Optimality conditions for mathematical programs with orthogonality type constraints

TL;DR

This paper establishes optimality conditions for mathematical programs with orthogonality constraints, linking T-stationarity to Morse theory, regularization convergence, and sparsity relaxation, providing a comprehensive theoretical framework.

Contribution

It introduces T-stationarity as a necessary optimality condition for MPOC and connects it with Morse theory, regularization convergence, and sparsity relaxation analysis.

Findings

T-stationarity captures the global structure of MPOC.

Karush-Kuhn-Tucker points of regularization converge to T-stationary points.

T-stationarity applied to sparsity relaxation leads to M-stationary points.

Abstract

We consider the class of mathematical programs with orthogonality type constraints (MPOC). Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For MPOC a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold. First, it allows to capture the global structure of MPOC in terms of Morse theory, i. e. deformation and cell-attachment results are established. For that, nondegeneracy for the T-stationary points is introduced and shown to hold at a generic MPOC. Second, we prove that Karush-Kuhn-Tucker points of the Scholtes-type regularization converge to T-stationary points of MPOC. This is done under the MPOC-tailored linear independence constraint qualification (LICQ), which turns out to be a generic property too. Third, we show that T-stationarity applied to the relaxation of sparsity constrained nonlinear optimization (SCNO) naturally leads to its M-stationary points. Moreover, we argue that all T-stationary points of this relaxation become degenerate.