Large-scale Nonlinear Programming Problem Research Articles

Successive approximation technique for a class of large-scale NLP problems and its application to dynamic programming

In this paper, two successive approximation techniques are presented for a class of large-scale nonlinear programming problems with decomposable constraints and a class of high-dimensional discrete optimal control problems, respectively. It is shown that: (a) the accumulation point of the sequence produced by the first method is a Kuhn-Tucker point if the constraint functions are decomposable and if the uniqueness condition holds; (b) the sequence converges to an optimum solution if the objective function is strictly pseudoconvex and if the constraint functions are decomposable and quasiconcave; and (c) similar conclusions for the second method hold also for a class of discrete optimal control problems under some assumptions.

Integrating modeling, algorithm design, and computational implementation to solve a large-scale non-linear mixed integer programming problem

This paper describes the formulation of a nonlinear mixed integer programming model for a large-scale product development and distribution problem and the design and computational implementation of a special purpose algorithm to solve the model. The results described demonstrate that integrating the art of modeling with the sciences of solution methodology and computer implementation provides a powerful approach for attacking difficult problems. The efforts described here were successful because they capitalized on the wealth of existing modeling technology and algorithm technology, the availability of efficient and reliable optimization, matrix generation and graphics software, and the speed of large-scale computer hardware. The model permitted the combined use of decomposition, general linear programming and network optimization within a branch and bound algorithm to overcome mathematical complexity. The computer system reliably found solutions with considerably better objective function values 30 to 50 times faster than had been achieved using general purpose optimization software alone. Throughout twenty months of daily use, the system was credited with providing insights and suggesting strategies that led to very large dollar savings.

Integrating modeling, algorithm design, and computational implementation to solve a large-scale nonlinear mixed integer programming problem

This paper describes the formulation of a nonlinear mixed integer programming model for a large-scale product development and distribution problem and the design and computational implementation of a special purpose algorithm to solve the model. The results described demonstrate that integrating the art of modeling with the sciences of solution methodology and computer implementation provides a powerful approach for attacking difficult problems. The efforts described here were successful because they capitalized on the wealth of existing modeling technology and algorithm technology, the availability of efficient and reliable optimization, matrix generation and graphics software, and the speed of large-scale computer hardware. The model permitted the combined use of decomposition, general linear programming and network optimization within a branch and bound algorithm to overcome mathematical complexity. The computer system reliably found solutions with considerably better objective function values 30 to 50 times faster than had been achieved using general purpose optimization software alone. Throughout twenty months of daily use, the system was credited with providing insights and suggesting strategies that led to very large dollar savings.

Discrete and continuous approaches to the optimal synthesis of distillation sequences

Recent developments in integer programming and in large-scale nonlinear programming allow the optimization of certain classes of chemical engineering plants. In this study two approaches for synthesizing distillation processes are presented and compared. In the discrete approach, an integer programming problem resulting from simplifying assumptions is solved with a branch-and-bound algorithm. In the continuous approach a large-scale nonlinear programming problem, whose formulation is rather difficult, is solved by means of a reduced-gradient algorithm. An example shows the complementary nature of the two approaches, the discrete approach providing a good initial feasible point for the continuous approach.

Reliability Optimization by Generalized Lagrangian-Function and Reduced-Gradient Methods

Nonlinear optimization problems for reliability of a complex system are solved using the generalized Lagrangian function (GLF) method and the generalized reduced gradient (GRG) method. GLF is twice continuously differentiable and closely related to the generalized penalty function which includes the interior and exterior penalty functions as a special case. GRG generalizes the Wolfe reduced gradient method and has been coded in FORTRAN title ``GREG'' by Abadie et al. Two system reliability optimization problems are solved. The first maximizes complex-system reliability with a tangent cost-function; the second minimizes the cost, with a minimum system reliability. The results are compared with those using the Sequential Unconstrained Minimization Technique (SUMT) and the direct search approach by Luus and Jaakola (LJ). Many algorithms have been proposed for solving the general nonlinear programming problem. Only a few have been demonstrated to be effective when applied to large-scale nonlinear programming problems, and none has proved to be so superior that it can be classified as a universal algorithm. Both GLF and GRG methods presented here have been successfully used in solving a number of general nonlinear programming problems in a variety of engineering applications and are better methods among the many algorithms.

A Piecewise Method for Optimal Load-Flow Problem

Power utilities are increasingly being interconnected to improve the reliability of power supply and achieve economy of operation. Optimal load-flow analysis of such interconnected power systems, in general, results in complex large scale nonlinear programming problem, the direct method of solution of which requires huge amounts of core and computer time. Decomposition techniques have been extensively investigated in the power system literature to alleviate these difficulties. A piecewise method based on graph theoretic concept is proposed in this paper. The decomposition of the optimal load-flow into an additively separable nonlinear programming problems is achieved through the artifact of equivalent networks which were introduced in the piecewise load-flow problem. Lasdon’s decomposition technique is adapted for its solution. The optimization of subsystems is carried out using a shifted penalty function algorithm imbedded with a recent Fletcher’s unconstrained function minimization routine. Few examples are presented to demonstrate the effectiveness of this approach over the others.

Algorithms for generating large scale nonlinear programming problems with known degree of difficulty

Convex-quadratic functions are defined based on real symmetric positive definite matrices. Gershgorin's Theorem is used for randomly generating positive definite matrices of arbitrary size. A computational algorithm is presented. The concept of degree of difficulty of a test function is introduced and it is used to define an algorithm to generate test problems with varying degree of difficulty. These algorithms when used in conjunction with the Rosen and Suzuki method may be used to generate large scale constrained nonlinear programming problems.

微分動的計画法による非線形計画問題の解法

Dynamic programming is one of the methods which utilize special structures of large-scale mathematical programming problems. Conventional dynamic programming, however, can hardly solve mathematical programming problems with many constraints. This paper proposes differential dynamic programming algorithms for solving large-scale nonlinear programming problems with many constraints and proves their local convergence. The present algorithms, based upon Kuhn-Tucker conditions for subproblems decomposed by dynamic programming, are composed of iterative methods for solving systems of nonlinear equations. It is shown that the convergence of the present algorithms with Newton's method is R-quadratic. Three numerical examples including the Rosen-Suzuki test problem show the efficiency of the present algorithms.