a b s t r a c tIn this work we briefly discuss some concepts of structural reliability as well as the optimization algorithm that is commonly used in this context, called HLRF. We show that the HLRF algorithm is a particular case of the SQP method, in which the Hessian of the Lagrangian is approximated by an identity matrix. Motivated by this fact, we propose the HLRF-BFGS algorithm that considers the BFGS update formula to approximate the Hessian. The algorithm proposed herein is as simple as the HLRF algorithm, since it requires just one function and gradient evaluation at each iteration and the new iterate is given by a recursive formula. Comparative numerical experiments on a set of problems selected from the literature are presented to illustrate the performance of the algorithm and the results indicate that the HLRF-BFGS algorithm has the advantage of being more robust and efficient with respect to the function and gradient evaluation, than HLRF.
We study a family of penalty functions for augmented Lagrangian methods, and concentrate on a penalty based on the modified logarithmic barrier function. The convex conjugate of this penalty induces a Bregman distance, and the dual iterates associated with the augmented Lagrangian algorithm correspond to the iterates produced by a proximal point algorithm based on this distance. The global convergence of the dual iterates is then proved. Moreover, the level curves of the quadratic approximation of the dual kernels associated with these penalty functions are the Dikin ellipsoids.Nowadays non-quadratic proximal point methods follow the same philosophy as in the quadratic case. The theory basically says that there is a close relationship between the sequences generated by a proximal point method applied to the dual of a convex programming problem and an augmented Lagrangian method applied to the primal.The theory of proximal point methods following the work of Moreau [3] for the quadratic kernel was extended to non-quadratic kernels. Censor and Zenios [4] have replaced the quadratic kernel by a D-function, also called the Bregman distance [5]. This algorithm has been further studied by Eckstein [6] and Chen and Teboulle [7] (also see [8][9][10][11][12]).Teboulle [13] has shown that the proximal point method for a non-quadratic kernel leads to an equivalent augmented Lagrangian method for the primal. In this case, the proximal point kernel comes from the conjugate of the primal penalty and is a -divergence.Tseng and Bertsekas [12] constructed an augmented Lagrangian method using an exponential penalty function. In the associated proximal point method, the kernel is the entropy function, which leads to an entropy minimization algorithm. Following this line, Polyak and Teboulle [11] constructed a family of augmented Lagrangian methods based on a so-called nonlinear rescaling principle. They have shown that this method naturally leads to the class of entropy-like distance functions introduced in [13] with the kernel given in terms of the conjugate of the scaling function.In this paper, we construct augmented Lagrangian methods based on a new family of penalty functions, and we study the particular penalty related to the modified logarithmic barrier function (MBF) of Polyak [14]. This modified MBF will be called M 2 BF. We show that the conjugate function of this barrier generates a Bregman distance in the dual. Therefore, the main contribution of this paper is a convergence proof for the proximal point method with the kernel given in terms of the conjugate of the modified logarithmic barrier. We also show that the kernels associated with this family of penalty functions have a nice property, that is, the level curves of their quadratic approximations are the Dikin ellipsoids, which does not occur with classical -divergences used in most convergence proofs for the proximal point method. We have implemented the new method and tested it with problems of the CUTE collection [15].The structure of the paper is as follows. In ...
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