Configuration Analysis and Design by Using Optimization Tools in Mathematica

In engineering, economic, and scientific studies, decisions are frequently modeled by applying optimization concepts and techniques. This article discusses global optimization in multiextremal models and tools to handle such models in Mathematica. Since we assume that not all readers are familiar with optimization models and methods, a general modeling framework is presented. We also review several built-in Mathematica optimization functions and then introduce MathOptimizer, an application package for continuous nonlinear (convex and global) optimization. To illustrate the usage of MathOptimizer, several configuration analysis and design models are formulated and solved. We also provide some comparative notes related to current Mathematica optimization functionality (namely, the function NMinimize) and to the recently introduced MathOptimizer Professional package.

[1]  P. M. Pardalos,et al.  Global minimization of nonconvex energy functions : molecular conformation and protein folding : DIMACS workshop, March 20-21, 1995 , 1996 .

[2]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[3]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[4]  Neil Gershenfeld,et al.  The nature of mathematical modeling , 1998 .

[5]  Bernd Hartke,et al.  Global cluster geometry optimization by a phenotype algorithm with Niches: Location of elusive minima, and low-order scaling with cluster size , 1999, J. Comput. Chem..

[6]  Rutherford Aris,et al.  Mathematical modeling : a chemical engineer's perspective , 1999 .

[7]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[8]  János D. Pintér,et al.  Finding elliptic Fekete points sets: two numerical solution approaches , 2001 .

[9]  János D. Pintér,et al.  Global Optimization: Software, Test Problems, and Applications , 2002 .

[10]  J D Pinter,et al.  Global Optimization in Action—Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications , 2010 .

[11]  D. Himmelblau,et al.  Optimization of Chemical Processes , 1987 .

[12]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[13]  Arnold Neumaier,et al.  Molecular Modeling of Proteins and Mathematical Prediction of Protein Structure , 1997, SIAM Rev..

[14]  G. Thompson,et al.  Optimal Control Theory: Applications to Management Science and Economics , 2000 .

[15]  Franz Rothlauf,et al.  Representations for genetic and evolutionary algorithms , 2002, Studies in Fuzziness and Soft Computing.

[16]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1992, Artificial Intelligence.

[17]  Michael W. Carter,et al.  Operations Research: A Practical Introduction , 2000 .

[18]  John L. Casti,et al.  Searching for Certainty , 1990 .

[19]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[20]  Christian Jacob,et al.  Illustrating Evolutionary Computation with Mathematica , 2001 .

[21]  Richard C. Larson,et al.  Model Building in Mathematical Programming , 1979 .

[22]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[23]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[24]  George B. Dantzig,et al.  Linear Programming 1: Introduction , 1997 .

[25]  János D. Pintér.  Globally Optimized Spherical Point Arrangements: Model Variants and Illustrative Results , 2001, Ann. Oper. Res..

[26]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[27]  S. Smale.  Mathematical problems for the next century , 1998 .

[28]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[29]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[30]  P. V. Remoortere.  Linear and combinatorial programming : K.G. Murty: 1976,J. Wiley, New York, 310 pp , 1979 .

[31]  M. A. Bhatti,et al.  Practical Optimization Methods with Mathematica Applications (& CD-ROM) , 2002, J. Oper. Res. Soc..

[32]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

[33]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[34]  Panos Y. Papalambros,et al.  Principles of Optimal Design: Author Index , 2000 .

[35]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[36]  H. Scheraga,et al.  Global optimization of clusters, crystals, and biomolecules. , 1999, Science.

[37]  Frederick S. Hillier,et al.  Introduction of Operations Research , 1967 .

[38]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[39]  Philip E. Gill,et al.  Practical optimization , 1981 .