I am not an expert on constrained optimization problems so I was wondering whether some of you could help me out.
Let $h_{\mu}$ stand for :
\begin{equation} h_{\mu} (\vec{X}) = \frac{1}{\sqrt{N}} \vec{X} \cdot \vec{\xi}_{\mu} - \sigma \end{equation}
Here $\vec{X}$ and $\vec{\xi}_{\mu}$ $\in \mathbb{R}^{N}$ with $\mu = 1..M$. Thus $\xi_{\mu}^{i} \in \mathbb{R}^{M \times N}$. $\sigma \in \mathbb{R}$ (can be negative)
I then define a constraint satisfaction problem to be the determination of all $\vec{X}$ such that $\forall \mu$ for given $\vec{\xi}_{\mu}$, we have
\begin{equation} h_{\mu}(\vec{X}) > 0 \end{equation}
The $\vec{\xi}_{\mu}$ are taken from a gaussian distribution $\mathcal{N}(0,1)$ and $\sigma$ is kept fixed. There are some constraints on $\vec{X}$:
\begin{align} \frac{1}{N}\sum_{i=1}^{N} X_{i} = m \\ \frac{1}{N} \sum_{i=1}^{N} X_{i}^2 =1 \end{align}
I define a cost function of the following form:
\begin{equation} \text{H}[\vec{X}] = \frac{1}{2} \sum_{\mu = 1}^{M} \Big( \frac{1}{\sqrt{N}}\vec{X} \cdot \vec{\xi}_{\mu} - \sigma \Big)^{2} \theta(-h_{\mu}) \end{equation}
In other words, I incur a cost only $\frac{1}{\sqrt{N}}\vec{X} \cdot \vec{\xi}_{\mu} - \sigma < 0$, i.e. whenever $h_{\mu} <0$.
The optimisation problem then is to minimize the cost function $\text{H}$ subject to the two constraints on $\vec{X}$. The other variables $\vec{\xi}_{\mu}$ and $\sigma$ are given and kept fixed.
My questions are :
- Is there a specific name for these types of optimisation problems?
- Are there analytical ways to treat such problems?
My attempt at computing the minimum of $\text{H}$ has essentially involved writing my own gradient descent routine which, as you may imagine, is not extremely performant leading to very long waiting times to find the solution. I wondered whether such problems have already been studied in the literature, so that I could use some existing knowledge to do this problem.
Thanks for your help. Cheers.