Let $x \in \mathbb{R}^n$ and let $P$ be a $n \times n$ positive definite symmetrix matrix. It is known that the maximum of $$\begin{array}{ll} \text{maximize} & x^T P \, x\\ \text{subject to} & x^T x \leq 1\end{array}$$ is $\lambda_{\text{max}}(P)$, the largest eigenvalue of $P$. Now consider the following problem $$\begin{array}{ll} \text{maximize} & x^T P \, […]

I would like to solve the following optimisation problem: $$\text{minimize} \quad x’Ax \qquad \qquad \text{subject to} \quad x’Bx = x’Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. Does anyone have a suggestion for an efficient way of solving this? Thank you.

It is easy to find the minimum of $\|Ax-b\|_2$, when $A$ has full column rank. But how is the case when we add the constraint $\|x\|_2=1$? Or, to be explicit, $$\min_{\|x\|_2=1}\|Ax-b\|_2=?$$ My idea is to construct the corresponding Lagrange function $$L(x,\lambda) = (Ax-b)^T(Ax-b) + \lambda(x^T x – 1)$$ Differentiate the function with respect to $x,\lambda$, […]

Consider the $n$-dimensional quadratically constrained quadratic optimization problem $$\begin{array}{ll} \text{maximize} & \frac12 x^T A x + b^T x\\ \text{subject to} & \| x \|_2 \le 1\end{array}$$ where $A$ is a symmetric $n\times n$ matrix that may be indefinite. Given the symmetry of the constraint, is there a nice closed-form solution, perhaps in terms of the […]

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