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I have a simple and stupid question if I have a convex quadratic optimization problem with polyhedral constraints as follows: $$ \begin{aligned} \inf_{x \in \mathbb{R}^{n}} & x^{\top} Ax + b^{\top}x\\ & \text{s.t. } Hx \le h. \end{aligned} $$

If I introduce the Lagrange function as below: $$ \inf_{x\in\mathbb{R}^{n}} \sup_{\beta \ge 0} x^{\top}Ax + b^{\top}x + \beta^{\top}(Hx-h), $$ under which optimality condition $$ \inf_{x\in\mathbb{R}^{n}} \sup_{\beta \ge 0} x^{\top}Ax + b^{\top}x + \beta^{\top}(Hx-h) = \sup_{\beta \ge 0} \inf_{x\in\mathbb{R}^{n}} x^{\top}Ax + b^{\top}x + \beta^{\top}(Hx-h) $$ holds?

As conditions of Von Neumann's Minimax theorem do not hold here, how should I consider the minimax theorem?

Any hints would be appreciated!

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  • $\begingroup$ If $x^TAx+b^Tx$ is convex, you can try using duality or the KKT conditions $\endgroup$ Commented Mar 7, 2024 at 17:31
  • $\begingroup$ Thanks for your reply. Lagrange duality works. Found condition in Theorem 4.14 of the book "Convex Analysis and Nonsmooth Optimization" by Dmitriy Drusvyatskiy. $\endgroup$ Commented Mar 13, 2024 at 12:24

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