An optimization written in standard form can be transformed via Generalized Lagrangian Function.

For example, for an original minimization problem,

; , ,

we can write the generalized Lagrangian function of the original problem as follows,

.

**The Lagrangian Transformation of the Original Problem**

The *original problem* can be rewritten as the *‘min-max problem of the generalized Lagrangian function’*, which equals to optimize the **sup (min-max) **of the function J(), in respective to x and Lagranian variables , like follows,

The above problem has exactly the same solution as the original problem.

**The Dual Form**

From the original problem, we can also write out the expression of the dual problem, in the form of the *‘max-min problem of the generalized Lagrangian function’* , which optimizes the **inf (max-min)** of J(),

However, this problem does not necessarily have the same solution as the original one. Normally, we have,

.

From the equation we see that the solution of the dual problem at least provides the lower-bound of the solution to the original problem, showing a positive dual gap . The equality holds if and only if the dual gap , when the the original problem is a convex one with Slater’s condition.

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