(Fenchel Duality) Consider the optimization problem $$ \begin{array}{ll} \displaystyle \min_x \ & f(x) \\ \text{s.t.} \ & g(x) \le 0, \\ & h(x) = 0. \end{array} $$ Its dual problem is $$ \begin{array}{ll} \displaystyle \max_{\lambda, \mu} \ & - ( f + \lambda^\top g + \mu^\top h )^*(0) \\ \text{s.t.} \ & \lambda\ge 0, \end{array} $$ which, assuming \( h(x) := Ax - b \), can be re-written as $$ \begin{array}{ll} \displaystyle \max_{\lambda,\mu,u} \ & \displaystyle -f^*(u^0) - \sum_{i=1}^m \lambda_i g_i^*\left( \frac{u^i}{\lambda_i} \right) + b^\top \mu \\ \text{s.t.} \ & \displaystyle \sum_{i=0}^m u^i + A^\top\mu = 0, \\ & \lambda\ge 0. \end{array} $$

(QP Duality) Consider the QP $$ \begin{align} \min_x \ & \frac 12 x^\top Q x + c^\top x \\ \text{s.t.} \ & Ax = b, \\ & x\ge 0. \end{align} $$ Its dual problem is $$ \begin{align} \max_{x,\lambda,\mu} \ & b^\top \lambda - \frac 12 x^\top Q x \\ \text{s.t.} \ & A^\top \lambda + \mu = c + Qx, \\ & \mu\ge 0. \end{align} $$ If, moreover, \( Q \) is invertible, the dual can be re-written as $$ \begin{align} \max_{\lambda, \mu} \ & b^\top \lambda - (A^\top\lambda + \mu - c)^\top Q (A^\top\lambda + \mu - c) \\ & \mu \ge 0. \end{align} $$

(Conic Duality) Consider the conic program $$ \begin{array}[t]{ll} \displaystyle \min_x \ & c^Tx \\ \textrm{s.t. } & Ax - b \in K. \end{array} $$ Its dual problem is $$ \begin{array}[t]{ll} \displaystyle \max_\lambda \ & b^T\lambda \\ \textrm{s.t. } & A^T\lambda = c \\ & \lambda\in K^*, \end{array} $$ in which \(K^*\) denotes the dual cone of \( K \).