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.     

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.     

求解带线性约束的凸优化的一类自适应不定线性化增广拉格朗日方法

增广拉格朗日方法是求解带线性约束的凸优化问题的有效算法.线性化增广拉格朗日方法通过线性化增广拉格朗日函数的二次罚项并加上一个临近正则项,使得子问题容易求解,其中正则项系数的恰当选取对算法的收敛性和收敛速度至关重要.较大的系数可保证算法收敛性,但容易导致小步长.较小的系数允许迭代步长增大,但容易导致算法不收敛.本文考虑求解带线性等式或不等式约束的凸优化问题.我们利用自适应技术设计了一类不定线性化增广拉格朗日方法,即利用当前迭代点的信息自适应选取合适的正则项系数,在保证收敛性的前提下尽量使得子问题步长选择范围更大,从而提高算法收敛速度.我们从理论上证明了算法的全局收敛性,并利用数值实验说明了算法的有效性.     

从常步长梯度方法的视角看不可微凸优化增广Lagrange方法的收敛性

增广Lagrange方法是求解非线性规划的一种有效方法.从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性.用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性,由此得到增广Lagrange方法乘子迭代的全局收敛性.     

A nonlinear augmented Lagrangian for constrained minimax problems

A novel smooth nonlinear augmented Lagrangian for solving minimax problems with inequality constraints, is proposed in this paper, which has the positive properties that the classical Lagrangian and the penalty function fail to possess. The corresponding algorithm mainly consists of minimizing the nonlinear augmented Lagrangian function and updating the Lagrange multipliers and controlling parameter. It is demonstrated that the algorithm converges Q-superlinearly when the controlling parameter is less than a threshold under the mild conditions. Furthermore, the condition number of the Hessian of the nonlinear augmented Lagrangian function is studied, which is very important for the efficiency of the algorithm. The theoretical results are validated further by the preliminary numerical experiments for several testing problems reported at last, which show that the nonlinear augmented Lagrangian is promising.     

A nonlinear Lagrangian for constrained optimization problems

A novel nonlinear Lagrangian is presented for constrained optimization problems with both inequality and equality constraints, which is nonlinear with respect to both functions in problem and Lagrange multipliers. The nonlinear Lagrangian inherits the smoothness of the objective and constraint functions and has positive properties. The algorithm on the nonlinear Lagrangian is demonstrated to possess local and linear convergence when the penalty parameter is less than a threshold (the penalty parameter in the penalty method has to approximate zero) under a set of suitable conditions, and be super-linearly convergent when the penalty parameter is decreased following Lagrange multiplier update. Furthermore, the dual problem based on the nonlinear Lagrangian is discussed and some important properties are proposed, which fail to hold for the dual problem based on the classical Lagrangian. At last, the preliminary and comparing numerical results for several typical test problems by using the new nonlinear Lagrangian algorithm and the other two related nonlinear Lagrangian algorithms, are reported, which show that the given nonlinear Lagrangian is promising.     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.     

Log-Sigmoid Multipliers Method in Constrained Optimization

In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for constrained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques.We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization.Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal–dual LS method was introduced.     

带有不等式约束的非线性规划问题的一个精确增广Lagrange函数

对求解带有不等式约束的非线性非凸规划问题的一个精确增广Lagrange函数进行了研究．在适当的假设下,给出了原约束问题的局部极小点与增广Lagrange函数,在原问题变量空间上的无约束局部极小点之间的对应关系．进一步地,在对全局解的一定假设下,还提供了原约束问题的全局最优解与增广Lagrange函数,在原问题变量空间的一个紧子集上的全局最优解之间的一些对应关系．因此,从理论上讲,采用该文给出的增广Lagrange函数作为辅助函数的乘子法,可以求得不等式约束非线性规划问题的最优解和对应的Lagrange乘子．     

A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization

We present a primal–dual algorithm for solving a constrained optimization problem. This method is based on a Newtonian method applied to a sequence of perturbed KKT systems. These systems follow from a reformulation of the initial problem under the form of a sequence of penalized problems, by introducing an augmented Lagrangian for handling the equality constraints and a log-barrier penalty for the inequalities. We detail the updating rules for monitoring the different parameters (Lagrange multiplier estimate, quadratic penalty and log-barrier parameter), in order to get strong global convergence properties. We show that one advantage of this approach is that it introduces a natural regularization of the linear system to solve at each iteration, for the solution of a problem with a rank deficient Jacobian of constraints. The numerical experiments show the good practical performances of the proposed method especially for degenerate problems.     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems

We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is a linearly constrained semismoothly differentiable (SC¹) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange 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. As the objective function of the dual problem is a SC¹ function involving the projection operator onto the cone of symmetrically semi-definite matrices, the analysis requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and properties of the projection operator in the symmetric-matrix space. Furthermore, the semismooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, which is proven to have global convergence and local quadratic rate. Finally numerical results, implemented by the augmented Lagrangian method, are reported.     

乘子方法中的参数选择

Di Pillo和Grippo提出的含参数C〉0的增广Lagrangian函数中，使用了最大函数，该函数可能在无穷多个点处不可微．为了克服这个问题，濮定国在2004年提出了一类带新的NCP函数的乘子法．该方法在增广Lagrangian函数和原问题之间存在很好的等价性；同时该方法具有全局收敛性，且在适当假设下，具有超线性收敛率．但是在该方法中，要求参数C充分大．为了实现算法及提高算法效率，本文给出了一个有效选择参数C的方法．     

Two-level primal-dual decomposition technique for large-scale nonconvex optimization problems with constraints

A two-level decomposition method for nonconvex separable optimization problems with additional local constraints of general inequality type is presented and thoroughly analyzed in the paper. The method is of primal-dual type, based on an augmentation of the Lagrange function. Previous methods of this type were in fact three-level, with adjustment of the Lagrange multipliers at one of the levels. This level is eliminated in the present approach by replacing the multipliers by a formula depending only on primal variables and Kuhn-Tucker multipliers for the local constraints. The primal variables and the Kuhn-Tucker multipliers are together the higher-level variables, which are updated simultaneously. Algorithms for this updating are proposed in the paper, together with their convergence analysis, which gives also indications on how to choose penalty coefficients of the augmented Lagrangian. Finally, numerical examples are presented.     

A Globally Convergent Lagrangian Barrier Algorithm for Optimization with General Inequality Constraints and Simple Bounds

We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combined in a Lagrangian barrier function. A sequence of such functions are approximately minimized within the domain defined by the simple bounds. Global convergence of the sequence of generated iterates to a first-order stationary point for the original problem is established. Furthermore, possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero. This paper is a companion to previous work of ours on augmented Lagrangian methods.

     

Canonical duality for solving general nonconvex constrained problems

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.     

Updating the Multipliers Associated with Inequality Constraints in an Augmented Lagrangian Multiplier Method

This paper contributes to the development of the field of augmented Lagrangian multiplier methods for general nonlinear programming by introducing a new update for the multipliers corresponding to inequality constraints. The update maintains naturally the nonnegativity of the multipliers without the need for a positive-orthant projection, as a result of the verification of the first-order necessary conditions for the minimization of a modified augmented Lagrangian penalty function.In the new multiplier method, the roles of the multipliers are interchanged: the multipliers corresponding to the inequality constraints are updated explicitly, whereas the multipliers corresponding to the equality constraints are approximated implicitly. It is shown that the basic properties of local convergence of the traditional multiplier method are valid also for the proposed method.