Solving Large Scale and Sparse Semidefinite Programs

2012 ◽  
Keyword(s):  
Large Scale ◽  
Semidefinite Programs

Large-scale semidefinite programs in electronic structure calculation

2006 ◽  
Vol 109 (2-3) ◽  
pp. 553-580 ◽  
Author(s):  
Mituhiro Fukuda ◽  
Bastiaan J. Braams ◽  
Maho Nakata ◽  
Michael L. Overton ◽  
Jerome K. Percus ◽  
...  
Keyword(s):  
Electronic Structure ◽  
Large Scale ◽  
Structure Calculation ◽  
Electronic Structure Calculation ◽  
Semidefinite Programs

scholarly journals Improving ADMMs for solving doubly nonnegative programs through dual factorization

4OR ◽  
2020 ◽  
Author(s):  
Martina Cerulli ◽  
Marianna De Santis ◽  
Elisabeth Gaar ◽  
Angelika Wiegele
Keyword(s):  
Objective Function ◽  
Large Scale ◽  
Objective Function Value ◽  
Optimal Solution ◽  
Stable Set ◽  
Dual Variable ◽  
Post Processing ◽  
Semidefinite Programs ◽  
The Matrix ◽  
The Impact

Abstract Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.

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