arXiv subjects

Defeng Sun

Publications and source records attributed to Defeng Sun.

At least 91 records · Page 5Linked to original sources

arXiv2019

On the Equivalence of Inexact Proximal ALM and ADMM for a Class of Convex Composite Programming

In this paper, we show that for a class of linearly constrained convex composite optimization problems, an (inexact) symmetric Gauss-Seidel based majorized multi-block proximal alternating direction method of multipliers (ADMM) is equivalent to an {\em inexact} proximal augmented Lagrangian method (ALM). This equivalence not only provides new perspectives for understanding some ADMM-type algorithms but also supplies meaningful guidelines on implementing them to achieve better computational efficiency. Even for the two-block case, a by-product of this equivalence is the convergence of the whole sequence generated by the classic ADMM with a step-length that exceeds the conventional upper bound of $(1+\sqrt{5})/2$, if one part of the objective is linear. This is exactly the problem setting in which the very first convergence analysis of ADMM was conducted by Gabay and Mercier in 1976, but, even under notably stronger assumptions, only the convergence of the primal sequence was known. A collection of illustrative examples are provided to demonstrate the breadth of applications for which our results can be used. Numerical experiments on solving a large number of linear and convex quadratic semidefinite programming problems are conducted to illustrate how the theoretical results established here can lead to improvements on the corresponding practical implementations.

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arXiv2018

A semi-proximal augmented Lagrangian based decomposition method for primal block angular convex composite quadratic conic programming problems

We propose a semi-proximal augmented Lagrangian based decomposition method for convex composite quadratic conic programming problems with primal block angular structures. Using our algorithmic framework, we are able to naturally derive several well known augmented Lagrangian based decomposition methods for stochastic programming such as the diagonal quadratic approximation method of Mulvey and Ruszczyński. Moreover, we are able to derive novel enhancements and generalizations of these well known methods. We also propose a semi-proximal symmetric Gauss-Seidel based alternating direction method of multipliers for solving the corresponding dual problem. Numerical results show that our algorithms can perform well even for very large instances of primal block angular convex QP problems. For example, one instance with more than $300,000$ linear constraints and $12,500,000$ nonnegative variables is solved in less than a minute whereas Gurobi took more than 3 hours, and another instance {\tt qp-gridgen1} with more than $331,000$ linear constraints and $986,000$ nonnegative variables is solved in about 5 minutes whereas Gurobi took more than 35 minutes.

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arXiv2018

Best Nonnegative Rank-One Approximations of Tensors

In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zero-th order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems under both the positive semidefinite and nonnegative conic constraints. A worst case approximation bound is given for this relaxation method. Then, the recent solver SDPNAL+ is adopted to solve this class of matrix optimization problems. Typically, the DNN relaxations are tight, and hence the best nonnegative rank-one approximation of a tensor can be revealed frequently. Extensive numerical experiments show that this approach is quite promising.

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arXiv2018

Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector function to all eigenvalues/singular values of the underlying matrices. Spectral operators play a crucial role in the study of various applications involving matrices such as matrix optimization problems (MOPs) {that include semidefinite programming as one of the most important example classes}. In this paper, we will study more fundamental first- and second-order properties of spectral operators, including the Lipschitz continuity, $ρ$-order B(ouligand)-differentiability ($0<ρ\le 1$), $ρ$-order G-semismoothness ($0<ρ\le 1$), and characterization of generalized Jacobians.

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arXiv2018

Convex Clustering: Model, Theoretical Guarantee and Efficient Algorithm

Clustering is a fundamental problem in unsupervised learning. Popular methods like K-means, may suffer from poor performance as they are prone to get stuck in its local minima. Recently, the sum-of-norms (SON) model (also known as the clustering path) has been proposed in Pelckmans et al. (2005), Lindsten et al. (2011) and Hocking et al. (2011). The perfect recovery properties of the convex clustering model with uniformly weighted all pairwise-differences regularization have been proved by Zhu et al. (2014) and Panahi et al. (2017). However, no theoretical guarantee has been established for the general weighted convex clustering model, where better empirical results have been observed. In the numerical optimization aspect, although algorithms like the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA) have been proposed to solve the convex clustering model (Chi and Lange, 2015), it still remains very challenging to solve large-scale problems. In this paper, we establish sufficient conditions for the perfect recovery guarantee of the general weighted convex clustering model, which include and improve existing theoretical results as special cases. In addition, we develop a semismooth Newton based augmented Lagrangian method for solving large-scale convex clustering problems. Extensive numerical experiments on both simulated and real data demonstrate that our algorithm is highly efficient and robust for solving large-scale problems. Moreover, the numerical results also show the superior performance and scalability of our algorithm comparing to the existing first-order methods. In particular, our algorithm is able to solve a convex clustering problem with 200,000 points in $\mathbb{R}^3$ in about 6 minutes.

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arXiv2018

On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope

We derive an explicit formula, as well as an efficient procedure, for constructing a generalized Jacobian for the projector of a given square matrix onto the Birkhoff polytope, i.e., the set of doubly stochastic matrices. To guarantee the high efficiency of our procedure, a semismooth Newton method for solving the dual of the projection problem is proposed and efficiently implemented. Extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software Gurobi and the academic code PPROJ [{\sc Hager and Zhang}, SIAM Journal on Optimization, 26 (2016), pp.~1773--1798]. In particular, our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 minutes on a modest desktop computer. More importantly, based on our efficient computation of the projections and their generalized Jacobians, we can design a highly efficient augmented Lagrangian method (ALM) for solving a class of convex quadratic programming (QP) problems constrained by the Birkhoff polytope. The resulted ALM is demonstrated to be much more efficient than Gurobi in solving a collection of QP problems arising from the relaxation of quadratic assignment problems.

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arXiv2018

Solving the OSCAR and SLOPE Models Using a Semismooth Newton-Based Augmented Lagrangian Method

The octagonal shrinkage and clustering algorithm for regression (OSCAR), equipped with the $\ell_1$-norm and a pair-wise $\ell_{\infty}$-norm regularizer, is a useful tool for feature selection and grouping in high-dimensional data analysis. The computational challenge posed by OSCAR, for high dimensional and/or large sample size data, has not yet been well resolved due to the non-smoothness and inseparability of the regularizer involved. In this paper, we successfully resolve this numerical challenge by proposing a sparse semismooth Newton-based augmented Lagrangian method to solve the more general SLOPE (the sorted L-one penalized estimation) model. By appropriately exploiting the inherent sparse and low-rank property of the generalized Jacobian of the semismooth Newton system in the augmented Lagrangian subproblem, we show how the computational complexity can be substantially reduced. Our algorithm presents a notable advantage in the high-dimensional statistical regression settings. Numerical experiments are conducted on real data sets, and the results demonstrate that our algorithm is far superior, in both speed and robustness, than the existing state-of-the-art algorithms based on first-order iterative schemes, including the widely used accelerated proximal gradient (APG) method and the alternating direction method of multipliers (ADMM).

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arXiv2018

Computing the Best Approximation Over the Intersection of a Polyhedral Set and the Doubly Nonnegative Cone

This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-)block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the efficient semismooth Newton method to solve the subproblems, which have no closed form solutions since the original four blocks are merged into two larger blocks. The $O(1/k^2)$ iteration complexity of the proposed algorithm is established. Extensive numerical results over various large scale semidefinite programming instances from relaxations of combinatorial problems demonstrate the effectiveness of the proposed algorithm.

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arXiv2018

An Efficient Semismooth Newton Based Algorithm for Convex Clustering

Clustering may be the most fundamental problem in unsupervised learning which is still active in machine learning research because its importance in many applications. Popular methods like K-means, may suffer from instability as they are prone to get stuck in its local minima. Recently, the sum-of-norms (SON) model (also known as clustering path), which is a convex relaxation of hierarchical clustering model, has been proposed in [7] and [5] Although numerical algorithms like ADMM and AMA are proposed to solve convex clustering model [2], it is known to be very challenging to solve large-scale problems. In this paper, we propose a semi-smooth Newton based augmented Lagrangian method for large-scale convex clustering problems. Extensive numerical experiments on both simulated and real data demonstrate that our algorithm is highly efficient and robust for solving large-scale problems. Moreover, the numerical results also show the superior performance and scalability of our algorithm compared to existing first-order methods.

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arXiv2017

An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems

The sparse group Lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level. In this paper, we develop an efficient augmented Lagrangian method for large-scale non-overlapping sparse group Lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth Newton method. Theoretically, we prove that, if the penalty parameter is chosen sufficiently large, the augmented Lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence, the dual infeasibility, and the duality gap of the primal and dual objective functions. Computationally, we derive explicitly the generalized Jacobian of the proximal mapping associated with the sparse group Lasso regularizer and exploit fully the underlying second order sparsity through the semismooth Newton method. The efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets.

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arXiv2017

Fast algorithms for large scale generalized distance weighted discrimination

High dimension low sample size statistical analysis is important in a wide range of applications. In such situations, the highly appealing discrimination method, support vector machine, can be improved to alleviate data piling at the margin. This leads naturally to the development of distance weighted discrimination (DWD), which can be modeled as a second-order cone programming problem and solved by interior-point methods when the scale (in sample size and feature dimension) of the data is moderate. Here, we design a scalable and robust algorithm for solving large scale generalized DWD problems. Numerical experiments on real data sets from the UCI repository demonstrate that our algorithm is highly efficient in solving large scale problems, and sometimes even more efficient than the highly optimized LIBLINEAR and LIBSVM for solving the corresponding SVM problems.

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arXiv2017

On efficiently solving the subproblems of a level-set method for fused lasso problems

In applying the level-set method developed in [Van den Berg and Friedlander, SIAM J. on Scientific Computing, 31 (2008), pp.~890--912 and SIAM J. on Optimization, 21 (2011), pp.~1201--1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop a highly efficient inexact semismooth Newton based augmented Lagrangian method for solving these subproblems. The efficiency of our approach is based on several ingredients that constitute the main contributions of this paper. Firstly, an explicit formula for constructing the generalized Jacobian of the proximal mapping of the fused lasso regularizer is derived. Secondly, the special structure of the generalized Jacobian is carefully extracted and analyzed for the efficient implementation of the semismooth Newton method. Finally, numerical results, including the comparison between our approach and several state-of-the-art solvers, on real data sets are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving challenging large-scale fused lasso problems.

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arXiv2017

On the R-superlinear convergence of the KKT residues generated by the augmented Lagrangian method for convex composite conic programming

Due to the possible lack of primal-dual-type error bounds, the superlinear convergence for the Karush-Kuhn-Tucker (KKT) residues of the sequence generated by augmented Lagrangian method (ALM) for solving convex composite conic programming (CCCP) has long been an outstanding open question. In this paper, we aim to resolve this issue by first conducting convergence rate analysis for the ALM with Rockafellar's stopping criteria under only a mild quadratic growth condition on the dual of CCCP. More importantly, by further assuming that the Robinson constraint qualification holds, we establish the R-superlinear convergence of the KKT residues of the iterative sequence under easy-to-implement stopping criteria {for} the augmented Lagrangian subproblems. Equipped with this discovery, we gain insightful interpretations on the impressive numerical performance of several recently developed semismooth Newton-CG based ALM solvers for solving linear and convex quadratic semidefinite programming.

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arXiv2017

A block symmetric Gauss-Seidel decomposition theorem for convex composite quadratic programming and its applications

For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $\frac{1}{2} \| {\bf x}-{\bf x}^k \|_{\mathcal T}^2$, where ${\mathcal T}$ is a symmetric positive semidefinite matrix related to the sGS decomposition and ${\bf x}^k$ is the previous iterate. By leveraging on such a connection to optimization, we are able to extend the result (which we name as the block sGS decomposition theorem) for solving a convex composite QP (CCQP) with an additional possibly nonsmooth term in $x_1$, i.e., $\min\{ p(x_1) + \frac{1}{2}\langle {\bf x},\, \mathcal{Q} {\bf x} \rangle -\langle {\bf b},\, {\bf x}\rangle\}$, where $p(\cdot)$ is a proper closed convex function. Based on the block sGS decomposition theorem, we are able to extend the classical block sGS method to solve a CCQP. In addition, our extended block sGS method has the flexibility of allowing for inexact computation in each step of the block sGS cycle. At the same time, we can also accelerate the inexact block sGS method to achieve an iteration complexity of $O(1/k^2)$ after performing $k$ block sGS cycles. As a {fundamental} building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the inexact semiproximal {ALM/ADMM} for linearly constrained multi-block convex composite conic programming (CCCP), and the accelerated block coordinate descent method for multi-block CCCP.

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arXiv2017

A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems

We develop a fast and robust algorithm for solving large scale convex composite optimization models with an emphasis on the $\ell_1$-regularized least squares regression (Lasso) problems. Despite the fact that there exist a large number of solvers in the literature for the Lasso problems, we found that no solver can efficiently handle difficult large scale regression problems with real data. By leveraging on available error bound results to realize the asymptotic superlinear convergence property of the augmented Lagrangian algorithm, and by exploiting the second order sparsity of the problem through the semismooth Newton method, we are able to propose an algorithm, called {\sc Ssnal}, to efficiently solve the aforementioned difficult problems. Under very mild conditions, which hold automatically for Lasso problems, both the primal and the dual iteration sequences generated by {\sc Ssnal} possess a fast linear convergence rate, which can even be superlinear asymptotically. Numerical comparisons between our approach and a number of state-of-the-art solvers, on real data sets, are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving difficult large scale Lasso problems.

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arXiv2017

A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery

This paper concerns with a noisy structured low-rank matrix recovery problem which can be modeled as a structured rank minimization problem. We reformulate this problem as a mathematical program with a generalized complementarity constraint (MPGCC), and show that its penalty version, yielded by moving the generalized complementarity constraint to the objective, has the same global optimal solution set as the MPGCC does whenever the penalty parameter is over a threshold. Then, by solving the exact penalty problem in an alternating way, we obtain a multi-stage convex relaxation approach. We provide theoretical guarantees for our approach under a mild restricted eigenvalue condition, by quantifying the reduction of the error and approximate rank bounds of the first stage convex relaxation (which is exactly the nuclear norm relaxation) in the subsequent stages and establishing the geometric convergence of the error sequence in a statistical sense. Numerical experiments are conducted for some structured low-rank matrix recovery examples to confirm our theoretical findings.

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arXiv2017

A complete characterization on the robust isolated calmness of the nuclear norm regularized convex optimization problems

In this paper, we provide a complete characterization on the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problem is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of the nuclear norm regularized convex optimization problems.

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arXiv2016

QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming

In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality, inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) where the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions.

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