Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming

We investigate in this paper global convergence properties of the augmented Lagrangian method for nonlinear semidefinite programming (NLSDP). Four modified augmented Lagrangian methods for solving NLSDP based on different algorithmic strategies are proposed. Possibly infeasible limit points of the proposed methods are characterized. It is proved that feasible limit points that satisfy the Mangasarian-Fromovitz constraint qualification are KKT points of NLSDP without requiring the boundedness condition of the multipliers. Preliminary numerical results are reported to compare the performance of the modified augmented Lagrangian methods.     

Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.     

On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

In this paper, we present new convergence properties of the augmented Lagrangian method for nonlinear semidefinite programs (NSDP). Convergence to the approximately global solutions and optimal values of NSDP is first established for a basic augmented Lagrangian scheme under mild conditions, without requiring the boundedness condition of the multipliers. We then propose four modified augmented Lagrangian methods for NSDP based on different algorithmic strategies. We show that the same convergence of the proposed methods can be ensured under weaker conditions.     

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.     

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.     

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.     

Using a factored dual in augmented Lagrangian methods for semidefinite programming

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.     

Exact augmented Lagrangian duality for mixed integer linear programming

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.     

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.     

On the approximate augmented Lagrangian for nonlinear symmetric cone programming

This paper studies the approximate augmented Lagrangian for nonlinear symmetric cone programming. The analysis is based on some results under the framework of Euclidean Jordan algebras. We formulate the approximate Lagrangian dual problem and study conditions for approximate strong duality results and an approximate exact penalty representation. We also show, under Robinson’s constraint qualification, that the sequence of stationary points of the approximate augmented Lagrangian problems converges to a stationary point of the original nonlinear symmetric cone programming.     

Duality in nonlinear programs using augmented Lagrangian functions

A generally nonconvex optimization problem with equality constraints is studied. The problem is introduced as an “inf sup” of a generalized augmented Lagrangian function. A dual problem is defined as the “sup inf” of the same generalized augmented Lagrangian. Sufficient conditions are derived for constructing the augmented Lagrangian function such that the extremal values of the primal and dual problems are equal. Characterization of a class of augmented Lagrangian functions which satisfy the sufficient conditions for strong duality is presented. Finally, some examples of functions and primal-dual problems in the above-mentioned class are presented.     

Global saddle points of nonlinear augmented Lagrangian functions

We notice that the results for the existence of global (local) saddle points of augmented Lagrangian functions in the literature were only sufficient conditions of some special types of augmented Lagrangian. In this paper, we introduce a general class of nonlinear augmented Lagrangian functions for constrained optimization problem. In two different cases, we present sufficient and necessary conditions for the existence of global saddle points. Moreover, as corollaries of the two results above, we not only obtain sufficient and necessary conditions for the existence of global saddle points of some special types of augmented Lagrangian functions mentioned in the literature, but also give some weaker sufficient conditions than the ones in the literature. Compared with our recent work (Wang et al. in Math Oper Res 38:740–760, 2013), the nonlinear augmented Lagrangian functions in this paper are more general and the results in this paper are original. We show that some examples (such as improved barrier augmented Lagrangian) satisfy the assumptions of this paper, but not available in Wang et al. (2013).     

Exact penalty functions in nonlinear programming

In this paper some new theoretic results on piecewise differentiable exact penalty functions are presented. Sufficient conditions are given for the existence of exact penalty functions for inequality constrained problems more general than concave and several classes of such functions are presented.This research was partially supported by a grant from the Office of Naval Research; contract number N00014-67-A-0321-0003 (NR047-095).     

Exact penalty functions in nonlinear programming

It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.     

Convergence of augmented Lagrangian methods in extensions beyond nonlinear programming

Mathematical Programming - The augmented Lagrangian method (ALM) is extended to a broader-than-ever setting of generalized nonlinear programming in convex and nonconvex optimization that is capable...     

A new augmented Lagrangian function for inequality constraints in nonlinear programming problems

In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.     

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.     

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.     

A homotopy method for nonlinear semidefinite programming

In this paper, for solving the nonlinear semidefinite programming problem, a homotopy is constructed by using the parameterized matrix inequality constraint. Existence of a smooth path determined by the homotopy equation, which starts from almost everywhere and converges to a Karush–Kuhn–Tucker point, is proven under mild conditions. A predictor-corrector algorithm is given for numerically tracing the smooth path. Numerical tests with nonlinear semidefinite programming formulations of several control design problems with the data contained in COMPl _e ib are done. Numerical results show that the proposed algorithm is feasible and applicable.