Two adaptive scaled gradient projection methods for Stiefel manifold constrained optimization

Computing the completely positive factorization via alternating minimization

“…We note that the equality constraints in (3) are known as orthogonality constraints and define the so called Stiefel manifold. [5][6][7] The method we propose, referred to as splitting alternating minimization (SAM), seeks a solution X, Y of (3) by generating sequences X k , Y k coming from the following alternating minimization scheme…”

“…Otherwise, we look for a solution $$$ X,Y $$$ of (3), and the correspondent $$$ B $$$ in (2) is given by $$$ B= LX $$$. We note that the equality constraints in (3) are known as orthogonality constraints and define the so called Stiefel manifold 5‐7 …”

Spectral Residual Method for Nonlinear Equations on Riemannian Manifolds

“…Different iterative methods have been developed for solving (2). Some popular schemes are based on gradient method [6,11,15,18,19,20,21], conjugate gradient methods [2,9,34], Newton's method [2,26], or quasi-Newton methods [2]. All these numerical methods can be used to find a zero of the following tangent vector field equation,…”

“…We remark that with this relaxed condition, Algorithm 1 is well defined. In fact, if at iteration k the procedure does not stop at Step 4, then Z k is a descent direction (see Lemma 2), and for all ρ 1 ∈ (0, 1) there exists t > 0 such that the non-monotone Zhang-Hager condition (20) holds by continuity, for τ > 0 sufficiently small(a proof of this fact appears in [31]). In addition, it follows form Step 4,(20) and Lemma 2 that…”