In this paper, we study parametric analysis of semidefinite optimization problems with respect to the perturbation of objective function. We investigate the behavior of the optimal partition and optimal set mapping in a so called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition, which have been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on the distance between the invariant subspaces spanned by the approximation of the optimal partition. where C, X, A i ∈ S n for i = 1, . . . , m, b ∈ R m ,C ∈ S n is a fixed direction, and S n denotes the vector space of symmetric n × n matrices endowed with the inner product C, X := Tr(CX). In this context, X 0 means that X belongs to the cone of positive semidefinite matrices, which is denoted by S n + . The primal and dual feasible CONTACT Ali Mohammad-Nezhad. Email: alm413@lehigh.edu; Tel.: +1(610)758-2903. set mappings are defined as Since P(.) is independent of ǫ, primal feasibility either holds or fails for all ǫ. Thus, we assume that P(ǫ) = ∅ for all ǫ.
Recently, Mohammad-Nezhad and Terlaky REF introduced the concepts of a nonlinearity interval and transition point for the optimal partition of parametric SDO problems.
Parametric analysis of semidefinite optimization
arXiv: Optimization and Control
Mathematics