How can we recognize
a local minimum
Constrained Optimization 31 H. H. Kha
Easy in this case
Constrained Optimization 32 H. H. Kha
This Tutorial Example has
an Inactive Constraint
Constrained Optimization 33 H. H. Kha
Tutorial example-Case 2
Constrained Optimization 34 H. H. Kha
Tutorial Example-Cost Function
Constrained Optimization 35 H. H. Kha
Tutorial Example-Feasible Region
Constrained Optimization 36 H. H. Kha
How to recognize a local minimum
Constrained Optimization 37 H. H. Kha
How to recognize a local minimum
Constrained Optimization 38 H. H. Kha
How to recognize a local minimum
Constrained Optimization 39 H. H. Kha
Given an equality constraint
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Want a constrained local minimum
Constrained Optimization 41 H. H. Kha
Summary of optimization
with one inequality constraint
Constrained Optimization 42 H. H. Kha
Karush-Kuhn-Tucker conditions
encode these conditions
Constrained Optimization 43 H. H. Kha
Let's check what
the KKT conditions imply
Constrained Optimization 44 H. H. Kha
KKT conditions for
multiple inequality constraints
Constrained Optimization 45 H. H. Kha
KKT for multiple equality & inequality
constraints
Constrained Optimization 46 H. H. Kha
Example 1
Chapter 47 H. H. Kha
Example 1
Chapter 48 H. H. Kha
Example 1
Chapter 49 H. H. Kha
Quadratic Ratio Optimization
Chapter 50 H. H. Kha
Quadratic Ratio Optimization
Chapter 51 H. H. Kha
Quadratic Ratio Optimization
Chapter 52 H. H. Kha
Example 2
Constrained Optimization 53 H. H. Kha
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