Optimal periodic control for distributed-parameter systems via reduced gradient methods

1994 ◽  
pp. 680-689
Author(s):  
Krystyn Styczeń ◽  
Krystyna Nitka-Styczeń
Keyword(s):  
Gradient Methods ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Periodic Control ◽  
Optimal Periodic Control ◽  
Reduced Gradient ◽  
Reduced Gradient Methods

Modified Reduced Gradient With Realizations Sorting for Hard Equality Constraints in Reliability-Based Design Optimization

2010 ◽  
Author(s):  
Chun-Min Ho ◽  
Kuei-Yuan Chan
Keyword(s):  
Design Optimization ◽  
Gradient Methods ◽  
Equality Constraints ◽  
Reliability Based Design Optimization ◽  
Reduced Gradient Method ◽  
Reliability Based Design ◽  
Reduced Gradient ◽  
Solution Efficiency ◽  
Inverse Functions ◽  
Reduced Gradient Methods

In this work, the presence of equality constraints in reliability-based design optimization (RBDO) problems is studied. Relaxation of soft equality constraints in RBDO and its challenges are briefly discussed while the main focus is on hard equalities that can not be violated even under uncertainty. Direct elimination of hard equalities to reduce problem dimensions is usually suggested; however, for nonlinear or black-box functions, variable elimination requires expensive root-finding processes or inverse functions that are generally unavailable. We extend the reduced gradient methods in deterministic optimization to handle hard equalities in RBDO. The efficiency and accuracy of the first and the second order predictions in reduced gradient methods are compared. Results show the first order prediction being more efficient when realizations of random variables are available. A gradient-weighted sorting with these random samples is proposed to further improve the solution efficiency of the reduced gradient method. Feasible design realizations subject to hard equality constraints are then available to be implemented with the state-of-the-art sampling techniques for RBDO problems. Numerical and engineering examples show the strength and simplicity of the proposed method.

A Unified Outline of Nonlinear Programming Algorithms

1985 ◽  
Vol 107 (4) ◽  
pp. 449-453 ◽  
Author(s):  
K. Schittkowski
Keyword(s):  
Nonlinear Programming ◽  
Sequential Quadratic Programming ◽  
Gradient Methods ◽  
Reduced Gradient ◽  
Numerical Performance ◽  
Constrained Nonlinear Programming ◽  
Generalized Reduced Gradient ◽  
Performance Results ◽  
Programming Algorithms ◽  
Reduced Gradient Methods

The four most successful approaches for solving the constrained nonlinear programming problem are the penalty, multiplier, sequential quadratic programming, and generalized reduced gradient methods. A general algorithmic frame will be presented, which realizes any of these methods only by specifying a search direction for the variables, a multiplier estimate, and some penalty parameters in each iteration. This approach allows one to illustrate common mathematical features and, on the other hand, serves to explain the different numerical performance results we observe in practice.

On the Identification of Hysteresis in Inelastic Systems

2001 ◽  
Author(s):  
Fai Ma
Keyword(s):  
Kalman Filter ◽  
Gradient Methods ◽  
Design Tool ◽  
Control Parameters ◽  
Internal Constraints ◽  
Inelastic Structures ◽  
Curve Fit ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient ◽  
Reduced Gradient Methods

Abstract The generalized model of differential hysteresis contains thirteen control parameters with which it can curve-fit practically any hysteretic trace. Three identification algorithms are developed to estimate the control parameters of hysteresis for different classes of inelastic structures. These algorithms are based upon the simplex, extended Kalman filter, and generalized reduced gradient methods. Novel techniques such as global search and internal constraints are incorporated to facilitate convergence and stability. Effectiveness of the devised algorithms is demonstrated through simulations of two inelastic systems with both pinching and degradation characteristics in their hysteretic traces. Due to very modest computing requirements, these identification algorithms may become acceptable as a design tool for mapping the hysteretic traces of inelastic structures.

REDUCED GRADIENT METHODS AND THEIR RELATION TO REACTION PATHS

2003 ◽  
Vol 02 (03) ◽  
pp. 385-417 ◽  
Author(s):  
WOLFGANG QUAPP
Keyword(s):  
Potential Energy ◽  
Potential Energy Surface ◽  
Energy Surface ◽  
Reaction Path ◽  
Gradient Methods ◽  
Theoretical Chemistry ◽  
Future Research ◽  
Branching Points ◽  
Reduced Gradient ◽  
Reduced Gradient Methods

The reaction path is an important concept in theoretical chemistry. We discuss different definitions, their merits as well as their drawbacks: IRC (steepest descent from saddle), reduced gradient following (RGF), gradient extremals, and some others. Many properties and problems are explained by two-dimensional figures. This paper is both a review and a pointer to future research. The branching points of RGF curves are valley-ridge inflection (VRI) points of the potential energy surface. These points may serve as indicators for bifurcations of the reaction path. The VRI points are calculated with the help of Branin's method. All the important features of the potential energy surface are independent of the coordinate system. Besides the theoretical definitions, we also discuss the numerical use of the methods.

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