A Feasible Level Proximal Point Method for Nonconvex Sparse Constrained Optimization

• URL: http://arxiv.org/abs/2010.12169v1
• Date: Fri, 23 Oct 2020 05:24:05 GMT
• Title: A Feasible Level Proximal Point Method for Nonconvex Sparse Constrained Optimization
• Authors: Digvijay Boob, Qi Deng, Guanghui Lan, Yilin Wang
• Abstract summary: We present a new model of a general convex or non objective machine machine objectives. We propose an algorithm that solves a constraint with gradually relaxed point levels of each subproblem. We demonstrate the effectiveness of our new numerical scale problems.
• Score: 25.73397307080647
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Nonconvex sparse models have received significant attention in high-dimensional machine learning. In this paper, we study a new model consisting of a general convex or nonconvex objectives and a variety of continuous nonconvex sparsity-inducing constraints. For this constrained model, we propose a novel proximal point algorithm that solves a sequence of convex subproblems with gradually relaxed constraint levels. Each subproblem, having a proximal point objective and a convex surrogate constraint, can be efficiently solved based on a fast routine for projection onto the surrogate constraint. We establish the asymptotic convergence of the proposed algorithm to the Karush-Kuhn-Tucker (KKT) solutions. We also establish new convergence complexities to achieve an approximate KKT solution when the objective can be smooth/nonsmooth, deterministic/stochastic and convex/nonconvex with complexity that is on a par with gradient descent for unconstrained optimization problems in respective cases. To the best of our knowledge, this is the first study of the first-order methods with complexity guarantee for nonconvex sparse-constrained problems. We perform numerical experiments to demonstrate the effectiveness of our new model and efficiency of the proposed algorithm for large scale problems.

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