An ADMM algorithm for semidefinite programming with application in economics

• Grant number: 23/13586-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2024
• End date: December 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Gabriel Haeser
• Grantee: Matias Oliveira Schwarz
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 18/24293-0 - Computational methods in optimization, AP.TEM

Abstract

The main objective of scientific initiation is the implementation of an Alternating Direction Method of Multipliers (ADMM) approach to solve the covariance matrix decomposition problem and explore its applications in various fields of economics.To apply ADMM to the covariance matrix decomposition problem, it is necessary for the matrix to be positive semidefinite (SDP), which is not always the case in practice. Sample limitations, measurement errors, and other factors can result in covariance matrices that are not SDP. Therefore, our initial approach involves investigating methods to make covariance matrices SDP. Our strategy is based on the work ''On the convergence of iterative schemes for solving a piecewise linear system of equations'' of N. Armijo, Y. Bello-Cruz and G. Haeser, where the authors address the correlation matrix, which also needs to be SDP, and use projections onto the SDP cone to achieve this goal.We then study the problem and its modeling through optimization, as well as the Principal Component Analysis (PCA) method, which has been the applied strategy. The ADMM-type method is subsequently implemented to solve the SDP problem.