An interior proximal gradient method for nonconvex optimization

Abstract

We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle complicated constraints, their combined usage is shown to successfully compensate the respective shortcomings. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems, thus bridging the gap with previous works that successfully addressed the convex case. Our interior proximal gradient algorithm benefits from warm starting, generates strictly feasible iterates with decreasing objective value, and returns after finitely many iterations a primal-dual pair approximately satisfying suitable optimality conditions. As a byproduct of our analysis of proximal gradient iterations we demonstrate that a slight refinement of traditional backtracking techniques waives the need for upper bounding the stepsize sequence, as required in existing results for the nonconvex setting.

References in corpus (16)

• Forward-backward envelope for the sum of two nonconvex functions: Further properties and nonmonotone line-search algorithms
• A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima
• Constrained composite optimization and augmented Lagrangian methods
• Proximal Gradient Algorithms under Local Lipschitz Gradient Continuity: A Convergence and Robustness Analysis of PANOC
• Proximal gradient methods beyond monotony
• Bregman Finito/MISO for nonconvex regularized finite sum minimization without Lipschitz gradient continuity
• Riemannian Interior Point Methods for Constrained Optimization on Manifolds
• An ADMM-Based Interior-Point Method for Large-Scale Linear Programming
• Simple algorithms for optimization on Riemannian manifolds with constraints
• Interior-proximal primal-dual methods
• Adaptive Proximal Gradient Method for Convex Optimization
• Solving Constrained Variational Inequalities via a First-order Interior Point-based Method
• On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms
• SPIRAL: A superlinearly convergent incremental proximal algorithm for nonconvex finite sum minimization
• Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited
• A Bregman inertial forward-reflected-backward method for nonconvex minimization