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Claudio_Maggioni_4/Claudio_Maggioni_4.md

@@ -217,7 +217,7 @@ $$L(x, \lambda) = c^T x - \lambda^T (Ax - b) - s^T x$$
 The KKT conditions are the following:
 
 1. The partial derivative of the lagrangian w.r.t. $x$ is 0:
-   $$\nabla_x L(x, \lambda) = c - A^T \lambda - s = 0 \Leftrightarrow A^T \lambda
+   $$\nabla_x L(x, \lambda, s) = c - A^T \lambda - s = 0 \Leftrightarrow A^T \lambda
    + s = c$$
 2. Equality constraints hold:
    $$Ax - b = 0 \Leftrightarrow Ax = b$$
@@ -233,7 +233,8 @@ The KKT conditions are the following:
 
 We define the dual problem is the following way:
 
-$$\max b^T \lambda \;\; \text{ s.t. } \;\; A^T \lambda \leq c \;$$
+$$\max b^T \lambda \;\; \text{ s.t. } \;\; c - A^T \lambda \geq 0
+\Leftrightarrow A^T \lambda \leq c \;$$
 
 We then introduce a slack variable $s$ to find the equality and inequality
 constraints:
@@ -250,12 +251,12 @@ $$\min - b^T \lambda \;\; \text{ s.t. } \;\; A^T \lambda + s = c \; \text{ and
 
 We then compute the Lagrangian of the dual problem:
 
-$$L(\lambda, x) = -b^T \lambda + x^T (A^T \lambda + s - c) - x^T s  = - b^T
+$$L(\lambda, x, s) = -b^T \lambda + x^T (A^T \lambda + s - c) - x^T s  = - b^T
 \lambda + x^T (A^T \lambda - c)$$
 
 The KKT conditions are the following:
 
-1. The partial derivative of the lagrangian w.r.t. $x$ is 0: $$\nabla_{\lambda}
+1. The partial derivative of the lagrangian w.r.t.\ $\lambda$ is 0: $$\nabla_{\lambda}
    L(\lambda, x) = - b^T + x^T A^T = 0 \Leftrightarrow Ax = b$$
 2. Equality constraints hold: $$A^T \lambda + s = c$$
 3. Inequality constraints hold: $$c - A^T \lambda \geq 0 \Leftrightarrow s \geq