  


On mixed integer reformulations of monotonic probabilistic programming problems with discrete distributions
Vladimir Norkin(norkini.com.ua) Abstract: The paper studies large scale mixed integer reformulation approach to stochastic programming problems containing probability and quantile functions, under assumption of discreteness of the probability distribution involved. Jointly with general sample approximation technique and contemporary mixed integer programming solvers the approach gives a regular framework to solution of practical probabilistic programming problems. In the literature this approach is applied basically to chance constrained problems with discrete probability distribution. In the present paper we extend it to more complicated problems, containing several probability/quantile functions both in the objective and constraints; in particular we consider probability and quantile functions optimization problems. First we show that the approach is applicable to probabilistic programming problems with objective and constraint functions monotonically depending on several probability functions. Next we remark that the general framework is applicable to optimization problems containing quantile functions, provided the last are uniformly bounded from below. As a solution technique beside general solvers we specialize branch and bound method for optimization of the probability function. To speed up the branch and bound method we derive dominance between integral variables basing on the dominance between probabilistic scenarios. Finally we note that some of probability functions may be defined on different probability spaces, this allows to treat within the considered framework some ambiguous probabilistic problems with uncertain probability distributions. As illustrations of the general framework we consider applications to (robust) safety first portfolio selection and quantile optimization. Keywords: stochastic programming, probabilistic programming, probability function, quantile function, mixed integer programming, branch and bound method Category 1: Stochastic Programming Category 2: Integer Programming ((Mixed) Integer Nonlinear Programming ) Category 3: Applications  OR and Management Sciences (Finance and Economics ) Citation: V.M.Glushkov Institute of Cybernetics, Glushkov avenue 40, 03178, Kiev, Ukraine; 17 May, 2010 Download: [PDF] Entry Submitted: 05/17/2010 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  