2024/05/07 by Rohan Rele, Rele, Rohan, Angelia Nedich +1
Computer Science · Mathematics · #46N10 #49K30 #65K05 #90C26 #Advanced Optimization Algorithms Research #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2405.04688
openalex publication_date 2024/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and sufficient conditions for the existence of solutions to a general problem of minimizing a proper closed function over a closed, possibly unbounded, set. Towards the result, we introduce cones of retractive directions for a set and a function, establishing some basic properties for them. We also investigate the relationships between the cone of retractive directions of a function and the cone of level sets of the function. Using the cones of retractive directions we provide necessary and sufficient conditions for the existence of solutions that require an asymptotically bounded decay of a function, and a relation between the cones of retractive directions of the constraint set and the asymptotic cone of the objective function. Finally we refine these conditions for more structured problems.