Nonlinear Systems and Optimization for the Chemical Engineer: Solving Numerical Problems -- Contents -- Preface -- 1 Function Root-Finding -- 1.1 Introduction -- 1.2 Substitution Algorithms -- 1.3 Bolzano's Algorithm -- 1.4 Function Approximation -- 1.4.1 Newton's Method -- 1.4.2 The Secant Method -- 1.4.3 Regula Falsi Method -- 1.4.4 Muller's Method or Parabolic Interpolation -- 1.4.5 Hyperbolic Interpolation Method -- 1.4.6 Inverse Polynomial Interpolation Method -- 1.4.7 Inverse Rational Interpolation Method -- 1.5 Use of a Multiprocessor Machine with a Known Interval of Uncertainty -- 1.6 Search for an Interval of Uncertainty -- 1.7 Stop Criteria -- 1.8 Classes for Function Root-Finding -- 1.9 Case Studies -- 1.9.1 Calculation of the Volume of a Nonideal Gas -- 1.9.2 Calculation of the Bubble Point of Vapor-Liquid Equilibrium -- 1.9.3 Zero-Crossing Problem -- 1.9.4 Stationary Condition in a Gravity-Flow Tank -- 1.10 Tests for BzzFunctionRoot and BzzFunctionRootMP Classes -- 1.11 Some Caveats -- 2 One-Dimensional Optimization -- 2.1 Introduction -- 2.2 Measuring the Efficiency of the Search for the Minimum -- 2.3 Comparison Methods -- 2.4 Parabolic Interpolation -- 2.5 Cubic Interpolation -- 2.6 Gradient-Based Methods -- 2.7 Combination of Algorithms in a General Program -- 2.8 Parallel Computations -- 2.9 Search for the Interval of Uncertainty -- 2.10 Stop Criteria -- 2.11 Classes for One-Dimensional Minimization -- 2.12 Case Studies -- 2.12.1 Optimization of Unimodal Functions -- 2.12.2 Optimization of a Batch Reactor -- 2.12.3 Maximum Level in a Gravity-Flow Tank in Transient Conditions -- 2.13 Tests -- 3 Unconstrained Optimization -- 3.1 Introduction -- 3.1.1 Necessary and Sufficient Conditions -- 3.1.2 Quadratic Functions -- 3.1.3 Directions of Function Decrease -- 3.1.4 Comparison with the One-Dimensional Case.

3.1.5 Classification of Methods -- 3.2 Heuristic Methods -- 3.2.1 Modified Hooke-Jeeves Method -- 3.2.2 The Rosenbrock Method -- 3.2.3 The Nelder-Mead Simplex Method -- 3.2.4 Robust Optnov Method Combined with the Simplex Method -- 3.3 Gradient-Based Methods -- 3.4 Conjugate Direction Methods -- 3.5 Newton's Method -- 3.6 Modified Newton Methods -- 3.6.1 Singular or Nonpositive Definite Hessian Matrix -- 3.6.2 Convergence Problems -- 3.6.3 One-Dimensional Search -- 3.6.4 Trust Region Methods -- 3.6.5 Use of Alternative Methods -- 3.7 Quasi-Newton Methods -- 3.8 Narrow Valley Effect -- 3.9 Stop Criteria -- 3.10 BzzMath Classes for Unconstrained Multidimensional Minimization -- 3.11 Case Study -- 3.11.1 Optimization of a Batch Reactor -- 3.11.2 Optimal Adiabatic Bed Reactors for Sulfur Dioxide with Cold Shot Cooling -- 3.11.3 Global Optimization -- 3.12 Tests -- 4 Large-Scale Unconstrained Optimization -- 4.1 Introduction -- 4.2 Collecting a Sparse Symmetric Matrix -- 4.3 Ordering the Hessian Rows and Columns -- 4.4 Quadratic Functions -- 4.5 Hessian Evaluation -- 4.6 Newton's Method -- 4.7 Inexact Newton Methods -- 4.8 Practical Preconditioners -- 4.9 openMP Parallelization -- 4.10 Class for Large-Scale Unconstrained Minimization -- 5 Robust Unconstrained Minimization -- 5.1 Introduction -- 5.2 One-Dimensional Minimization -- 5.3 Classes for One-Dimensional Robust Minimization -- 5.4 Examples in One-Dimensional Space -- 5.5 Examples in Multidimensional Space -- 5.6 Two-Dimensional Minimization -- 5.7 Classes for Robust Two-Dimensional Minimization -- 5.8 Examples for BzzMinimizationTwoVeryRobust Class -- 5.9 Multidimensional Robust Minimization -- 5.9.1 Outer Optimizer -- 5.9.2 Inner Optimizer -- 5.10 Class for Robust Multidimensional Minimization -- 6 Robust Function Root-Finding -- 6.1 Introduction -- 6.2 Class and Examples -- 7 Nonlinear Systems.

7.1 Introduction -- 7.2 Comparing Nonlinear Systems to Other Iterative Problems -- 7.2.1 Comparison with Function Root-Finding -- 7.2.2 Comparison with the Multidimensional Optimization -- 7.3 Convergence Test -- 7.4 Substitution Methods -- 7.5 Minimization Methods -- 7.6 Jacobian Evaluation -- 7.7 Newton's Method -- 7.8 Gauss-Newton Method -- 7.9 Modified Newton Methods -- 7.9.1 Singular or Ill-Conditioned Jacobian -- 7.9.2 Convergence Problem -- 7.10 Newton's Method and Parallel Computations -- 7.11 Quasi-Newton Methods -- 7.12 Quasi-Newton Methods and Parallel Computing -- 7.13 Stop Criteria -- 7.13.1 Bounds, Constraints, and Discontinuities -- 7.14 Classes for Nonlinear System Solution with Dense Matrices -- 7.15 Tests for the BzzNonLinearSystem Class -- 7.16 Sparse and Large-Scale Systems -- 7.17 Large Linear System Solution with Iterative Methods -- 7.18 Classes for Nonlinear System Solution with Sparse Matrices -- 7.19 Continuation Methods -- 7.20 Solution of Certain Equations with Respect to Certain Variables -- 7.21 Case Studies -- 7.21.1 Heat Exchange in a Thermal Furnace -- 7.21.2 Calculation of Chemical Equilibria -- 7.21.3 Multiple Solutions in a CSTR -- 7.21.4 Critical Size of a Nuclear Reactor -- 7.21.5 Stationary Conditions for a Nonisothermal Continuous Stirred Tank Reactor -- 7.21.6 Vapor-Liquid Equilibrium: The Flash Separator -- 7.21.7 Boundary Value Problems -- 7.22 Special Cases -- 7.22.1 Vapor-Liquid Equilibrium: Distillation Column -- 7.22.2 Kinetic Postprocessor -- 7.23 Some Caveats -- 8 Underdimensioned Nonlinear Systems -- 8.1 Introduction -- 8.2 Underdimensioned Linear Systems -- 8.2.1 Null Space -- 8.2.2 Determination of One Solution -- 8.2.3 Projection Methods -- 8.2.4 Stable Gauss Factorization -- 8.3 Class for Underdimensioned Nonlinear System Solution -- 9 Constrained Minimization -- 9.1 Introduction.

9.2 Equality Constraints -- 9.3 Equality and Inequality Constraints -- 9.4 Lagrangian Dual Problem -- 10 Linear Programming -- 10.1 Introduction -- 10.2 Basic Attic Method Concepts -- 10.3 Attic Method -- 10.3.1 Certain Important Peculiarities of the Attic Method -- 10.3.2 What Happens in a Generic Iteration -- 10.3.3 Selecting the New Feasible Point and the New Inequality Constraint -- 10.3.4 Selection of the Search Direction -- 10.4 Differences between the Attic Method and Traditional Approaches -- 10.4.1 A Simple Application of the Attic Method -- 10.4.2 Certain Advantages to the Attic Method -- 10.5 Explosion in the Number of Iterations -- 10.5.1 Selecting Constraints Rather Than Vertices -- 10.5.2 The Tile Effect -- 10.5.3 Wall Constraints and Roof Constraints -- 10.5.4 When a Constraint with λ>0 Should also be Removed -- 10.6 Degeneracy -- 10.7 Duality -- 10.8 General Considerations -- 11 Quadratic Programming -- 11.1 Introduction -- 11.2 KKT Conditions for a QP Problem -- 11.3 Equality-Constrained QP -- 11.3.1 Solving the Full KKT System -- 11.3.2 Shur-Complement Method -- 11.3.3 Null Space Methods -- 11.4 Equality- and Inequality-Constrained Problems -- 11.5 Class for QP -- 11.6 Projection or Reduced Direction Search Methods for Bound-Constrained Problems -- 11.7 Equality, Inequality, and Bound Constraints -- 11.8 Tests -- 12 Constrained Minimization: Penalty and Barrier Functions -- 12.1 Introduction -- 12.2 Penalty Function Methods -- 12.2.1 Quadratic Penalty Function -- 12.2.2 Nonsmooth Exact Penalty Function -- 12.2.3 The Maratos Effect -- 12.2.4 Augmented Lagrangian Penalty Function -- 12.2.5 Bound-Constrained Formulation for Lagrangian Penalty Function -- 12.3 Barrier Function Methods -- 12.4 Mixed Penalty-Barrier Function Methods -- 13 Constrained Minimization: Active Set Methods -- 13.1 Introduction.

13.2 Class for Constrained Minimization -- 13.3 Successive Linear Programming -- 13.4 Projection Methods -- 13.5 Reduced Direction Search Methods -- 13.6 Projection or Reduced Direction Search Methods for Bound-Constrained Problems -- 13.7 Successive Quadratic Programming or Projected Lagrangian Method -- 13.7.1 Selection of the Merit Function -- 13.7.2 Updating the Jacobian of the System -- 13.8 Narrow Valley Effect -- 13.9 The Nonlinear Constraints Effect -- 13.10 Tests -- 14 Parametric Continuation in Optimization and Process Control -- 14.1 Introduction -- 14.2 Algebraic Constraints -- 14.2.1 Distillation Column -- References -- Appendix A: Copyrights -- Index.