Development of linear and nonlinear optimization Theories, Methods and Applications.

Follow. optimization Research includes Theories, Methods, and applications. the optimization theory mainly studies the problem solution's optimum conditions, sensitivity analysis, solution existence and general complexity. optimization methods include constructing new algorithms, proving the convergence of solutions, comparison and complexity of algorithms. optimization application research includes Algorithm Implementation, algorithm program, software package and commercialization, and application in practical problems. this section describes the development of linear and nonlinear optimization Theories, Methods, and applications.

1. Linear Optimization

Linear Optimization, also known as linear planning, is the most widely used branch in operations research. this is because many problems in natural sciences and social sciences can be converted into linear programming problems. the Research and Development of Linear Programming theories and algorithms have gone through three climax, each of which has aroused great social attention. the first climax of Linear Planning Research is the famous study of the pure shape method. this method was proposed by dantzig in 1947. It has been using mature algorithm theories and improved algorithms and software to govern linear programming for more than thirty years. with the research of computational complexity theory developed in 1960s, the simple form method was challenged at the end of 1970s. in 1979, khachiyan, a mathematician of the former Soviet Union, proposed the first so-called polynomial time algorithm, the elliptical method, which is theoretically better than the simple method. It was a sensation and set off the second climax of linear planning. however, it is regrettable that extensive numerical experiments show that the calculation of the elliptical algorithm is worse than that of the pure method.

　　

Karmarkar proposed another polynomial time Algorithm for Linear Programming in 1984. this algorithm is superior to the elliptical method in both theory and value, which has aroused great attention in the academic world and thus sets off the third climax of linear programming. since then, many scholars have made great efforts to improve and improve this algorithm and have obtained many improved algorithms. these algorithms use different ideas and methods to obtain iterative point columns within a feasible region. Therefore, these algorithms are collectively called internal point Algorithms for Solving Linear Programming Problems. at present, the internal point algorithm is beyond and replaces the simple form method with an irresistible trend.

　　

Software that can be accessed on the Internet to solve linear and integer programming problems include: eqps (linear, integer, and nonlinear programming), FMP (linear and Mixed Integer Programming ), HS/lplo (Linear Planning), korbx (Linear Planning), lamps (linear and integer planning), lpblp (Linear Planning), milp (Mixed Integer planning ), minto (Mixed Integer planning), mpsiii (linear and Mixed Integer planning), OML (linear and Mixed Integer planning), OSL (linear, quadratic, and mixed integer planning ), proclp (linear and Integer Programming), WB (linear and Mixed Integer Programming), whizard (linear and Mixed Integer Programming), xpressmp (linear and Mixed Integer Programming), etc.

An important theory of nonlinear planning is the establishment of the Kuhn-Tucker optimal condition (KT condition) in 1951. since then, in 1950s, the gradient method and Newton method were mainly studied. starting from the DFP methods proposed by David on (1959), Fletcher and Powell (1963), 1960s was the active period of the quasi-Newton method. At the same time, there was a good research on the conjugate gradient method. in 1970, the BFGS method proposed by Broyden, Fletcher, Goldfarb, and Shanno from different perspectives was the most effective quasi-Newton method. broyden, Dennis, and more have improved the theory of the quasi-Newton method. during the rapid development of non-linear planning in 1970s, the constrained variable scale (SQP) method (represented by Han and Powell) and the Laplace multiplier method (represented by Powell and hestenes) is the main research result of this period. the rapid development of computers makes the study of nonlinear planning even more powerful. in 1980s, the trust domain method, sparse quasi-Newton method, large-scale problem method and parallel computing were studied. In 1990s, the internal point method and finite storage method for solving nonlinear programming problems were studied. it is no exaggeration to say that this half century is the prime time for optimal development.

　　

Compared with linear programming, the nonlinear programming software is not perfect. however, a large number of software programs are available to solve non-linear programming problems, and a considerable portion of them can be downloaded for free from the Internet. BTN is a software that uses the block truncation Newton method of the line search technique to solve the non-constraint problem. The approximate Newton direction is obtained through the block Gradient Method to Solve the Newton equation. the block structure facilitates parallel processing of linear algebra equations and function compute. BTN has two versions: simplified version and user version. the simplified version does not require parallel technology, but the user version allows a variety of complex operations, including parallel processing. this software can use

　　

Lancelot is a software package developed by Conn, gowould, and toint to solve large-scale optimization problems. It is suitable for solving non-constrained optimization, Non-Linear Least Square, boundary constraint optimization, and general constraint optimization problems. the basic idea of this software is to use the augmented Laplace function to deal with constraints. In each iteration, A subproblem of boundary constraint optimization is solved. The methods used are combined with trust domain and projection gradient technologies. minpack is a software package developed by Argonne National Laboratory. It is suitable for solving nonlinear equations and Nonlinear Least Squares. The basic method used is damping least squares. This software can be obtained from the library. proc NLP is a program of the or module in the SAS commercial software developed by SAS software company, this program is suitable for solving non-constrained optimization, non-linear least squares, Linear Constrained Optimization, quadratic programming, and general constrained optimization. tenmin is a tensor method software developed by Schnabel to solve small and medium scale problems ($ n <100 $. Software that can be accessed on the Internet to solve non-linear optimization problems include conopt (nonlinear planning), dot (optimization design Toolbox), Excel and Quattro Pro solvers (linear, integer and Nonlinear Programming), fsqp (nonlinear programming and extremely minor problems), grg2 (Nonlinear Programming), lbfgs (finite storage method ), lindo (linear, quadratic, and mixed integer planning), lssol (Least Square and quadratic planning), Minos (linear and nonlinear planning), nlpjob (nonlinear multi-objective Planning ), optpack (constrained and unrestricted optimization), pets (Parallel Algorithms for solving nonlinear equations and unrestricted problems), qpopt (linear and Quadratic Programming ), sqopt (Large-Scale Linear and convex quadratic planning), snopt (Large-Scale Linear, quadratic, and nonlinear planning), sprnlp (sparse Least Squares, sparse and dense nonlinear planning ), sysfit, tensolve, and ve10.