Augmented Lagrangian Methods for Nonlinear Semidefinite Programming with Complementarity Constraints

• Grant number: 25/22124-0
• Support Opportunities: Scholarships abroad - Research
• Start date: August 01, 2026
• End date: July 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Daiana Oliveira dos Santos
• Grantee: Daiana Oliveira dos Santos
• Host Investigator: Stephen Andrew Vavasis
• Host Institution: Universidade Federal de São Paulo (UNIFESP). Campus Osasco. Osasco , SP, Brazil
• Institution abroad: University of Waterloo, Canada
• Associated research grant: 23/08706-1 - Numerical optimization, AP.TEM

Abstract

Mathematical programming problems with complementarity constraints (MPCC) have attracted significant interest due to their wide range of applications in engineering, economics, and optimization. These problems, which belong to the class of nonlinear programming problems with equilibrium constraints, typically violate the classical constraint qualification conditions. As a result, the validity of traditional optimality conditions is compromised, making the development of efficient algorithms more challenging. In this context, sequential optimality conditions have emerged as promising alternatives, as they provide genuine necessary criteria without relying on regularity assumptions.This project aims to extend these ideas to the framework of nonlinear semidefinite programming with conic complementarity constraints (SDCMPCC), which arises in relevant applications such as correlation matrix problems with rank constraints, bilinear matrix inequalities, and energy market models.Moreover, we propose a new augmented Lagrangian method tailored to this setting, with both theoretical convergence analysis and computational implementation. The goal is to achieve theoretical advances in the characterization of solutions, as well as to design robust algorithms capable of handling degenerate problems, thereby broadening practical applications in financial optimization, control, and equilibrium models. (AU)