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High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

Xiaojun Chen(xiaojun.chen***at***polyu.edu.hk)
Philippe L. Toint(philippe.toint***at***unamur.be)

Abstract: This paper studies high-order evaluation complexity for partially separable convexly-constrained optimization involving non-Lipschitzian group sparsity terms in a nonconvex objective function. We propose a partially separable adaptive regularization algorithm using a $p$-th order Taylor model and show that the algorithm can produce an (epsilon,delta)-approximate q-th-order stationary point in at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of the objective function and its first p derivatives (whenever they exist) Our model uses the underlying rotational symmetry of the Euclidean norm function to build a Lipschitzian approximation for the non-Lipschitzian group sparsity terms, which are defined by the group \ell_2-\ell_a norm with a in (0,1). The new result shows that the partially-separable structure and non-Lipschitzian group sparsity terms in the objective function may not affect the worst-case evaluation complexity order.

Keywords: complexity theory, nonlinear optimization, non-Lipschitz functions, partially-separable problems, group sparsity, isotropic model

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Applications -- Science and Engineering (Data-Mining )

Category 3: Applications -- Science and Engineering (Statistics )

Citation:

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Entry Submitted: 02/27/2019
Entry Accepted: 02/27/2019
Entry Last Modified: 02/27/2019

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