The Forward-Backward Splitting Algorithm

AbstractIn this paper, we consider a cognitive radio network scenario, where two primary users want to exchange information with each other and meanwhile, one secondary node wishes to send messages to a cognitive base station. To meet the target quality of service (QoS) of the primary users and raise the communication opportunity of the secondary nodes, a cognitive bidirectional relaying (BDR) scheme is examined. First, system outage probabilities of conventional direct transmission and BDR schemes are presented. Next, a new system parameter called operating region is defined and calculated, which indicates in which position a secondary node can be a cognitive relay to assist the primary users. Then, a cognitive BDR scheme is proposed, giving a transmission protocol along with a time-slot splitting algorithm between the primary and secondary transmissions. Information-theoretic metric of ergodic capacity is studied for the cognitive BDR scheme to evaluate its performance. Simulation results show that with the proposed scheme, the target QoS of the primary users can be guaranteed, while increasing the communication opportunity for the secondary nodes.

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A multi-level splitting algorithm based on differential evolution

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962318500216 ◽

2018 ◽

Vol 09 (02) ◽

pp. 1850021

Author(s):

Liping Wang ◽

Wenhui Fan

Keyword(s):

Differential Evolution ◽

Rare Event ◽

Sampling Strategy ◽

Splitting Algorithm ◽

Algorithm Efficiency ◽

Event Simulation ◽

Sample Paths ◽

Level Splitting ◽

Key Factor ◽

Multi Level

Multi-level splitting algorithm is a famous rare event simulation (RES) method which reaches rare set through splitting samples during simulation. Since choosing sample paths is a key factor of the method, this paper embeds differential evolution in multi-level splitting mechanism to improve the sampling strategy and precision, so as to improve the algorithm efficiency. Examples of rare event probability estimation demonstrate that the new proposed algorithm performs well in convergence rate and precision for a set of benchmark cases.

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On Tuy’s 1964 Cone Splitting Algorithm for Concave Minimization

Nonconvex Optimization and Its Applications - From Local to Global Optimization ◽

10.1007/978-1-4757-5284-7_7 ◽

2001 ◽

pp. 133-153

Author(s):

Christophe Meyer

Keyword(s):

Concave Minimization ◽

Splitting Algorithm

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COSMO: A Conic Operator Splitting Method for Convex Conic Problems

Journal of Optimization Theory and Applications ◽

10.1007/s10957-021-01896-x ◽

2021 ◽

Vol 190 (3) ◽

pp. 779-810

Author(s):

Michael Garstka ◽

Mark Cannon ◽

Paul Goulart

Keyword(s):

Convex Sets ◽

Operator Splitting ◽

Splitting Method ◽

Computational Cost ◽

Coefficient Matrix ◽

Benchmark Problems ◽

Portfolio Optimisation ◽

Splitting Algorithm ◽

Semidefinite Programs ◽

Operator Splitting Method

AbstractThis paper describes the conic operator splitting method (COSMO) solver, an operator splitting algorithm and associated software package for convex optimisation problems with quadratic objective function and conic constraints. At each step, the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. Numerical comparisons with other state-of-the-art solvers for a variety of benchmark problems show the effectiveness of our approach. Our Julia implementation is open source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.

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