Maximize $c^t x + x^T A x$ subject to $x^T x = 1$ where $A \succeq 0$

Question

The vector Bingham-von Mises-Fisher distribution is defined on on the sphere $S^{p-1}$ and has density $$p(x \vert c, A) \propto \text{exp}\{c^Tx + x^TAx\}$$ with respect to the uniform measure on $S^{p-1}.$ Assume $c\in\mathbb{R}^p\setminus \{0\}$ and $A \succeq 0.$ The modal set of the distribution is the set of solutions to the optimization problem in the title. How can I characterize this set and under what conditions does it consist of a single point?

Answer 1

We have a non-convex quadratically constrained quadratic program (QCQP). Let $\mathrm b := -\frac 12 \mathrm c$ and

$$\mathcal L (\mathrm x, \lambda) := \mathrm x^{\top} \mathrm A \, \mathrm x - 2 \mathrm b^{\top} \mathrm x - \lambda (\mathrm x^{\top} \mathrm x - 1)$$

be the Lagrangian. Taking the partial derivatives and finding where they vanish, we obtain

$$\left( \mathrm A - \lambda \mathrm I \right) \mathrm x = \mathrm b \qquad\qquad\qquad \| \mathrm x \|_2 = 1$$

We have two cases to consider. If

• $\lambda$ is an eigenvalue of $\mathrm A$, then the linear system $\left( \mathrm A - \lambda \mathrm I \right) \mathrm x = \mathrm b$ has either no solution at all or infinitely many solutions. In the latter case, we intersect the affine solution space with the unit Euclidean sphere to find solutions of the QCQP.

• $\lambda$ is not an eigenvalue of $\mathrm A$, then the linear system $\left( \mathrm A - \lambda \mathrm I \right) \mathrm x = \mathrm b$ has a unique solution. If this unique solution has unit Euclidean norm, we have found the unique solution of the QCQP.