stiefel manifold

A Class of Smooth Exact Penalty Function Methods for Optimization Problems with Orthogonality Constraints

  • Nachuan Xiao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization Tags , ,

    Updating the augmented Lagrangian multiplier by closed-form expression yields efficient first-order infeasible approach for optimization problems with orthogonality constraints. Hence, parallelization becomes tractable in solving this type of problems. Inspired by this closed-form updating scheme, we propose an exact penalty function model with compact convex constraints (PenC). We show that PenC can act as an … Read more

    A Riemannian Conjugate Gradient Algorithm with Implicit Vector Transport for Optimization on the Stiefel Manifold

  • Hugo Lara
  • Harry Oviedo
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In this paper, a reliable curvilinear search algorithm for solving optimization problems over the Stiefel manifold is presented. This method is inspired by the conjugate gradient method, with the purpose of obtain a new direction search that guarantees descent of the objective function in each iteration. The merit of this algorithm lies in the fact … Read more

    A New First-order Algorithmic Framework for Optimization Problems with Orthogonality Constraints

  • Xiaojun Chen
  • Bin Gao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization Tags , , , , ,

    In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean … Read more

    A Riemannian conjugate gradient method for optimization on the Stiefel manifold

  • Xiaojing Zhu
    • Categories Applications - Science and Engineering, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , ,

    In this paper we propose a new Riemannian conjugate gradient method for optimization on the Stiefel manifold. We introduce two novel vector transports associated with the retraction constructed by the Cayley transform. Both of them satisfy the Ring-Wirth nonexpansive condition, which is fundamental for convergence analysis of Riemannian conjugate gradient methods, and one of them … Read more

    A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold

  • Yu-Hong Dai
  • Bo Jiang
    • Categories Applications - Science and Engineering, Nonlinear Optimization Tags , , , , , , , , , , ,

    This paper considers optimization problems on the Stiefel manifold $X^TX=I_p$, where $X\in \mathbb{R}^{n \times p}$ is the variable and $I_p$ is the $p$-by-$p$ identity matrix. A framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of $X$ and the null space of $X^T$. While this general framework … Read more

    A Feasible method for Optimization with Orthogonality Constraints

  • Zaiwen Wen
  • Wotao Yin
    • Categories Approximation Algorithms, Nonlinear Optimization Tags , , , , , , , , , ,

    Minimization with orthogonality constraints (e.g., $X^\top X = I$) and/or spherical constraints (e.g., $\|x\|_2 = 1$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. … Read more