Chance constrained stochastic programming

Question

A stochastic programming optimizes the expectation of a cost function with respect to values.

\begin{cases} {\boldsymbol x}=\text{argmin}~ E(f({\boldsymbol x}))\\ {\boldsymbol g}({\boldsymbol x})<{\boldsymbol 0} \end{cases} where $E$ refers to expectation.

A chance constrained programming is a programming with constrained chance

\begin{cases} {\boldsymbol x}=\text{argmin}~ f({\boldsymbol x})\\ P({\boldsymbol g}({\boldsymbol x})<{\boldsymbol 0})>\alpha \end{cases} where $P$ refers to probability and $\alpha$ refers to the confidence level.

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But I am looking for a different programming with both expectation and chance constrained properties like

\begin{cases} {\boldsymbol x}=\text{argmin}~ E(f({\boldsymbol x}))\\ P({\boldsymbol g}({\boldsymbol x})<{\boldsymbol 0})>\alpha \end{cases}

• Does such an optimization exist?

• Is such an optimization common?

• If yes, what is the name of this optimization?

• Is there any academic publication about this optimization?

Answer 1

I think your definition of a stochastic program is suspect (I have never seen it defined this way). It seems that you're saying that a stochastic program is an optimization problem of the form

\begin{equation} \begin{array}{rl} \min\ & \mathbb{E}_\xi[f(x,\xi)]\\ \text{s.t.}\ & g(x)\leqslant0 \end{array} \end{equation}

where $\xi$ is some random variable. In many cases, such a problem would be uninteresting. For example, suppose (as is often done in the literature) that $f$ has the form

\begin{equation} f(x,\xi)=\xi_1x_1+\dots+\xi_nx_n. \end{equation}

where $\xi_i$ are independent random variables. Then the objective function is given by

\begin{equation} \mathbb{E}_\xi[f(x,\xi)]=\mathbb{E}_\xi[\xi_1x_i+\dots+\xi_nx_n]=\mathbb{E}[\xi_1]x_1+\dots+\mathbb{E}[\xi_n]x_n. \end{equation}

Since the values $\mathbb{E}[\xi_i]$ are just constants, we've reduced the objective function to a (deterministic) affine function of $x$. This isn't really a stochastic program at all--we just replaced some random variables with their expected values.

This confusion makes it hard to address the rest of your question. In principle, chance constraints can be combined with lots of different models--it really depends on what you're modeling.

I would suggest two books to read up on this subject:

1. Birge and Louveaux have a very good textbook (published by Springer) that introduces the fundamentals of stochastic programming. In particular, they introduce the concept of linear programming with recourse, which is at the heart of most stochastic programming as it is studied today.
2. Prekopa wrote a seminal text (called simply "Stochastic Programming") which is much more technical than Birge and Louveaux, but treats chance constraints much more thoroughly. In particular, they explore different places where chance constraints can appear.