Dissolving Constraints for Riemannian Optimization

Nachuan, Xiao,; Liu, Xin; Toh, Kim-Chuan

doi:10.48550/arxiv.2203.10319

Cited by 2 publications

(3 citation statements)

References 38 publications

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“…This method assumes that the feasible set of ( 2) is a Riemannian manifold. Thus, one can use Riemannian optimization method such as Riemannian gradient descent and Riemannian trust region method to solve the problem [1,14,16,33,54,55]. Apart from ALM and Riemannian optimization methods, Bellavia et al [9] developed a rank-adaptive interior point method for low-rank linear SDP problems and applied it to matrix completion problems.…”

Section: Feasible Methods and Its Limitationmentioning

confidence: 99%

A Feasible Method for Solving an SDP Relaxation of the Quadratic Knapsack Problem

Tang

Toh

2024

Mathematics of OR

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In this paper, we consider a semidefinite programming (SDP) relaxation of the quadratic knapsack problem. After applying low-rank factorization, we get a nonconvex problem, whose feasible region is an algebraic variety with certain good geometric properties, which we analyze. We derive a rank condition under which these two formulations are equivalent. This rank condition is much weaker than the classical rank condition if the coefficient matrix has certain special structures. We also prove that, under an appropriate rank condition, the nonconvex problem has no spurious local minima without assuming linearly independent constraint qualification. We design a feasible method that can escape from nonoptimal nonregular points. Numerical experiments are conducted to verify the high efficiency and robustness of our algorithm as compared with other solvers. In particular, our algorithm is able to solve a one-million-dimensional sparse SDP problem accurately in about 20 minutes on a modest computer. Funding: The research of K.-C. Toh is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010].

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Section: Feasible Methods and Its Limitationmentioning

confidence: 99%

A Feasible Method for Solving an SDP Relaxation of the Quadratic Knapsack Problem

Tang

Toh

2024

Mathematics of OR

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“…Very recently, [64] proposes constraint dissolving approaches for minimizing smooth functions over a closed Riemannian manifold. In their proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a corresponding constraint dissolving function.…”

Section: Motivationmentioning

confidence: 99%

“…In their proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a corresponding constraint dissolving function. According to [64], the constraint dissolving function for OCP can be expressed as…”

Section: Motivationmentioning

confidence: 99%

A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold

Hu¹,

Nachuan²,

Liu³

et al. 2022

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This paper focus on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems on the Stiefel manifold. We propose a framework for developing subgradient-based methods and establish their convergence properties based on prior works. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach enables the efficient and direct implementations of various unconstrained solvers to nonsmooth optimization problems over the Stiefel manifold.

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Dissolving Constraints for Riemannian Optimization

Cited by 2 publications

References 38 publications

A Feasible Method for Solving an SDP Relaxation of the Quadratic Knapsack Problem

A Feasible Method for Solving an SDP Relaxation of the Quadratic Knapsack Problem

A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold

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