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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 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.
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 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.
Original language | English |
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Pages (from-to) | 1-27 |
Number of pages | 27 |
Journal | Mathematical Programming |
Volume | 1902.10767 |
DOIs | |
Publication status | Accepted/In press - 1 Jan 2020 |
Keywords
- complexity theory, nonlinear optimization, non-Lipschitz functions, partially-separable problems, group sparsity, isotropic model
- nonlinear optimization
- non-Lipschitz functions
- partially-separable problems
- group sparsity
- isotropic model
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Projects
- 1 Active
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Complexity in nonlinear optimization
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Project: Research
Activities
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Recent results in worst-case evaluation complexity for smooth and non-smooth, exact and inexact, nonconvex optimization
Philippe TOINT (Speaker)
8 May 2020Activity: Talk or presentation types › Invited talk
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Departement of Applied Mathematics, Polytechnic University of Hong Kong
Philippe Toint (Visiting researcher)
15 Nov 2018 → 15 Dec 2018Activity: Visiting an external institution types › Visiting an external academic institution