Projecting onto the Intersection of a Cone and a Sphere

Bauschke, Heinz H.; Bùi, Minh N.; Wang, Xianfu

doi:10.1137/17m1141849

Cited by 32 publications

(19 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, changing the Lorentz cone by an arbitrary closed convex cone K would lead to a more general concept of K-copositivity, thus our study is anticipated to initialise new perspectives for investigating the general copositivity of a symmetric matrix. In general, exploiting the specific intrinsic geometric and algebraic structure of problems posed on the sphere can significantly lower down the cost of finding solutions; see [1,9,10,[19][20][21][22]. We know that a strict local minimizer of a spherically quasi-convex quadratic function is also a strict global minimizer, which makes interesting and natural to refer the problem about characterizing the spherically quasi-convex quadratic functions on spherically convex sets.…”

Section: Introductionmentioning

confidence: 99%

On the spherical quasi-convexity of quadratic functions

Ferreira¹,

Németh

Xiao

2019

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

On the spherical quasi-convexity of quadratic functions

Ferreira¹,

Németh

Xiao

2019

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…We are in a position to describe our proposed algorithm for solving (8). To proceed we first perform a change of variable θ mk = √ η mk γ mk and rewrite the considered problem with respect to the new variable.…”

Section: Proposed Methods a Problem Reformulationmentioning

confidence: 99%

First-Order Methods for Energy-Efficient Power Control in Cell-Free Massive MIMO : Invited Paper

Tran

Ngo

2019

2019 53rd Asilomar Conference on Signals, Systems, and Computers

View full text Add to dashboard Cite

This paper considers a cell-free massive MIMO system with multiple-antenna access points and single-antenna users. The APs use conjugate beamforming to beamform the data to all users in the network. Total energy efficiency maximization is investigated. This optimization problem is nonconvex and thus difficult to solve. Existing solutions are based on second-order optimization methods in connection with convex approximations. These methods have been shown to perform very well but their complexity does not scale favorably with the network size. To tackle this issue, in this paper we propose to use a first-order method for nonconvex programming to our energy efficiency problem. Compared to the second-order methods, the proposed method achieves the same performance, while its run time is much faster. Thus, it is considered as a feasible solution for resource allocation in cell-free massive MIMO systems.

show abstract

“…Example Let ε > 0 and K be a nonempty closed convex cone in

ℝ^{n \times r}

. Then the projection onto the intersection

K \cap 𝔹_{F} (0, ε)

is given by (see Reference 47, Theorem 7.1)

\Pr_{K \cap 𝔹_{F} (0, ε)} (X) = \Pr_{𝔹_{F} (0, ε)} \circ \Pr_{K} (X) = \frac{ε}{\max \{{‖\Pr_{K} (X)‖}_{F}, ε\}} \Pr_{K} (X) \forall X \in ℝ^{n \times r} .

Notice that in general

\Pr_{𝔹_{F} (0, ε)} \circ \Pr_{K} (X) \neq \Pr_{K} (X) \circ \Pr_{𝔹_{F} (0, ε)}

(see Reference 47, Example 7.5).…”

Section: Preliminariesmentioning

confidence: 99%

Factorization of completely positive matrices using iterative projected gradient steps

Boţ

Nguyen

2021

Numerical Linear Algebra App

View full text Add to dashboard Cite

We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient steps are simple in the sense that they have explicit formulas and do not require inner loops. Furthermore, no expensive procedure to find an appropriate starting point is needed. The convergence analysis shows that the whole sequence of generated iterates converges to a critical point of the objective function and it makes use of the Łojasiewicz inequality. Its rate of convergence expressed in terms of the Łojasiewicz exponent of a regularization of the objective function is also provided. Numerical experiments demonstrate the efficiency of the proposed method, in particular in comparison to other factorization algorithms, and emphasize the role of the relaxation and inertial parameters.

show abstract

Projecting onto the Intersection of a Cone and a Sphere

Cited by 32 publications

References 15 publications

On the spherical quasi-convexity of quadratic functions

On the spherical quasi-convexity of quadratic functions

First-Order Methods for Energy-Efficient Power Control in Cell-Free Massive MIMO : Invited Paper

Factorization of completely positive matrices using iterative projected gradient steps

Contact Info

Product

Resources

About