scholarly journals Linear programming via a quadratic penalty function

1996 ◽  
Vol 44 (3) ◽  
pp. 345-370 ◽  
Author(s):  
Mustafa �. Pinar
Keyword(s):  
Linear Programming ◽  
Penalty Function ◽  
Quadratic Penalty Function

ANALYSING THE IMPACT OF PENALTY CONSTANT ON PENALTY FUNCTION THROUGH PARTICE SWARM OPTIMIZATION

2018 ◽  
Vol 6 (2) ◽  
pp. 01-06
Author(s):  
Raju Prajapati ◽  
Om Prakash Dubey
Keyword(s):  
Linear Programming ◽  
Penalty Function ◽  
Inequality Constraint ◽  
Test Problems ◽  
Swarm Optimization ◽  
Non Linear Programming ◽  
Linear Programming Problems ◽  
The Impact ◽  
The Given ◽  
Computational Purpose

Non Linear Programming Problems (NLPP) are tedious to solve as compared to Linear Programming Problem (LPP).  The present paper is an attempt to analyze the impact of penalty constant over the penalty function, which is used to solve the NLPP with inequality constraint(s). The improved version of famous meta heuristic Particle Swarm Optimization (PSO) is used for this purpose. The scilab programming language is used for computational purpose. The impact of penalty constant is studied by considering five test problems. Different values of penalty constant are taken to prepare the unconstraint NLPP from the given constraint NLPP with inequality constraint. These different unconstraint NLPP is then solved by improved PSO, and the superior one is noted. It has been shown that, In all the five cases, the superior one is due to the higher penalty constant. The computational results for performance are shown in the respective sections.

Application of a New Penalty Function Method to Design Optimization

1977 ◽  
Vol 99 (1) ◽  
pp. 31-36 ◽  
Author(s):  
S. B. Schuldt ◽  
G. A. Gabriele ◽  
R. R. Root ◽  
E. Sandgren ◽  
K. M. Ragsdell
Keyword(s):  
Linear Programming ◽  
Nonlinear Programming ◽  
Design Optimization ◽  
Penalty Function ◽  
Function Method ◽  
Test Problems ◽  
Penalty Function Method ◽  
Method Of Multipliers ◽  
Non Linear Programming ◽  
Linear Programming Problems

This paper presents Schuldt’s Method of Multipliers for nonlinear programming problems. The basics of this new exterior penalty function method are discussed with emphasis upon the ease of implementation. The merit of the technique for medium to large non-linear programming problems is evaluated, and demonstrated using the Eason and Fenton test problems.

scholarly journals Use of the Shor’s r-Algorithm in Linear Robust Optimization Problems

2021 ◽  
pp. 29-42
Author(s):  
P. Stetsyuk ◽  
М. Stetsyuk ◽  
D. Bragin ◽  
N. Мolodyk
Keyword(s):  
Linear Programming ◽  
Robust Optimization ◽  
Convex Function ◽  
Linear Function ◽  
Penalty Function ◽  
Piecewise Linear ◽  
Auxiliary Problem ◽  
Unconstrained Minimization ◽  
Piecewise Linear Function ◽  
Nonsmooth Penalty

The paper is devoted to the description of a new approach to the construction of algorithms for solving linear programming problems (LP-problems), in which the number of constraints is much greater than the number of variables. It is based on the use of a modification of the r-algorithm to solve the problem of minimizing a nonsmooth function, which is equivalent to LP problem. The advantages of the approach are demonstrated on the linear robust optimization problem and the robust parameters estimation problem using the least moduli method. The developed octave programs are designed to solve LP problems with a very large number of constraints, for which the use of standard software from linear programming is either impossible or impractical, because it requires significant computing resources. The material of the paper is presented in three sections. In the first section for the problem of minimizing a convex function we describe a modification of the r-algorithm with a constant coefficient of space dilation in the direction of the difference of two successive subgradients and an adaptive method for step size adjustment in the direction of the antisubgradient in the transformed space of variables. The software implementation of this modification is presented in the form of Octave function ralgb5a, which allows to find or approximation of the minimum point of a convex function, or approximation of the maximum point of the concave function. The code of the ralgb5a function is given with a brief description of its input and output parameters. In the second section, a method for solving the LP problem is presented using a nonsmooth penalty function in the form of maximum function and the construction of an auxiliary problem of unconstrained minimization of a convex piecewise linear function. The choice of the finite penalty coefficient ensures equivalence between the LP-problem and the auxiliary problem, and the latter is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test LP-problems with the number of constraints from two hundred thousand to fifty million and the number of variables from ten to fifty are presented. The third section presents least moduli method that is robust to abnormal observations or "outliers". The method uses the problem of unconstrained minimization of a convex piecewise linear function, and is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test problems with a large number of observations (from two hundred thousand to five million) and a small number of unknown parameters (from ten to one hundred) are presented. They demonstrate the superiority of the developed programs over well-known linear programming software such as the GLPK package. Keywords: robust optimization, linear programming problem, nonsmooth penalty function, r-algorithm, least modulus method, GNU Octave.

scholarly journals State-Constrained Sub-Optimal Tracking Controller for Continuous-Time Linear Time-Invariant (CT-LTI) Systems and Its Application for DC Motor Servo Systems

2020 ◽  
Vol 10 (16) ◽  
pp. 5724
Author(s):  
Jihwan Kim ◽  
Ung Jon ◽  
Hyeongcheol Lee
Keyword(s):  
Penalty Function ◽  
Continuous Time ◽  
Analytic Solution ◽  
Linear Time ◽  
Dc Motor ◽  
Optimal Tracking ◽  
Linear Time Invariant ◽  
Time Invariant ◽  
Quadratic Penalty Function ◽  
Time Linear

In this paper, we propose an analytic solution of state-constrained optimal tracking control problems for continuous-time linear time-invariant (CT-LTI) systems that are based on model-based prediction, the quadratic penalty function, and the variational approach. Model-based prediction is a concept taken from model-predictive control (MPC) and this is essential to change the direction of calculation for the solution from backward to forward. The quadratic penalty function plays an important role in deriving the analytic solution since it can transform the problem into a form that does not have inequality constraints. For computational convenience, we also propose a sub-optimal controller derived from the steady-state approximation of the analytic solution and show that the proposed controller satisfies the Lyapunov stability. The main advantage of the proposed controller is that it can be implemented in real time with a lower computational load compared to the implicit MPC. Finally, the simulation results for a DC motor servo system are shown and compared with the results of the direct multi-shooting method and the implicit MPC to verify the effectiveness of the proposed controller.

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