A hierarchy of polyhedral approximations of robust semidefinite programs

2016 ◽  
Author(s):  
Raphael Louca ◽  
Eilyan Bitar
Keyword(s):  
Semidefinite Programs ◽  
Polyhedral Approximations

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

scholarly journals Polyhedral approximations of the semidefinite cone and their application

2021 ◽  
Author(s):  
Yuzhu Wang ◽  
Akihiro Tanaka ◽  
Akiko Yoshise
Keyword(s):  
Initial Approximation ◽  
Semidefinite Relaxation ◽  
Stable Set ◽  
Cutting Plane Methods ◽  
Diagonally Dominant Matrices ◽  
Maximum Stable Set ◽  
Simple Expansion ◽  
Diagonally Dominant ◽  
Polyhedral Approximations ◽  
Semidefinite Cone

AbstractWe develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

scholarly journals An Interior Point Method for Solving Semidefinite Programs Using Cutting Planes and Weighted Analytic Centers

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
John Machacek ◽  
Shafiu Jibrin
Keyword(s):  
Newton’S Method ◽  
Interior Point ◽  
Newton's Method ◽  
Interior Point Method ◽  
Optimal Solution ◽  
Point Method ◽  
Feasible Region ◽  
Test Problems ◽  
Analytic Centers ◽  
Semidefinite Programs

We investigate solving semidefinite programs (SDPs) with an interior point method called SDP-CUT, which utilizes weighted analytic centers and cutting plane constraints. SDP-CUT iteratively refines the feasible region to achieve the optimal solution. The algorithm uses Newton’s method to compute the weighted analytic center. We investigate different stepsize determining techniques. We found that using Newton's method with exact line search is generally the best implementation of the algorithm. We have also compared our algorithm to the SDPT3 method and found that SDP-CUT initially gets into the neighborhood of the optimal solution in less iterations on all our test problems. SDP-CUT also took less iterations to reach optimality on many of the problems. However, SDPT3 required less iterations on most of the test problems and less time on all the problems. Some theoretical properties of the convergence of SDP-CUT are also discussed.

scholarly journals COSMO: A Conic Operator Splitting Method for Convex Conic Problems

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.

scholarly journals An inexact dual logarithmic barrier method for solving sparse semidefinite programs

2018 ◽  
Vol 178 (1-2) ◽  
pp. 109-143 ◽  
Author(s):  
Stefania Bellavia ◽  
Jacek Gondzio ◽  
Margherita Porcelli
Keyword(s):  
Barrier Method ◽  
Semidefinite Programs ◽  
Logarithmic Barrier
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