Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations
One of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form . Here is the vector of physical parameters, is a random vector describing the uncertainty of the model, is the constraints mapping, and is the vector of constraint levels. In this paper specific Monte Carlo tools for the estimations of the gradient and Hessian of P(x) are proposed when the input random vector [Lambda] has a multivariate normal distribution and small variances. Using the small variance hypothesis, approximate expressions for the first- and second-order derivatives are obtained, whose Monte Carlo estimations have low computational costs. The number of calls of the constraints mapping g for the proposed estimators of the gradient and Hessian of P(x) is only 1+2Nx+2N[Lambda]. These tools are implemented in penalized optimization routines adapted to stochastic optimization, and are shown to reduce the computational cost of chance constrained programming substantially.
Applied probability Monte Carlo methods Stochastic programming Optimization with constraints Random constraints
In order to set up a list of libraries that you have access to,
you must first login
or sign up.
Then set up a personal list of libraries from your profile page by
clicking on your user name at the top right of any screen.