https://shika-b.github.io/tags/mathematics/Recent content in Mathematics on Slow DataHugoen-usSun, 09 Nov 2025 13:44:57 +0100https://shika-b.github.io/blog/lagrangian_duality/Sun, 09 Nov 2025 13:44:57 +0100https://shika-b.github.io/blog/lagrangian_duality/<h2 id="optimization-context">Optimization context</h2> <p>Let&rsquo;s imagine for a moment we are trying to solve a class of <em>constrained</em> optimization problems of the form </p> $$\begin{align*} &\inf_x f(x) \\ &\text{subject to } &g_i(x) \leq 0 \text{ for } 1 \leq i \leq m \\ &&h_j(x) = 0, \text{ for } 1 \leq j \leq n \end{align*} $$<p> where $f, g_1, \ldots, g_m, h_1, \ldots, h_n \in \mathcal{C}$ for $\mathcal{C}$ a fixed class of smooth functions. We fix such a problem $(P)$. What is a reasonable strategy to solve this?</p>