Contents. Preface. 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3

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2 Contents Preface ix 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3 2 Mathematical modeling, cases Introduction Enclosing a set of points Dynamic decision strategies A black box design; a sugar centrifugal screen Design and factorial or quadratic regression Nonlinear optimization in economic models Spatial economic-ecological model Neoclassical dynamic investment model for cattle ranching Several optima in environmental economics Parameter estimation, model calibration, nonlinear regression Learning of neural nets, seen as parameter estimation Summary and discussion points Exercises 27 3 NLP optimality conditions Intuition with some examples Derivative information Derivatives Directional derivative Gradient Second-order derivative Taylor Quadratic functions 41

3 3.4 Optimality conditions, no binding constraints First-order conditions Second-order conditions Optimality conditions, binding constraints Lagrange multiplier method Karush- Kuhn Tucker conditions Convexity First-order conditions are sufficient Local minimum point is global minimum point Maximum point at the boundary of the feasible area Summary and discussion points Exercises Appendix: Solvers for Examples 3.2 and Goodness of optimization algorithms Effectiveness and efficiency of algorithms Effectiveness Efficiency Some basic algorithms and their goodness Introduction NLP local optimization: Bisection and Newton Deterministic GO: Grid search, Piyavskii-Shubert Stochastic GO: PRS, Multistart, Simulated Annealing Investigating algorithms Characteristics Comparison of algorithms Summary and discussion points Exercises 89 5 Nonlinear Programming algorithms Introduction General NLP problem Algorithms Minimizing functions of one variable Bracketing Bisection Golden Section search Quadratic interpolation Cubic interpolation Method of Newton Algorithms not using derivative information Method of Nelder and Mead Method of Powell Algorithms using derivative information Steepest descent method 107

4 5.4.2 Newton method Conjugate gradient method Quasi-Newton method Inexact line search Trust region methods Algorithms for nonlinear regression Linear regression methods Gauss-Newton and Levenberg-Marquardt Algorithms for constrained optimization Penalty and barrier function methods Gradient projection method Sequential quadratic programming Summary and discussion points Exercises Deterministic GO algorithms Introduction Deterministic heuristic, DIRECT Selection for refinement Choice for sampling and updating rectangles Algorithm and illustration Stochastic models and response surfaces Mathematical structures Concavity Difference of convex functions, d.c Lipschitz continuity and bounds on derivatives Quadratic functions Bilinear functions Multiplicative and fractional functions Interval arithmetic Global Optimization branch and bound Examples from nonconvex quadratic programming Example concave quadratic programming Example indefinite quadratic programming Cutting planes Summary and discussion points Exercises Stochastic GO algorithms Introduction Random sampling in higher dimensions All volume to the boundary Loneliness in high dimensions PRS- and Multistart-based methods Pure Random Search as benchmark 174

5 7.3.2 Multistart as benchmark Clustering to save on local searches Tunneling and filled functions Ideal and real, PAS and Hit and Run Population algorithms Controlled Random Search and Raspberries Genetic algorithms Particle swarms Summary and discussion points Exercises 197 References 199 Index 205