Optimization Theory Primal Optimization Problem subject to: Primal Optimal Value:

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Presentation transcript:

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 KKT Complementarity Condition

Optimization Theory Dual Lagrangian Function

Optimization Theory Dual Optimization problem subject to: Primal subject to: Dual For convex optimization problem:

Support Vector Machine (SVM) SVM Classification Regression Linear SVM Nonlinear SVM

Support Vector Machine (SVM) Linear SVM Training Prediction

Support Vector Machine (SVM) Linear SVM Training Training dataset: Optimal Separating Hyperplane:

Support Vector Machine (SVM) Linear SVM Prediction Testing dataset:

Support Vector Machine (SVM) Linear SVM: Separable case The optimal hyperplane is obtained by maximizing the margin Support vectors

Support Vector Machine (SVM) Linear SVM: Separable case Primal Problem

Support Vector Machine (SVM) Linear SVM: Separable case

Support Vector Machine (SVM) Linear SVM: Separable case

Support Vector Machine (SVM) Linear SVM: Separable case

Support Vector Machine (SVM) Linear SVM: Separable case Dual Problem

Support Vector Machine (SVM) Linear SVM: Separable case

Support Vector Machine (SVM) Linear SVM: Non-separable case

Support Vector Machine (SVM) Linear SVM: Non-separable case

Support Vector Machine (SVM) Linear SVM: Non-separable case

Support Vector Machine (SVM) Linear SVM: Non-separable case (Primal Problem) Subject to:

Support Vector Machine (SVM) Linear SVM: Non-separable case (Primal Problem)

Support Vector Machine (SVM) Linear SVM: Non-Separable case

Support Vector Machine (SVM) Linear SVM: Non-separable case (Implementation) Quadratic programming Problem