Constrained Mathematical Programming

The general constrained mathematical programming problem is to find an x ∈ ℝ n $$ \boldsymbol {x}\in \mathbb {R}^{n}$$ to minimize the objective function f x $$\displaystyle \ f\left ( \boldsymbol {x}\right ) {} $$ subject to the constraints h i x = 0 , i = 1 , … , m , $$\displaystyle h_{i}\left ( \boldsymbol {x}\right ) =0,\quad i=1,\ldots ,m, $$ h i x ≤ 0 , i = m + 1 , … , m + r , $$\displaystyle h_{i}\left ( \boldsymbol {x}\right ) \leq 0,\quad i=m+1,\ldots ,m+r, $$ where f and h i, i = 1, …, m + r, are continuously differentiable functions of the n-vector variable x.