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A NewtonCG augmented Lagrangian method for semidefinite programming
 SIAM J. Optim
"... Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresp ..."
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Cited by 63 (12 self)
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Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth NewtonCG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts.
Calibrating least squares covariance matrix problems with equality and inequality constraints
 SIAM Journal on Matrix Analysis and Applications
"... In many applications in finance, insurance, and reinsurance, one seeks a solution of finding a covariance matrix satisfying a large number of given linear equality and inequality constraints in a way that it deviates the least from a given symmetric matrix. One difficulty in finding an efficient met ..."
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Cited by 13 (3 self)
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In many applications in finance, insurance, and reinsurance, one seeks a solution of finding a covariance matrix satisfying a large number of given linear equality and inequality constraints in a way that it deviates the least from a given symmetric matrix. One difficulty in finding an efficient method for solving this problem is due to the presence of the inequality constraints. In this paper, we propose to overcome this difficulty by reformulating the problem as a system of semismooth equations with two level metric projection operators. We then design an inexact smoothing Newton method to solve the resulted semismooth system. At each iteration, we use the BiCGStab iterative solver to obtain an approximate solution to the generated smoothing Newton linear system. Our numerical experiments confirm the high efficiency of the proposed method.
Calibrating least squares semidefinite programming with equality and inequality constraints
 SIAM Journal on Matrix Analysis and Applications
"... Abstract. In this paper, we consider the least squares semidefinite programming with a large number of equality and inequality constraints. One difficulty in finding an efficient method for solving this problem is due to the presence of the inequality constraints. In this paper, we propose to overco ..."
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Cited by 9 (2 self)
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Abstract. In this paper, we consider the least squares semidefinite programming with a large number of equality and inequality constraints. One difficulty in finding an efficient method for solving this problem is due to the presence of the inequality constraints. In this paper, we propose to overcome this difficulty by reformulating the problem as a system of semismooth equations with two level metric projection operators. We then design an inexact smoothing Newton method to solve the resulting semismooth system. At each iteration, we use the BiCGStab iterative solver to obtain an approximate solution to the generated smoothing Newton linear system. Our numerical experiments confirm the high efficiency of the proposed method.
Positive semidefinite matrix completions on chordal graphs and the constraint nondegeneracy in semidefinite programming
, 2008
"... LetG = (V,E) be a graph. In matrix completion theory, it is known that the following two conditions are equivalent: (i) G is a chordal graph; (ii) Every Gpartial positive semidefinite matrix has a positive semidefinite matrix completion. In this paper, we relate these two conditions to constraint n ..."
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Cited by 4 (0 self)
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LetG = (V,E) be a graph. In matrix completion theory, it is known that the following two conditions are equivalent: (i) G is a chordal graph; (ii) Every Gpartial positive semidefinite matrix has a positive semidefinite matrix completion. In this paper, we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to (iii) For any Gpartial positive definite matrix that has a positive semidefinite completion, constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions.
Glare Point
 Applied Optics
, 1991
"... Solving logdeterminant optimization problems by a NewtonCG primal proximal ..."
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Cited by 3 (0 self)
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Solving logdeterminant optimization problems by a NewtonCG primal proximal
A PROXIMAL POINT ALGORITHM FOR LOGDETERMINANT OPTIMIZATION WITH GROUP LASSO REGULARIZATION
"... We consider the covariance selection problem where variables are clustered into groups and the inverse covariance matrix is expected to have a blockwise sparse structure. This problem is realized via penalizing the maximum likelihood estimation of the inverse covariance matrix by group Lasso regul ..."
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Cited by 3 (0 self)
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We consider the covariance selection problem where variables are clustered into groups and the inverse covariance matrix is expected to have a blockwise sparse structure. This problem is realized via penalizing the maximum likelihood estimation of the inverse covariance matrix by group Lasso regularization. We propose to solve the resulting logdeterminant optimization problem by the classical proximal point algorithm (PPA). At each iteration, as it is difficult to update the primal variables directly, we first solve the dual subproblem by a NewtonCG method and then update the primal variables by explicit formulas based on the computed dual variables. We also propose to accelerate the PPA by an inexact generalized Newton’s method when the iterate is close to the solution. Theoretically, we prove that, at the optimal solution, the negative definiteness of the generalized Hessian matrices of the dual objective function is equivalent to the constraint nondegeneracy condition for the primal problem. Global and local convergence results are also presented for the proposed PPA. Moreover, based on the augmented Lagrangian function of the dual problem we derive an alternating direction method (ADM), which is easily implementable, and demonstrated to be efficient for some random problems. Numerical results, including comparisons with the ADM, are presented to demonstrate that the proposed NewtonCG based PPA is stable, efficient and, in particular, outperforms the ADM, especially when higher accuracy is required.
Clarke Generalized Jacobian of the Projection onto Symmetric Cones
, 2008
"... In this paper, we give an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, which generalizes and unifies the existing related results on secondorder cones and the cones of symmetric positive semidefinite matrices over the reals. Our characterization of the Cl ..."
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Cited by 2 (2 self)
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In this paper, we give an exact expression for Clarke generalized Jacobian of the projection onto symmetric cones, which generalizes and unifies the existing related results on secondorder cones and the cones of symmetric positive semidefinite matrices over the reals. Our characterization of the Clarke generalized Jacobian exposes a connection to rankone matrices.
Numerical Algorithms for a Class of Matrix Norm Approximation Problems
, 2012
"... This thesis focuses on designing robust and efficient algorithms for a class of matrix norm approximation (MNA) problems that are to find an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems a ..."
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Cited by 2 (2 self)
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This thesis focuses on designing robust and efficient algorithms for a class of matrix norm approximation (MNA) problems that are to find an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems arise often in numerical algebra,
Complementarity Problems over Symmetric Cones: A Survey of Recent Developments in Several Aspects
, 2010
"... The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the S ..."
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Cited by 2 (0 self)
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The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for the SCCP, and various properties of the SCCP. In this paper, we will provide a brief survey on the recent developments on these three themes.
Exact regularization in an augmented lagrangian method for semidefinite programming
 J. Scheduling
, 2008
"... Augmented Lagrangian method is emerging as an important class of methods in semidefinite programming, especially when there are many constraints. This is done by applying the standard augmented Lagrangian method to the dual problem. The resulting method can be cast as MoreauYosida regularization o ..."
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Cited by 1 (0 self)
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Augmented Lagrangian method is emerging as an important class of methods in semidefinite programming, especially when there are many constraints. This is done by applying the standard augmented Lagrangian method to the dual problem. The resulting method can be cast as MoreauYosida regularization of the primal problem. Despite its numerical success, there still lacks theoretical justification on the penalty parameter selection, which often in practice does not have to be big to achieve good approximate solutions. This phenomenon is related to exact regularization of convex programs. This paper studies when the regularization in the augmented Lagrangian method is exact. Under strict complementarity condition, a necessary and sufficient condition is that the gradient of the quadratic regularization function at a concerned optimal solution belongs to the positive polar cone of the critical cone at the concerned solution. In the absence of strict complementarity condition, singleton of the critical cone proves to be necessary and sufficient for uniqueness of optimal solutions and exactness of arbitrary quadratic functions. We also characterize this latter condition via directional regularity of a perturbed normal condition. Key words. Exact regularization, augmented Lagrangian method, critical cone, directional regularity, semidefinite programming. AMS subject classifications. 49M45, 90C25, 90C33 1