I am trying to find the minimum of $-x_1$ with restrictions $\bar g\leq\bar 0$ so that $$\bar g=\begin{pmatrix} (x_1+2)^2+(x_2-4)^2-20\\ (x_1+2)^2+x_2^2-20\\ -x_1\end{pmatrix}\leq \begin{pmatrix}0\\0\\0\\\end{pmatrix}=\bar0$$ I used KKT conditions to solve this puzzle below so $x_1=\frac{4\mu_1+4\mu_2-1-\mu_3}{2(\mu_1+\mu_2)}$ and $x_2=\frac{4\mu_1}{\mu_1+\mu_2}$ where $\mu_i\in\mathbb R\forall i$. I know from the graphical plot that the solution is something like $(1.5,1.5)$ but I cannot […]

I am trying to understand visually what this condition actually mean. It is the optimality condition in KKT. It means something like that constraint -set, objective -set and hyperplane -set has no common directions. I have seen only abstract definitions without examples so I have very vague understanding about this statement. Please, explain it, perhaps […]

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and $g_i(x)$ is the $i$th constraint function. $I$ denotes the number of constraints. Consider now a point $x^\star$ that satisfies the Karush-Kuhn-Tucker conditions. In general, in non-convex optimization, a KKT point can be […]

I have the following nonlinear optimization problem: $$ \begin{align*} \text{Find } x \text{ that maximizes } & \frac{1}{\|Ax\|} (Ax)^{\top} y \\ \text{Subject to } & \sum_{i=1}^n x_i = 1 \\ & x_i \geq 0 \; \forall \: i \in \{1\dots n\} \\ \text{Where } & A \in \mathbb{R}^{m \, \times \, n} \\ & x […]

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