Lagrange Dual Theory for NLP

1. Classic form of nonlinear programming
   F1: \(f,h,g\) are arbitrary (not necessarily diferentiable or continuous) functions.
   
   F2:
   
   F3:
   \[\begin{align*} \min \; & f(x)\\ \textrm{s.t.} \; & g(x)\leq 0\\ & h(x)=0 \\ & x\in X; \end{align*}\]
   As \(h(x)=0\) can be equivalently written as two inequality constraints \(h(x)\leq 0\) and \(-h(x)\leq 0\), we only consider
   

2. Lagrange function and its dual
   1) Lagrange function:\(\mu \geq 0\) is called the Lagrange multiplier.
   
   2)Lagrange dual function