Descent directions and efficient solutions in discretely distributed stochastic programs (Q1210734)

In this book the author considers the problem \[ \text{minimize }Eu(A(\omega)x-b(\omega))\text{ subject to }x\in D, \] where \((A(\omega),b(\omega))\) is an \(m\times (n+1)\) random marix, E denotes expectation and D is a convex subset of \(R^ n\). It is assumed that either A(\(\omega)\) or both A(\(\omega)\), b(\(\omega)\) have a discrete distribution. The consideration may be divided into the following two parts. 1. Construction of feasible descent directions is done using a notion of stochastic dominance. In the case of a discrete distribution it leads to a system of linear equalities and inequalities. 2. The notion of a stationary point is defined based on the notion of feasible descent direction. Stationarity is carefully studied i.e. it is characterized in several ways and in many particular cases. Links between stationarity and optimality are studied. Results in both parts are obtained without any derivative of the objective function.