op05 constrained optimization

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: [email protected]

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


Optimization


Unconstrained Optimization


Unconstrained Minimization


Unconstrained Maximization


Constrained Optimization:

Equality Constraints


Tutorial Example


Tutorial Example-Cost function


Tutorial Example-Feasible Region


Feasible region


Given a pint xF on the constraint surface

Condition to decrease the cost function


Condition to remain on

the constraint surface


To minimize function

with a equality constraint


Condition for a local optimum


From this fact Lagrange Multipliers

make sense


Multiple Equality Constraints


Constrained Optimization:

Inequality Constraints


Tutorial Example-Case 1


Tutorial Example-Cost function


How can we recognize

a local minimum


Easy in this case


This Tutorial Example has

an Inactive Constraint


Tutorial example-Case 2


Tutorial Example-Cost Function


How to recognize a local minimum


Given an equality constraint


Want a constrained local minimum


Summary of optimization

with one inequality constraint


Karush-Kuhn-Tucker conditions

encode these conditions


Let's check what

the KKT conditions imply


KKT conditions for

multiple inequality constraints


KKT for multiple equality & inequality

constraints


Example 1

Chapter 47 H. H. Kha

Example 1


Quadratic Ratio Optimization


Example 2


op05 constrained optimization

Documents