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# Optimization Theory - PowerPoint PPT Presentation

Optimization Theory. Primal Optimization Problem. subject to:. Primal Optimal Value:. Optimization Theory. Convex Optimization Problem. subject to:. : convex functions. Optimization Theory. Primal Lagrangian function. subject to:. Optimization Theory. Kuhn-Tucker Theory.

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Primal Optimization Problem

subject to:

Primal Optimal Value:

Convex Optimization Problem

subject to:

: convex functions

Primal Lagrangian function

subject to:

Kuhn-Tucker Theory

KKT Complementarity Condition

Dual Lagrangian Function

Dual Optimization problem

subject to:

subject to:

Primal

Dual

For convex optimization problem:

• SVM

• Classification

• Regression

• Linear SVM

• Nonlinear SVM

• Linear SVM

• Training

• Prediction

• Linear SVM Training

Training dataset:

Optimal Separating Hyperplane:

• Linear SVM Prediction

Testing dataset:

• Linear SVM: Separable case

• The optimal hyperplane is obtained by maximizing the margin

• Support vectors

• Linear SVM: Separable case

Primal Problem

• Linear SVM: Separable case

• Linear SVM: Separable case

• Linear SVM: Separable case

• Linear SVM: Separable case

Dual Problem

• Linear SVM: Separable case

• Linear SVM: Non-separable case

• Linear SVM: Non-separable case

• Linear SVM: Non-separable case

• Linear SVM: Non-separable case (Primal Problem)

Subject to:

• Linear SVM: Non-separable case (Primal Problem)

• Linear SVM: Non-Separable case

• Linear SVM: Non-separable case (Implementation)

Quadratic programming Problem