We consider a Newton-CG 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 Newton-CG 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 SDP problems 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 in previous attempts.

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A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsź(x)=0 or as the problem of minimizing the merit functionź=1/2źźź22, we extend results which hold for sufficiently smooth functionsF to the nonsmooth case.

In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 źźź(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesź is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.

Solution of monotone complementarity problems with locally Lipschitzian functions

The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.

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A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a $P_0$-function, we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform $P$-function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of Mangasarian--Solodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new simple active-set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.

A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm

In this paper, we introduce a constant positive linear dependence condition (CPLD), which is weaker than the Mangasarian--Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). We show that a limit point of a sequence of approximating Karush--Kuhn--Tucker (KKT) points is a KKT point if the CPLD holds there. We show that a KKT point satisfying the CPLD and the strong second-order sufficiency conditions (SSOSC) is an isolated KKT point. We then establish convergence of a general sequential quadratical programming (SQP) method under the CPLD and the SSOSC. Finally, we apply these results to analyze the feasible SQP method proposed by Panier and Tits in 1993 for inequality constrained optimization problems. We establish its global convergence under the SSOSC and a condition slightly weaker than the Mangasarian--Fromovitz constraint qualification, and we prove superlinear convergence of a modified version of this algorithm under the SSOSC and a condition slightly weaker than the linear independence constraint qualification.

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On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods

This paper presents a globally convergent and locally superlinearly convergent method for solving a convex minimization problem whose objective function has a semismooth but nondifferentiable gradient. Applications to nonlinear minimax problems, stochastic programs with recourse, and their extensions are discussed.

A globally convergent Newton method for convex SC 1 minimization problems

In this paper, we introduce and explore augmented quaternions and augmented unit quaternions, and present an augmented unit quaternion optimization model. An augmented quaternion consist of a quaternion and a translation vector. The multiplication rule of augmented quaternion is defined. An augmented unit quaternion consists of a unit quaternion and a translation vector. The augmented unit quaternions form a Lie group. By means of augmented unit quaternions, we study the error model and kinematics. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as augmented unit quaternion optimization problems, which are actually real smooth spherical equality constrained optimization problems. Comparing with the corresponding unit dual quaternion optimization model, the augmented unit quaternion optimization model has less variables and removes the orthogonality constraints.

Augmented Quaternion and Augmented Unit Quaternion Optimization

We introduce rotation addition and motion addition. In this way, motions replace unit dual quaternions to represent rigid body movements in the 3D space. The inﬁnitesimal unit is no longer needed. By means of motion addition, we formulate two classical problems in robot research, i.e

Motion Addition and Motion Optimization

Motion, Dual Quaternion Optimization and Motion Optimization

Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an n × n dual complex matrix is diagonalizable if and only if it has exactly n eigenvalues with n appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. Further exploration on the Jordan form of dual complex matrices will establish a base of the dual complex matrix theory.

Li Qi

Papers

A globally convergent Newton method for convex SC 1 minimization problems

Augmented Quaternion and Augmented Unit Quaternion Optimization

Motion Addition and Motion Optimization

Motion, Dual Quaternion Optimization and Motion Optimization

Eigenvalues of Dual Complex Matrices