## Karush-Kuhn-Tucker (KKT) Conditions

The necessary conditions for a constrained local optimum are called the Kuhn-Tucker Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development.

#### Part 1: 5 Minute Tutorial on the KKT Conditions

This 5 minute introductory video reviews the 4 KKT conditions and applies them to solve a simple quadratic programming (QP) problem with:

- 1 Quadratic objective function
- 2 Linear equality constraints
- 3 Variables (x
_{1}, x_{2}, x_{3})

Download the following worksheet on KKT conditions. The video below reviews the solution to this worksheet.

#### Part 2: 5 Minute Tutorial with KKT Conditions and Inequality Constraints

This next 5 minute introductory is similar to the previous one but solves a problem with inequality constraints instead of equality constraints. The problem is a simple quadratic programming (QP) problem with:

- 1 Quadratic objective function
- 2 Linear inequality constraints
- 3 Variables (x
_{1}, x_{2}, x_{3})

Download the following worksheet on KKT conditions with inequality constraints. The video below reviews the solution to this worksheet.

#### Part 3: 5 Minute KKT Exercise with both Inequality and Equality Constraints

This 5 minute exercise is similar to the previous ones but solves a problem with both equality and inequality constraints.

Download the following worksheet on KKT conditions with inequality and equality constraints. The video below reviews the solution to this worksheet.

#### Part 4: 5 Minute Application Exercise for the Optimal Volume of a Tank

This 5 minute exercise covers an application to a tank volume optimization. In this case, we specify the final Lagrange multiplier of $8/ft^{3}.

Download the following worksheet on this application of the KKT conditions. The video below reviews the solution to this worksheet.

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