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.

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.

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 A Symmetric Rank-One Quasi-Newton Method for Nonnegative Matrix Factorization

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Shu-Zhen Lai ◽  
Hou-Biao Li ◽  
Zu-Tao Zhang
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Hessian Matrix ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Matrix Approximation ◽  
Active Set ◽  
Rank One ◽  
Symmetric Rank One ◽  
Quasi Newton

As is well known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing, signal processing, and so forth. In this paper, an algorithm on nonnegative matrix approximation is proposed. This method is mainly based on a relaxed active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. The method improves some recent results. In addition, some numerical experiments are presented in the synthetic data, imaging processing, and text clustering. By comparing with the other six nonnegative matrix approximation methods, this method is more robust in almost all cases.

Optimization With Discrete Variables via Recursive Quadratic Programming: Part II — Algorithm and Results

1987 ◽  
Author(s):  
J. C. Cha ◽  
R. W. Wayne
Keyword(s):  
Quadratic Programming ◽  
Performance Comparison ◽  
Programming Algorithm ◽  
Discrete Variables ◽  
Contour Analysis ◽  
Recursive Quadratic Programming ◽  
Analysis Technique ◽  
Rank One ◽  
Symmetric Rank One ◽  
Monotonicity Analysis

Abstract A discrete recursive quadratic programming algorithm is developed for mixed discrete constrained nonlinear progrmming (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 strategy 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 this 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.

A Hybrid Variable Metric Update for the Recursive Quadratic Programming Method

1989 ◽  
Author(s):  
T. J. Beltracchi ◽  
G. A. Gabriele
Keyword(s):  
Quadratic Programming ◽  
Optimization Problems ◽  
Engineering Optimization ◽  
Variable Metric ◽  
Hybrid Variable ◽  
Recursive Quadratic Programming ◽  
Rank One ◽  
Sensitivity Studies ◽  
Symmetric Rank One ◽  
Quadratic Programming Method

Abstract The Recursive Quadratic Programming (RQP) method has been shown to be 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 convergence of the Hessian approximation is affected by the choice of the variable metric update. Most of the research that has been performed with the RQP method uses the Broyden Fletcher Shanno (BFS) or Symmetric Rank One (SR1) variable metric update. The SR1 update has been shown to yield better estimates of the Hessian of the Lagrangian than those found when the BFS update is used, though there are cases where the SR1 update becomes unstable. This paper describes a hybrid variable metric update that is shown to yield good approximations of the Hessian of the Lagrangian. The hybrid update combines the best features of the SRI and BFS updates and is more stable than the SR1 update. Testing of the method shows that the efficiency of the RQP method is not affected by the new update, but more accurate Hessian approximations are produced. This should increase the accuracy of the solutions and provide more reliable information for post optimality analyses, such as parameter sensitivity studies.

On Legendre transformations and umbilic catastrophes

1978 ◽  
Vol 83 (2) ◽  
pp. 273-288 ◽  
Author(s):  
M. J. Sewell
Keyword(s):  
Elementary Mathematics ◽  
Hessian Matrix ◽  
Full Rank ◽  
Rank Zero ◽  
Rank One ◽  
Dimension One

In a previous paper this author used elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and those elementary catastrophes having one ‘behaviour’ variable, which belong to a family of polynomials called cuspoids (see Sewell (13)). The Legendre singularities permitted there are isolated and of dimension one, i.e. they exist where a Hessian matrix exceptionally has co-rank one (not more) and elsewhere has full rank (co-rank zero).

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