Question related to optimization of nature $\text{min} ~~ Q(x) ~~ \text{subjected to} ~~C(x, y) \geq C_0$

This is little off-the-track question than the conventional ones asked in this forum. This is a problem related to my research. I am stuck to an optimization problem of the following nature:

$\text{min} ~~ Q(x) ~~ \text{subjected to} ~~C(x, y) \geq C_0$, where $y \sim \mathcal{N}(\mu, \sigma^2)$.

I have thought of the approach is to integrate out the $y$ from $C(x, y)$ and make it a function of $x, \mu$ and $\sigma^2$ like $\int_{-\infty}^{\infty} C(x, y) \mathcal{N}(y; \mu, \sigma^2) dy$. Is this approach mathematically correct? What problems can come on the value $C_0$?

All the functions, $Q(x)$, $C(x, y)$ are exponential in nature and contains $\text{erf}$ functions as components.

• Are we trying to satisfy the constraint with some probability $p$, or must it be always satisfied? If always, why not take $y^*$ to be the $y$ which minimizes $C$? Commented Jul 30, 2017 at 16:35
• Thanks a lot for your comment. Yes that can be a possibility to fix $y$ at $y^*$ and get different values of $C(x,y)$. A range of values of $y$ can be chosen. But the problem in my case is $x \in \mathbb{R}^{33}$ and $y \in \mathbb{R}^{33}$ also. So, a prohibitively large number of $y$ values will be required which will require a prohibitively large number of evaluation of $C(x,y)$. Thats why I wanted to have a closed form expression of $C(x,y)$. And for the first question, the constraint must always be satisfied. Moreover, the $C(x,y)$ has a closed form expression. Commented Jul 30, 2017 at 16:55

Because $y$ is random, you have to specify in what manner you want the constraint to hold.
Your "integrating out" thought apparently corresponds to having the constraint satisfied in expectation, i.e., $E(C(x,y)) \ge 0$. This renders the optimization problem a deterministic optimization problem.
An alternative possibility would be to specify that the constraint is satisfied with some minimum probability. i.e, $P(C(x,y) \ge C_0) \ge p$, where $p$ is part of the specification of the optimization problem. This is known as a chance constraint, making your optimization problem a "chance-constrained program", for which there is a large literature. This is a type of stochastic optimization (programming) problem. See for example https://web.stanford.edu/class/ee364a/lectures/chance_constr.pdf .
• Thanks a lot for your comment. The constraint is deterministic in nature and I have derived a closed form expression based upon the distribution of $x$. Moreover for the second question the constraint must be satisfied every time. The type of constraint (chance constraint) you have written is extensively used in reliability based design optimization (RBDO). I think if I keep the value of $p$ at $0.999$ then also it will be a kind of deterministic optimization. Commented Jul 30, 2017 at 16:59
• Pardon my ignorance. If $x$ is a decision variable then can it not have a distribution? I mean can a decision variable not be a random variable? Second thing is, suppose $x$ and $y$ both are deterministic variables then is there any optimization procedure for this type of problem? Because, the objective does not have the other set of variable $y$. Commented Jul 30, 2017 at 17:14
• In particular, I am having this problem in robust design of parameters and tolerances in quality design. $x$ are tolerance variables and $y$ are parameter variables i.e. any system parameter can be described as $y \pm 3x$. The objective function is cost of assigning tolerances $x$ and the cost does not depend on parameter $y$ values. But in constraint we have to satisfy the quality yield which is a function of both parameter and tolerance values. Now in robust design the objective is to find the parameter values which will be least sensitive to both external and internal errors. Commented Jul 30, 2017 at 17:44