On the Global Convergence of Nonlinear Programming Algorithms

1985 ◽  
Vol 107 (4) ◽  
pp. 454-458 ◽  
Author(s):  
K. Schittkowski
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Initial Point ◽  
Convergence Result ◽  
Gradient Algorithms ◽  
Reduced Gradient ◽  
Numerical Performance ◽  
Generalized Reduced Gradient ◽  
Nonlinear Programming Methods ◽  
Programming Algorithms

In a previous paper a unified outline of some of the most successful nonlinear programming methods was presented by the author, i.e. of penalty, multiplier, sequential quadratic programming, and generalized reduced gradient algorithms, to illustrate their common mathematical features and to explain the different numerical performance observed in practice. By defining a general algorithmic frame for all these approaches, a global convergence result can be achieved in the sense that starting from an arbitrary initial point, a stationary solution will be approximated.

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.

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.

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.

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.

Aggregate Constraint Homotopy Method for Nonlinear Programming on Unbounded Set

2011 ◽  
Vol 50-51 ◽  
pp. 283-287
Author(s):  
Yu Xiao ◽  
Hui Juan Xiong ◽  
Zhi Gang Yan
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Additional Assumption ◽  
Normal Cone ◽  
Constraint Qualifications ◽  
Convergence Result ◽  
Homotopy Method ◽  
Cone Condition ◽  
Feasible Set

In [1], an aggregate constraint aggregate (ACH) method for nonconvex nonlinear programming problems was presented and global convergence result was obtained when the feasible set is bounded and satisfies a weak normal cone condition with some standard constraint qualifications. In this paper, without assuming the boundedness of feasible set, the global convergence of ACH method is proven under a suitable additional assumption.

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.

scholarly journals Computer Aided Optimal Design of Compressed Air Energy Storage Systems

1980 ◽  
Vol 102 (3) ◽  
pp. 437-445 ◽  
Author(s):  
F. W. Ahrens ◽  
A. Sharma ◽  
K. M. Ragsdell
Keyword(s):  
Energy Storage ◽  
Numerical Models ◽  
Operating Cost ◽  
Optimization Methods ◽  
Compressed Air ◽  
Compressed Air Energy Storage ◽  
Reduced Gradient ◽  
The Media ◽  
Generalized Reduced Gradient ◽  
Programming Algorithms

An automated procedure for the design of Compressed Air Energy Storage (CAES) systems is presented. The procedure relies upon modern nonlinear programming algorithms, decomposition theory, and numerical models of the various system components. Two modern optimization methods are employed; BIAS, a Method of Multipliers code and OPT, a Generalized Reduced Gradient code. The procedure is demonstrated by the design of a CAES facility employing the Media, Illinois Galesville aquifer as the reservoir. The methods employed produced significant reduction in capital and operating cost, and in number of aquifer wells required.

scholarly journals A comparative study on optimization methods for the constrained nonlinear programming problems

2005 ◽  
Vol 2005 (2) ◽  
pp. 165-173 ◽  
Author(s):  
Ozgur Yeniay
Keyword(s):  
Nonlinear Programming ◽  
Comparative Study ◽  
Sequential Quadratic Programming ◽  
Gradient Methods ◽  
Optimization Methods ◽  
Test Problems ◽  
Reduced Gradient ◽  
Constrained Nonlinear Programming ◽  
Generalized Reduced Gradient ◽  
Reduced Gradient Methods

Constrained nonlinear programming problems often arise in many engineering applications. The most well-known optimization methods for solving these problems are sequential quadratic programming methods and generalized reduced gradient methods. This study compares the performance of these methods with the genetic algorithms which gained popularity in recent years due to advantages in speed and robustness. We present a comparative study that is performed on fifteen test problems selected from the literature.

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