arXiv subjects

Jinyan Fan

Publications and source records attributed to Jinyan Fan.

10 recordsLinked to original sources

arXiv2026

A Unified Zeroth-Order Proximal Newton-Type Framework for Composite Optimization

We propose a unified derivative-free proximal Newton-type algorithm framework for solving composite optimization problems formulated as the sum of a black-box function and a known regularization term. We establish the iteration and oracle complexity bounds for the algorithm to attain an $\epsilon$-optimal solution under both nonconvex and strongly convex settings. We also establish its local R-superlinear convergence based on the Dennis--Mor\'{e} condition, and theoretically address an open problem by showing that the BFGS scheme is more compatible with finite-difference gradient estimators than with smoothing-based ones. Numerical experiments are further presented to demonstrate the efficiency of the proposed method.

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arXiv2025

A derivative-free Levenberg-Marquardt method for sparse nonlinear least squares problems

This paper studies sparse nonlinear least squares problems, where the Jacobian matrices are unavailable or expensive to compute, yet have some underlying sparse structures. We construct the Jacobian models by the $ \ell_1 $ minimization subject to a small number of interpolation constraints with interpolation points generated from some certain distributions,and propose a derivative-free Levenberg-Marquardt algorithm based on such Jacobian models.It is proved that the Jacobian models are probabilistically first-order accurate and the algorithm converges globally almost surely.Numerical experiments are presented to show the efficiency of the proposed algorithm for sparse nonlinear least squares problems.

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arXiv2025

A derivative-free regularization algorithm for equality constrained nonlinear least squares problems

In this paper, we study the equality constrained nonlinear least squares problem, where the Jacobian matrices of the objective function and constraints are unavailable or expensive to compute. We approximate the Jacobian matrices via orthogonal spherical smoothing and propose a derivative-free regularization algorithm for solving the problem. At each iteration, a regularized augmented Lagrangian subproblem is solved to obtain a Newton-like step. If a sufficient decrease in the merit function of the approximate KKT system is achieved, the step is accepted, otherwise a derivative-free LM algorithm is applied to get another step to satisfy the sufficient decrease condition. It is shown that the algorithm either finds an approximate KKT point with arbitrary high probability or converges to a stationary point of constraints violation almost surely.

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arXiv2024

On the global complexity of a derivative-free Levenberg-Marquardt algorithm via orthogonal spherical smoothing

In this paper, we propose a derivative-free Levenberg-Marquardt algorithm for nonlinear least squares problems, where the Jacobian matrices are approximated via orthogonal spherical smoothing. It is shown that the gradient models which use the approximate Jacobian matrices are probabilistically first-order accurate, and the high probability complexity bound of the algorithm is also given.

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arXiv2018

Completely Positive Binary Tensors

A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear matrix inequalities. This result can be used to determine whether a binary tensor is completely positive or not. When it is, we give an algorithm for computing its cp-rank and the decomposition. When the order is odd, we show that the cp-rank decomposition is unique. When the order is even, we completely characterize when the cp-rank decomposition is unique. We also discuss how to compute the nearest cp-approximation when a binary tensor is not completely positive.

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arXiv2017

Tensor Eigenvalue Complementarity Problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre's hierarchy of semidefinite relaxations. We show that it has finite convergence for general tensors. Numerical experiments are presented to show the efficiency of proposed methods.

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arXiv2014

Completely positive tensor decomposition

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a symmetric tensor is completely positive. If it is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.

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arXiv2014

The CP-matrix Approximation Problem

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = V V ^T$. In this paper, we study the CP-matrix approximation problem of projecting a matrix onto the intersection of a set of linear constraints and the cone of CP matrices. We formulate the problem as the linear optimization with the norm cone and the cone of moments. A semidefinite algorithm is presented for the problem. A CP-decomposition of the projection matrix can also be obtained if the problem is feasible.

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arXiv2014

Interiors of completely positive cones

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $A = BB^T$. We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented.

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arXiv2013

The CP-matrix completion problem

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems, and study its properties. When all the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved.

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