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 and 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) wherein 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. The software reviewed as part of this submission was given the DOI (Digital Object Identifier)  https://doi.org/10.5281/zenodo.1206980.     

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with cone constraints, and its dual, which has fewer variables than the original one, is a semismoothly differentiable (SC ¹) convex programming problem with both a linear inequality constraint and a linear second-order cone constraint. We demonstrate the global convergence of the augmented Lagrangian method with an exact solution to the subproblem and prove that the convergence rate of primal iterates, generated by the augmented Lagrangian method, is proportional to 1/r, and the rate of multiplier iterates is proportional to $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. Furthermore, a semismooth Newton method with Armijo line search is constructed to solve the subproblems in the augmented Lagrangian approach. Finally, numerical results are reported to show the effectiveness of the augmented Lagrangian method with both an exact solution and an inexact solution to the subproblem for solving the inverse second-order cone quadratic programming problem.     

Exact augmented Lagrangian functions for nonlinear semidefinite programming

In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi’s work from 1996, that was aimed at solving nonlinear programming problems. Then, through our framework, we propose a practical augmented Lagrangian function for NSDP, proving that it is continuously differentiable and exact under the so-called nondegeneracy condition. We also present some preliminary numerical experiments.     

A sequential quadratically constrained quadratic programming method with an augmented Lagrangian line search function

Based on an augmented Lagrangian line search function, a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems. Compared to quadratic programming solved in the traditional SQP methods, a convex quadratically constrained quadratic programming is solved here to obtain a search direction, and the Maratos effect does not occur without any other corrections. The “active set” strategy used in this subproblem can avoid recalculating the unnecessary gradients and (approximate) Hessian matrices of the constraints. Under certain assumptions, the proposed method is proved to be globally, superlinearly, and quadratically convergent. As an extension, general problems with inequality and equality constraints as well as nonmonotone line search are also considered.     

Recursive quadratic programming algorithm that uses an exact augmented Lagrangian function

An algorithm for nonlinear programming problems with equality constraints is presented which is globally and superlinearly convergent. The algorithm employs a recursive quadratic programming scheme to obtain a search direction and uses a differentiable exact augmented Lagrangian as line search function to determine the steplength along this direction. It incorporates an automatic adjustment rule for the selection of the penalty parameter and avoids the need to evaluate second-order derivatives of the problem functions. Some numerical results are reported.     

Duality in nonlinear multiobjective programming using augmented Lagrangian functions

A vector-valued generalized Lagrangian is constructed for a nonlinear multiobjective programming problem. Using the Lagrangian, a multiobjective dual is considered. Without assuming differentiability, weak and strong duality theorems are established using Pareto efficiency.The research of the second author was partially supported a GTE/SLU grant while visiting St. Lawrence University in the summer of 1991.     

An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.      

An augmented Lagrangian fish swarm based method for global optimization

This paper presents an augmented Lagrangian methodology with a stochastic population based algorithm for solving nonlinear constrained global optimization problems. The method approximately solves a sequence of simple bound global optimization subproblems using a fish swarm intelligent algorithm. A stochastic convergence analysis of the fish swarm iterative process is included. Numerical results with a benchmark set of problems are shown, including a comparison with other stochastic-type algorithms.     

A class of continuous quadratic programming problems

Summary The present paper deals with some duality relations of a class of continuous quadratic programming problems. The analysis is developed by use of technique ofDorn for quadratic programming problems.

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Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit verschiedenen Dualitätsrelationen für eine Klasse von stetigen quadratischen Optimierungsproblemen. Die Untersuchung basiert auf einem vonDorn entwickelten Verfahren für quadratische Optimierungsprobleme.

     

An augmented Lagrangian approach with a variable transformation in nonlinear programming

Tangent cone and (regular) normal cone of a closed set under an invertible variable transformation around a given point are investigated, which lead to the concepts of θ⁻¹-tangent cone of a set and θ⁻¹-subderivative of a function. When the notion of θ⁻¹-subderivative is applied to perturbation functions, a class of augmented Lagrangians involving an invertible mapping of perturbation variables are obtained, in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables. A necessary and sufficient condition for the exact penalty representation under the proposed augmented Lagrangian scheme is obtained. For an augmenting function with an Euclidean norm, a sufficient condition (resp., a sufficient and necessary condition) for an arbitrary vector (resp., 0) to support an exact penalty representation is given in terms of θ⁻¹-subderivatives. An example of the variable transformation applied to constrained optimization problems is given, which yields several exact penalization results in the literature.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

A truncated Newton method in an augmented Lagrangian framework for nonlinear programming

In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, and is thus suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear convergence rate.     

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.     

An interior Newton method for quadratic programming

We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential. Received October 11, 1993 / Revised version received February 20, 1996 Published online July 19, 1999     

The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

We analyze the rate of local convergence of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold . The research of Defeng Sun is partly supported by the Academic Research Fund from the National University of Singapore. The research of Jie Sun and Liwei Zhang is partly supported by Singapore–MIT Alliance and by Grants RP314000-042/057-112 of the National University of Singapore. The research of Liwei Zhang is also supported by the National Natural Science Foundation of China under project grant no. 10471015 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.