scholarly journals A QP-Free Algorithm for Finite Minimax Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Daolan Han ◽  
Jinbao Jian ◽  
Qinfeng Zhang
Keyword(s):  
Minimax Problems ◽  
Linear Equations ◽  
Inequality Constraints ◽  
Coefficient Matrix ◽  
Systems Of Linear Equations ◽  
Full Dimension ◽  
Finite Minimax ◽  
The Stability ◽  
Nonlinear Minimax Problems ◽  
Nonlinear Minimax

The nonlinear minimax problems without constraints are discussed. Due to the expensive computation for solving QP subproblems with inequality constraints of SQP algorithms, in this paper, a QP-free algorithm which is also called sequential systems of linear equations algorithm is presented. At each iteration, only two systems of linear equations with the same coefficient matrix need to be solved, and the dimension of each subproblem is not of full dimension. The proposed algorithm does not need any penalty parameters and barrier parameters, and it has small computation cost. In addition, the parameters in the proposed algorithm are few, and the stability of the algorithm is well. Convergence property is described and some numerical results are provided.

scholarly journals A sequential quadratic programming algorithm for nonlinear minimax problems

2007 ◽  
Vol 76 (3) ◽  
pp. 353-368 ◽  
Author(s):  
Qing-Jie Hu ◽  
Ju-Zhou Hu
Keyword(s):  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Minimax Problems ◽  
Linear Equations ◽  
Programming Algorithm ◽  
Quadratic Program ◽  
Active Set ◽  
Nonlinear Minimax Problems ◽  
Sequential Quadratic Programming Algorithm ◽  
Nonlinear Minimax

In this paper, an active set sequential quadratic programming algorithm with non-monotone line search for nonlinear minmax problems is presented. At each iteration of the proposed algorithm, a main search direction is obtained by solving a reduced quadratic program which always has a solution. In order to avoid the Maratos effect, a correction direction is yielded by solving the reduced system of linear equations. Under mild conditions without the strict complementarity, the global and superlinear convergence can be achieved. Finally, some preliminary numerical experiments are reported.

scholarly journals A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhijun Luo ◽  
Lirong Wang
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Descent Direction ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Globally Convergent ◽  
Variable Distribution ◽  
Systems Of Linear Equations

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

Numerical Determination of the Response of a Linear Vibration System With a Singular Mass Matrix

1972 ◽  
Vol 94 (1) ◽  
pp. 64-69
Author(s):  
K. D. Willmert
Keyword(s):  
Multistep Methods ◽  
Mass Matrix ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Vibrational System ◽  
Wide Range ◽  
Mass Matrices ◽  
The Stability ◽  
The Many

In numerically determining the response of a linear second-order multidegree-of-freedom vibrational system subjected to a general excitation, the common approach of applying one of the many multistep methods of numerical analysis (e.g., Milne-Simpson, Adams-Bashforth, etc.) leads ultimately to the solution of a system of linear equations. However, when the mass matrix of the original vibrational system is singular, the coefficient matrix of the system of equations also becomes singular and thus the response cannot be determined. Presented is a means of applying these multistep methods to vibrational systems which results in a method that is capable of obtaining the response independent of the singularity of the mass matrix. This technique is particularly useful in optimization where the values of the parameters of the system are unknown in advance, and thus the method of determining the response must be applicable for a wide range of values of the parameters. In the development and investigation of this technique, the causes of the stability problems which develop from the application of multistep methods to systems with nearly singular mass matrices become apparent.

3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

2014 ◽  
Vol 26 (5) ◽  
pp. 566-572 ◽  
Author(s):  
Ailan Liu ◽  
◽  
Dingguo Pu ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Ncp Function ◽  
Line Searches ◽  
Complementarity Condition ◽  
Linear Independence Constraint Qualification ◽  
Systems Of Linear Equations

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00260005/04.jpg"" width=""300"" />Algorithm flow chart</div> We propose a nonmonotone QP-free infeasible method for inequality-constrained nonlinear optimization problems based on a 3-1 piecewise linear NCP function. This nonmonotone QP-free infeasible method is iterative and is based on nonsmooth reformulation of KKT first-order optimality conditions. It does not use a penalty function or a filter in nonmonotone line searches. This algorithm solves only two systems of linear equations with the same nonsingular coefficient matrix, and is implementable and globally convergent without a linear independence constraint qualification or a strict complementarity condition. Preliminary numerical results are presented. </span>

scholarly journals On the stability of ADI methods

2017 ◽  
Vol 42 (598) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Linear Equations ◽  
Implicit Methods ◽  
Stability Properties ◽  
Require Time ◽  
Jump Discontinuities ◽  
Systems Of Linear Equations ◽  
Two Space Dimensions ◽  
Large Systems ◽  
Adi Methods ◽  
The Stability

When solving parabolic equations in two space dimensions implicit methods are preferred to the explicit method because of their better stability properties. Straightforward implementation of implicit methods require time-consuming solution of large systems of linear equations, and ADI methods are preferred instead. We expect the ADI methods to inherit the stability properties of the implicit methods they are derived from, and we demonstrate that this is partly true. The Douglas-Rachford and Peaceman-Rachford methods are absolutely stable in the sense that their growth factors are ≤ 1 in absolute value. Near jump discontinuities, however, there are differences w.r.t. how the ADI methods react to the situation: do they produce oscillations and how effectively do they damp them. We demonstrate the behaviour on two simple examples.

AN INFEASIBLE SSLE FILTER ALGORITHM FOR GENERAL CONSTRAINED OPTIMIZATION WITHOUT STRICT COMPLEMENTARITY

2011 ◽  
Vol 28 (03) ◽  
pp. 361-399 ◽  
Author(s):  
CHUNGEN SHEN ◽  
WENJUAN XUE ◽  
DINGGUO PU
Keyword(s):  
Linear Independence ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Active Set ◽  
Strict Complementarity ◽  
Complementarity Condition ◽  
Sequential Systems ◽  
Computational Effect ◽  
Systems Of Linear Equations ◽  
One Step

In this paper, we propose a new sequential systems of linear equations (SSLE) filter algorithm, which is an infeasible QP-free method. The new algorithm needs to solve a few reduced systems of linear equations with the same nonsingular coefficient matrix, and after finitely many iterations, only two linear systems need to be solved. Furthermore, the nearly active set technique is used to improve the computational effect. Under the linear independence condition, the global convergence is proved. In particular, the rate of convergence is proved to be one-step superlinear without assuming the strict complementarity condition. Numerical results and comparison with other algorithms indicate that the new algorithm is promising.

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