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1. Introduction
In this paper, we propose a class of optimization problems called stochastic convex semidefinite programs (SCSDPs):where is a smooth convex function for every realization of on , is a random matrix whose probability distribution is supported on set , is a linear map, and mean that , and is the space of real symmetric endowed with the standard trace inner product and Frobenius norm . is the set of positive semidefinite matrices in .
SCSDPs may be viewed as an extension of the following stochastic models:
(1) Stochastic (Linear) Semidefinite Programs (SLSDPs) ([1–5])
where is the minimum of the problem where denotes the Frobenius inner product between and .
(2) Stochastic Convex Quadratic Semidefinite Programs (SCQSDPs)
where
