New variable-metric algorithms for nondifferentiable optimization problems

Abstract Design optimization of aircraft engine-mount systems for vibration isolation is presented. The engine is modeled as a rigid body connected to a flexible base representing the nacelle. The base is modeled with mass and stiffness matrices and structural damping using finite element modeling. The mounts are modeled as three-dimensional springs with hysteresis damping. The objective is to select the stiffness coefficients and orientation angles of the individual mounts to minimize the transmitted forces from the engine to the base. Meanwhile, the mounts have to be stiff enough not allowing engine deflection to exceed its limits under static and low frequency loadings. It is shown that with an optimal system the transmitted forces may be reduced significantly particularly when mount orientation angles are also treated as design variables. The optimization problems are solved using a Constraint Variable Metric approach. The closed form derivatives of the engine vibrational amplitudes with respect to design variables are derived in order to achieve a more effective optimization search technique.

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Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-011-0024-z ◽

2011 ◽

Vol 21 (2) ◽

pp. 317-329 ◽

Cited By ~ 3

Author(s):

Abdelkrim El Mouatasim ◽

Rachid Ellaia ◽

Eduardo de Cursi

Keyword(s):

Nonconvex Optimization ◽

Optimization Problems ◽

Random Perturbation ◽

Linear Constraints ◽

Nonconvex Optimization Problems ◽

Variable Metric ◽

Lipschitz Continuous ◽

Nonsmooth Nonconvex Optimization ◽

Variable Metric Method ◽

Set Of Points

Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraintsWe present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.

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A Hybrid Variable Metric Update for the Recursive Quadratic Programming Method

Journal of Mechanical Design ◽

10.1115/1.2912780 ◽

1991 ◽

Vol 113 (3) ◽

pp. 280-285 ◽

Cited By ~ 2

Author(s):

T. J. Beltracchi ◽

G. A. Gabriele

Keyword(s):

Quadratic Programming ◽

Optimization Problems ◽

Variable Metric ◽

Hybrid Variable ◽

Recursive Quadratic Programming ◽

Line Searches ◽

Inexact Line Searches ◽

Rank One ◽

Symmetric Rank One ◽

Quadratic Programming Method

The Recursive Quadratic Programming (RQP) method has become known as one of the most effective and efficient algorithms for solving engineering optimization problems. The RQP method uses variable metric updates to build approximations of the Hessian of the Lagrangian. If the approximation of the Hessian of the Lagrangian converges to the true Hessian of the Lagrangian, then the RQP method converges quadratically. The choice of a variable metric update has a direct effect on the convergence of the Hessian approximation. Most of the research performed with the RQP method uses some modification of the Broyden-Fletcher-Shanno (BFS) variable metric update. This paper describes a hybrid variable metric update that yields good approximations to the Hessian of the Lagrangian. The hybrid update combines the best features of the Symmetric Rank One and BFS updates, but is less sensitive to inexact line searches than the BFS update, and is more stable than the SR1 update. Testing of the method shows that the efficiency of the RQP method is unaffected by the new update but more accurate Hessian approximations are produced. This should increase the accuracy of the solutions obtained with the RQP method, and more importantly, provide more reliable information for post optimality analyses, such as parameter sensitivity studies.

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Conditions for variable-metric algorithms to be conjugate-gradient algorithms

Journal of Optimization Theory and Applications ◽

10.1007/bf00935363 ◽

1983 ◽

Vol 41 (3) ◽

pp. 439-450

Author(s):

D. G. McDowell

Keyword(s):

Conjugate Gradient ◽

Variable Metric ◽

Conjugate Gradient Algorithms ◽

Gradient Algorithms ◽

Variable Metric Algorithms

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A New Homotopy Proximal Variable-Metric Framework for Composite Convex Minimization

Mathematics of Operations Research ◽

10.1287/moor.2021.1138 ◽

2021 ◽

Author(s):

Quoc Tran-Dinh ◽

Ling Liang ◽

Kim-Chuan Toh

Keyword(s):

Convergence Rates ◽

Optimization Problems ◽

Linear Convergence ◽

Convex Minimization ◽

Variable Metric ◽

Complexity Bounds ◽

Convex Problems ◽

Convex Minimization Problems ◽

Global Iteration ◽

Primal Dual

This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is to utilize a new parameterization strategy of the optimality condition to design a class of homotopy proximal variable-metric algorithms that can achieve linear convergence and finite global iteration-complexity bounds. We identify at least three subclasses of convex problems in which our approach can apply to achieve linear convergence rates. The second idea is a new primal-dual-primal framework for implementing proximal Newton methods that has attractive computational features for a subclass of nonsmooth composite convex minimization problems. We specialize the proposed algorithm to solve a covariance estimation problem in order to demonstrate its computational advantages. Numerical experiments on the four concrete applications are given to illustrate the theoretical and computational advances of the new methods compared with other state-of-the-art algorithms.

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