Dear Professor,

I would like to solve an optimization problem \max_{x\in \mathbb{R}^n,x\geq0} \varphi(x)=\mathbb{P}[\xi\leq b+Ax], where \xi \in\mathbb{R}^m is a Gaussian random vector with variance-covariance matrix \Sigma and mean \mu. I formed a problem with standard normal as follows,

\mathbb{P}[\xi\leq b+Ax]=\mathbb{P}[W(\xi - \mu)\leq W(Ax+b-\mu)], where W is the whitening matrix.

Therefore, the r.v. follows the standard multivariate normal distribution. The problem becomes: \max_{x\in \mathbb{R}^n,x\geq0} \Phi(WAx+W(b-\mu)), where \Phi is the CDF of the standard multivariate normal. As \Phi is quasiconcave, I planned to solve this problem by posing a series of convex feasibility problems.

The original problem is equivalent to the feasibility problem.

\begin{array}{*{20}{c}}
{{\rm{find}}}&x\\
{{\rm{s}}{\rm{.t}}{\rm{.}}}&{\Phi \left( {WAx + W\left( {b - \mu } \right)} \right) \ge \eta }\\
{}&{x \ge 0}
\end{array}

I wanted to solve this quasiconcave problem by the bisection method. However, when I tried to solve the value of the first constraint with the function of ** mvncdf**, it reported an error.! My program is shown below:

Is there any way to solve this problem? Do you have any suggestions to deal with this problem?

Thanks a lot!

Best wishes,

Xin