Chance Constrained Programming (CCP) is a mathematical optimization approach used to handle problems under uncertainty. It was first introduced by Charnes and Cooper in 1959 and further developed by Miller and Wagner in 1965. CCP is widely used in various fields, including finance, engineering, and operations research, to optimize decision-making processes where certain constraints need to be satisfied with a specified probability.

Chance Constrained Programming involves the use of probability and confidence levels to handle uncertainty in optimization problems. It distinguishes between single and joint chance constraints:

A general chance constrained optimization problem can be formulated as follows:

${\displaystyle \min f(x,u,\xi ){\text{s.t. }}g(x,u,\xi )=0,\Pr\{h(x,u,\xi )\geq 0\}\geq \alpha }$

Here, ${\displaystyle f}$ is the objective function, ${\displaystyle g}$ represents the equality constraints, ${\displaystyle h}$ represents the inequality constraints, ${\displaystyle x}$ represents the state variables, ${\displaystyle u}$ represents the control variables, ${\displaystyle \xi }$ represents the uncertain parameters, and ${\displaystyle \alpha }$ is the confidence level.

Common objective functions in CCP involve minimizing the expected value of a cost function, possibly combined with minimizing the variance of the cost function.

To solve CCP problems, the stochastic optimization problem is often relaxed into an equivalent deterministic problem. There are different approaches depending on the nature of the problem:

Chance constrained programming is used in engineering for process optimisation under uncertainty and production planning and in finance for portfolio selection. It has been applied to renewable energy integration, generating flight trajectory for UAVs, and robotic space exploration.