scholarly journals Approximation of Sequences of Symmetric Matrices with the Symmetric Rank-One Algorithm and Applications

2015 ◽  
Vol 36 (1) ◽  
pp. 329-347 ◽  
Author(s):  
Sylvain Arguillère
Keyword(s):  
Symmetric Matrices ◽  
Rank One ◽  
Symmetric Rank One

Optimization With Discrete Variables Via Recursive Quadratic Programming: Part 2—Algorithm and Results

1989 ◽  
Vol 111 (1) ◽  
pp. 130-136 ◽  
Author(s):  
J. Z. Cha ◽  
R. W. Mayne
Keyword(s):  
Quadratic Programming ◽  
Performance Comparison ◽  
Programming Algorithm ◽  
Discrete Variables ◽  
Contour Analysis ◽  
Recursive Quadratic Programming ◽  
Analysis Technique ◽  
Rank One ◽  
Symmetric Rank One ◽  
Monotonicity Analysis

A discrete recursive quadratic programming algorithm is developed for a class of mixed discrete constrained nonlinear programming (MDCNP) problems. The symmetric rank one (SR1) Hessian update formula is used to generate second order information. Also, strategies, such as the watchdog technique (WT), the monotonicity analysis technique (MA), the contour analysis technique (CA), and the restoration of feasibility have been considered. Heuristic aspects of handling discrete variables are treated via the concepts and convergence discussions of Part I. This paper summarizes the details of the algorithm and its implementation. Test results for 25 different problems are presented to allow evaluation of the approach and provide a basis for performance comparison. The results show that the suggested method is a promising one, efficient and robust for the MDCNP problem.

The Symmetric Rank One Formula and its Application in Discrete Nonlinear Optimization

1988 ◽  
Author(s):  
J. Z. Cha ◽  
R. W. Mayne
Keyword(s):  
Hessian Matrix ◽  
Attractive Feature ◽  
Recursive Quadratic Programming ◽  
Rank One ◽  
Hereditary Properties ◽  
Symmetric Rank One ◽  
Derivative Information ◽  
Theoretical Considerations ◽  
Specific Search ◽  
Search Directions

Abstract The hereditary properties of the Symmetric Rank One (SRI) update formula for numerically accumulating second order derivative information are studied. The unique advantage of the SR1 formula is that it does not require specific search directions for development of the Hessian matrix. This is an attractive feature for optimization applications where arbitrary search directions may be necessary. This paper explores the use of the SR1 formula within a procedure based on recursive quadratic programming (RQP) for solving a class of mixed discrete constrained nonlinear programming (MDCNP) problems. Theoretical considerations are presented along with numerical examples which illustrate the procedure and the utility of SR1.

The Symmetric Rank One Formula and its Application in Discrete Nonlinear Optimization

1991 ◽  
Vol 113 (3) ◽  
pp. 312-317 ◽  
Author(s):  
J. Z. Cha ◽  
R. W. Mayne
Keyword(s):  
Hessian Matrix ◽  
Attractive Feature ◽  
Recursive Quadratic Programming ◽  
Rank One ◽  
Constrained Nonlinear Programming ◽  
Symmetric Rank One ◽  
Derivative Information ◽  
Theoretical Considerations ◽  
Specific Search ◽  
Search Directions

The Symmetric Rank One (SR1) update formula is studied for its use in numerically accumulating second order derivative information for optimization. The unique advantage of the SR1 formula is that it does not require specific search directions for development of the Hessian matrix. This is an attractive feature for optimization applications where arbitrary search directions may be necessary. This paper explores the use of the SR1 formula within a procedure based on recursive quadratic programming (RQP) for solving a class of mixed discrete constrained nonlinear programming (MDCNP) problems. Theoretical considerations are presented along with numerical examples which illustrate the procedure and the utility of SR1.

A Hybrid Variable Metric Update for the Recursive Quadratic Programming Method

1991 ◽  
Vol 113 (3) ◽  
pp. 280-285 ◽  
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.

scholarly journals Structured symmetric rank-one method for unconstrained optimization

2011 ◽  
Vol 88 (12) ◽  
pp. 2608-2617 ◽  
Author(s):  
Farzin Modarres ◽  
Malik Abu Hassan ◽  
Wah June Leong
Keyword(s):  
Unconstrained Optimization ◽  
Rank One ◽  
Symmetric Rank One
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