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Appendix C

# Hints on Constrained Optimization

Chapter Outline

C.1 Equality Constraints 1023

C.2 Inequality Constraints 1025

The Karush-Kuhn-Tucker (KKT) conditions 1025

Min-Max duality 1026

Saddle point condition 1027

Lagrangian duality 1027

Convex programming 1028

Wolfe dual representation 1029

References 1029

## C.1 Equality Constraints

We will first focus on linear equality constraints and then generalize to the nonlinear case. The problem is cast as

$\begin{array}{ll}\hfill \underset{\mathbit{\theta }}{min}& J\left(\mathbit{\theta }\right),\hfill \\ \hfill \text{s.t.}& A\mathbit{\theta }=\mathbit{b},\hfill \end{array}$ where A is an m × l matrix and b, θ are m × 1 and l × 1 vectors, respectively. It is assumed that the cost function J(θ) is twice continuously differentiable and it is, in general, ...

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