Large Scale Nonlinear Programming Using The Generalized Reduced Gradient Method

1980 ◽  
Vol 102 (3) ◽  
pp. 566-573 ◽  
Author(s):  
G. A. Gabriele ◽  
K. M. Ragsdell
Keyword(s):  
Nonlinear Programming ◽  
Large Scale ◽  
Sparse Matrices ◽  
Minimum Weight ◽  
Nonlinear Constraints ◽  
Reduced Gradient Method ◽  
Nonlinear Programs ◽  
Structural Problems ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

An extension of the generalized reduced gradient (GRG) method to large scale nonlinear programs with nonlinear constraints is discussed. The approach presented here represents the adoption of efficient methods for sparse matrices within the framework of the GRG algorithm. The resulting code, LGOPT, is described, and experience with a class of minimum weight structural problems is given.

scholarly journals A Mathematical Model for Optimal Strength and Alignment of a Marine Shafting System

1985 ◽  
Vol 29 (03) ◽  
pp. 212-222
Author(s):  
Zissimos Mourelatos ◽  
Panos Papalambros
Keyword(s):  
Mathematical Model ◽  
Nonlinear Programming ◽  
Programming Formulation ◽  
Solution Strategy ◽  
Reduced Gradient Method ◽  
Trade Offs ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient ◽  
Optimal Strength ◽  
Vertical Positioning

The design of a marine shafting system is modeled mathematically in order to perform optimization studies with respect to shaft strength as well as longitudinal and vertical positioning of the bearings. The objective criteria used are minimization of the bearing reaction influence numbers and even distribution of the bearing loading. Design trade-offs can be thus established. The problem is posed in a nonlinear programming formulation and is solved using a standard generalized reduced gradient method (GRG2), but in a specialized solution strategy. Two examples from actual ship designs are presented.

The Use of Constraint Management Techniques in the Generalized Reduced Gradient Method

1991 ◽  
Author(s):  
Rajiv Agrawal ◽  
Junhua Cheng ◽  
Gary L. Kinzel
Keyword(s):  
Nonlinear Programming ◽  
Computational Effort ◽  
State Variables ◽  
Reduced Gradient Method ◽  
Constraint Management ◽  
Reduced Gradient ◽  
Management Techniques ◽  
Generalized Reduced Gradient ◽  
Linearized Model ◽  
Occurrence Matrix

Abstract The Generalized Reduced Gradient (GRG) method has proven to be an effective strategy for solving constrained nonlinear programming problems, and this paper proposes the use of constraint management techniques to enhance the GRG method. Some of the troublesome issues in the method such as the selection of the initial basis and the problem of basis interchange can be resolved if the constraints are represented in the form of a nonlinear occurrence matrix. A heuristic algorithm is presented to determine an initial partition of the state and decision variables. The procedure is aimed at minimizing the nonlinear component in the set of state variables so that computational effort during the numerical iterations is reduced. Another algorithm is presented to automate the basis interchange by using a linearized model of the governing equations and a backward dependency procedure. An improved canonical form for the general nonlinear programming problem that has the objective function embedded in the set of equality constraints is also presented. Constraint management ideas are also used to decompose the basis Jacobian matrix into smaller irreducible subsets. Several mathematical and design problems are posed as test cases for the Constraint Management Based Generalized Reduced Gradient (CMB-GRG) procedure, and representative results are presented.

scholarly journals Receding Horizon Control of Type 1 Diabetes Mellitus by Using Nonlinear Programming

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Hamza Khan ◽  
József K. Tar ◽  
Imre Rudas ◽  
Levente Kovács ◽  
György Eigner
Keyword(s):  
Diabetes Mellitus ◽  
Type 1 Diabetes ◽  
Nonlinear Programming ◽  
Type 1 Diabetes Mellitus ◽  
Receding Horizon ◽  
Reduced Gradient Method ◽  
Optimization Task ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

Receding Horizon Controllers are one of the mostly used advanced control solutions in the industry. By utilizing their possibilities we are able to predict the possible future behavior of our system; moreover, we are able to intervene in its operation as well. In this paper we have investigated the possibilities of the design of a Receding Horizon Controller by using Nonlinear Programming. We have applied the developed solution in order to control Type 1 Diabetes Mellitus. The nonlinear optimization task was solved by the Generalized Reduced Gradient method. In order to investigate the performance of our solution two scenarios were examined. In the first scenario, we applied “soft” disturbance—namely, smaller amount of external carbohydrate—in order to be sure that the proposed method operates well and the solution that appeared through optimization is acceptable. In the second scenario, we have used “unfavorable” disturbance signal—a highly oscillating external excitation with cyclic peaks. We have found that the performance of the realized controller was satisfactory and it was able to keep the blood glucose level in the desired healthy range—by considering the restrictions for the usable control action.

Synthesis of a Large Oscillation Four-Bar Mechanism

1997 ◽  
Author(s):  
Gloria K. Starns ◽  
Donald R. Flugrad
Keyword(s):  
Reduced Gradient Method ◽  
Large Oscillation ◽  
Oscillation Mechanism ◽  
Straight Line ◽  
Transmission Angle ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient ◽  
Angular Oscillations ◽  
Four Bar Linkage ◽  
Effective Transmission

Abstract This paper demonstrates procedures implemented for the synthesis of a four-bar mechanism that produces large angular oscillations of the output member while maintaining effective transmission angles. The mechanisms are modeled as being driven by a force applied at the coupler link. Additionally this force’s line of action is constrained to occur along an approximate straight line. This research was conducted out of the need for a device that is capable of retraction of the horizontal tool bar housed on the back of a tractor. The tool bars accommodate the implements required to accomplish the numerous tasks of the farmer, i.e. row markers, sprayer arms, planters, etc. Upon retraction of the tool bar so that it is parallel to ground, the appropriate tools are lowered to their working position. As the length of these bars increases, a savings of time and increased productivity is realized. Kurt Hain makes the following observation regarding large oscillation mechanisms in [1]: “It would be very difficult to solve this problem with one four-bar linkage, because it is difficult to design a four-bar linkage having such a large oscillation of a crank without running into problems of poor transmission angle characteristics; it might be possible to use linkages in combinations with gears, but this would make the mechanism more expensive, less efficient, and probably noisier.” In this study simulated annealing, a genetic algorithm and the generalized reduced gradient method are used to produce mechanisms with large angular oscillations of the output member and transmission angles that vary by as little as 20° from 90°. A comparative analysis of each of the optimization procedures is presented with observations regarding the efficacy of each method in the solution of the large oscillation mechanism.

An inventory model of a three parameter Weibull distributed deteriorating item with variable demand dependent on price and frequency of advertisement under trade credit

2019 ◽  
Vol 53 (3) ◽  
pp. 903-916 ◽  
Author(s):  
Ali Akbar Shaikh ◽  
Leopoldo Eduardo Cárdenas–Barrón ◽  
Asoke Kumar Bhunia ◽  
Sunil Tiwari
Keyword(s):  
Trade Credit ◽  
Inventory Model ◽  
Selling Price ◽  
Credit Policy ◽  
Variable Demand ◽  
Reduced Gradient Method ◽  
Deterioration Rate ◽  
Rate Dependent ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

This paper develops an inventory model for a deteriorating item with variable demand dependent on the selling price and frequency of advertisement of the item under the financial trade credit policy. Shortages are allowed and these are partially backlogged with a variable rate dependent on the duration of waiting time until to the arrival of next order. In this inventory model, the deterioration rate follows a three-parameter Weibull distribution. The corresponding inventory model is formulated and solved by using the well-known generalized reduced gradient method along with an algorithm. To validate the inventory model, two numerical examples are considered and solved. Finally, based on one numerical example, the impacts of different parameters are studied by a sensitivity analysis considering one parameter at a time and leaving the other parameters fixed.

scholarly journals The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zhengyong Zhou ◽  
Bo Yu
Keyword(s):  
Nonlinear Programming ◽  
Interior Point Method ◽  
Large Scale ◽  
State Of The Art ◽  
Numerical Procedure ◽  
Homotopy Method ◽  
Nonlinear Constraints ◽  
Computational Results ◽  
Homotopy Methods ◽  
Constraint Function

The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

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