Nonlinear and Mixed-Integer Optimization Fundamentals and Applications

by Floudas, Christodoulos A.

• ISBN13: 9780195100563
• ISBN10: 0195100565
• Format: Hardcover
• Copyright: 1995-10-05
• Publisher: Oxford University Press

Summary

Filling a void in chemical engineering and optimization literature, this book presents the theory and methods for nonlinear and mixed-integer optimization, and their applications in the important area of process synthesis. Other topics include modeling issues in process synthesis, andoptimization-based approaches in the synthesis of heat recovery systems, distillation-based systems, and reactor-based systems. The basics of convex analysis and nonlinear optimization are also covered and the elementary concepts of mixed-integer linear optimization are introduced. All chapters haveseveral illustrations and geometrical interpretations of the material as well as suggested problems. Nonlinear and Mixed-Integer Optimization will prove to be an invaluable source--either as a textbook or a reference--for researchers and graduate students interested in continuous and discretenonlinear optimization issues in engineering design, process synthesis, process operations, applied mathematics, operations research, industrial management, and systems engineering.

Table of Contents

Introduction	3	(14)
Mathematical and Optimization Models	3	(1)
Structure of Nonlinear and Mixed-Integer Optimization Models	4	(1)
Illustrative Applications	5	(7)
Binary Distillation Design	6	(2)
Retrofit Design of Multiproduct Batch Plants	8	(3)
Multicommodity Facility Location--Allocation	11	(1)
Scope of the Book	12	(5)
PART 1 FUNDAMENTALS OF CONVEX ANALYSIS AND NONLINEAR OPTIMIZATION
Convex Analysis	17	(28)
Convex Sets	17	(7)
Basic Definitions	17	(3)
Convex Combination and Convex Hull	20	(2)
Separation of Convex Sets	22	(2)
Support of Convex Sets	24	(1)
Convex and Concave Functions	24	(13)
Basic Definitions	25	(1)
Properties of Convex and Concave Functions	25	(2)
Continuity and Semicontinuity	27	(3)
Directional Derivative and Subgradients	30	(1)
Differentiable Convex and Concave Functions	31	(3)
Minimum (Infimum) and Maximum (Supremum)	34	(2)
Feasible Solution, Local and Global Minimum	36	(1)
Generalizations of Convex and Concave Functions	37	(8)
Quasi-convex and Quasi-concave Functions	37	(2)
Properties of Quasi-convex and Quasi-concave Functions	39	(1)
Differentiable Quasi-convex, Quasi-concave Functions	40	(1)
Pseudo-convex and Pseudo-concave Functions	40	(1)
Properties of Pseudo-convex and Pseudo-concave Functions	40	(1)
Relationships among Convex, Quasi-convex and Pseudo-convex Functions	41	(4)
Fundamentals of Nonlinear Optimization	45	(30)
Unconstrained Nonlinear Optimization	45	(4)
Formulation and Definitions	45	(1)
Necessary Optimality Conditions	46	(1)
Sufficient Optimality Conditions	47	(1)
Necessary and Sufficient Optimality Conditions	48	(1)
Constrained Nonlinear Optimization	49	(26)
Formulation and Definitions	49	(2)
Lagrange Functions and Multipliers	51	(1)
Interpretation of Lagrange Multipliers	52	(2)
Existence of Lagrange Multipliers	54	(2)
Weak Lagrange Functions	56	(1)
First-Order Necessary Optimality Conditions	56	(5)
First-Order Sufficient Optimality Conditions	61	(1)
Saddle Point and Optimality Conditions	62	(2)
Second-Order Necessary Optimality Conditions	64	(3)
Second-Order Sufficient Optimality Conditions	67	(1)
Outline of Nonlinear Algorithmic Methods	68	(7)
Duality Theory	75	(20)
Primal Problem	75	(2)
Formulation	75	(1)
Perturbation Function and Its Properties	76	(1)
Stability of Primal Problem	76	(1)
Existence of Optimal Multipliers	77	(1)
Dual Problem	77	(5)
Formulation	78	(1)
Dual Function and Its Properties	78	(1)
Illustration of Primal-Dual Problems	79	(1)
Geometrical Interpretation of Dual Problem	80	(2)
Weak and Strong Duality	82	(5)
Illustration of Strong Duality	84	(1)
Illustration of Weak and Strong Duality	85	(1)
Illustration of Weak Duality	86	(1)
Duality Gap and Continuity of Perturbation Function	87	(8)
Illustration of Duality Gap	88	(7)
PART 2 FUNDAMENTALS OF MIXED-INTEGER OPTIMIZATION
Mixed-Integer Linear Optimization	95	(14)
Motivation	95	(1)
Formulation	96	(2)
Mathematical Description	96	(1)
Complexity Issues in MILP	96	(1)
Outline of MILP Algorithms	97	(1)
Branch and Bound Method	98	(11)
Basic Notions	98	(3)
General Branch and Bound Framework	101	(2)
Branch and Bound Based on Linear Programming Relaxation	103	(6)
Mixed-Integer Nonlinear Optimization	109	(116)
Motivation	109	(1)
Formulation	110	(4)
Mathematical Description	111	(1)
Challenges/Difficulties in MINLP	112	(1)
Overview of MINLP Algorithms	112	(2)
Generalized Benders Decomposition, GBD	114	(30)
Formulation	114	(1)
Basic Idea	115	(1)
Theoretical Development	116
Algorithmic Development	112	(13)
Variants of GBD	125	(15)
GBD in Continuous and Discrete-Continuous Optimization	140	(4)
Outer Approximations, OA	144	(11)
Formulation	144	(1)
Basic Idea	145	(1)
Theoretical Development	145	(6)
Algorithmic Development	151	(4)
Outer Approximation with Equality Relaxation, OA/ER	155	(13)
Formulation	155	(1)
Basic Idea	156	(1)
Theoretical Development	156	(4)
Algorithmic Development	160	(1)
Illustration	161	(7)
Outer Approximation with Equality Relaxation and Augmented Penalty, OA/ER/AP	168	(7)
Formulation	168	(1)
Basic Idea	169	(1)
Theoretical Development	169	(1)
Algorithm Development	170	(1)
Illustration	171	(4)
Generalized Outer Approximation, GOA	175	(8)
Formulation	175	(1)
Basic Idea	175	(1)
Theoretical Development	176	(3)
Algorithmic Development	179	(1)
Worst-Case Analysis of GOA	180	(1)
Generalized Outer Approximation with Exact Penalty, GOA/EP	181	(2)
Comparison of GBD and OA-based Algorithms	183	(7)
Formulation	183	(1)
Nonlinear Equality Constraints	184	(1)
Nonlinearities in y and Joint x-y	184	(2)
The Primal Problem	186	(1)
The Master Problem	187	(2)
Lower Bounds	189	(1)
Generalized Cross Decomposition, GCD	190	(35)
Formulation	190	(1)
Basic Idea	191	(1)
Theoretical Development	191	(8)
Algorithmic Development	199	(4)
GCD under Separability	203	(5)
GCD In Continuous and Discrete-Continuous Optimization	208	(17)
PART 3 APPLICATIONS IN PROCESS SYNTHESIS
Process Synthesis	225	(34)
Introduction	225	(4)
The Overall Process System	226	(3)
Definition	229	(3)
Difficulties/Challenges in Process Synthesis	230	(2)
Approaches in Process Synthesis	232	(1)
Optimization Approach in Process Synthesis	233	(24)
Outline	233	(1)
Representation of Alternatives	234	(1)
Mathematical Model of Superstructure	235	(21)
Algorithmic Development	256	(1)
Application Areas	257	(2)
Heat Exchanger Network Synthesis	259	(120)
Introduction	259	(2)
Problem Statement	261	(1)
Definition of Temperature Approaches	262	(1)
Targets for HEN Synthesis	262	(42)
Minimum Utility Cost	262	(18)
Minimum Number of Matches	280	(14)
Minimum Number of Matches for Vertical Heat Transfer	294	(10)
Decomposition-based HEN Synthesis Approaches	304	(19)
Heat Exchanger Network Derivation	305	(16)
HEN Synthesis Strategy	321	(2)
Simultaneous HEN Synthesis Approaches	323	(56)
Simultaneous Matches-Network Optimization	324	(14)
Pseudo-Pinch	338	(4)
Synthesis of HENs Without Decomposition	342	(14)
Simultaneous Optimization Models for HEN Synthesis	356	(23)
Distillation-based Separation Systems Synthesis	379	(28)
Introduction	379	(2)
Synthesis of Heat-integrated Sharp Distillation Sequences	381	(12)
Problem Statement	382	(1)
Basic Idea	382	(1)
Derivation of Superstructure	383	(2)
Mathematical Formulation of Superstructure	385	(8)
Synthesis of Nonsharp Distillation Sequences	393	(14)
Problem Statement	396	(1)
Basic Idea	396	(1)
Nonsharp Separation Superstructure	397	(3)
Mathematical Formulation of Nonsharp Separation Superstructure	400	(7)
Synthesis of Reactor Networks and Reactor-Separator-Recycle Systems	407	(28)
Introduction	407	(4)
Synthesis of Isothermal Reactor Networks	411	(11)
Problem Statement	411	(1)
Basic Idea	412	(1)
Reactor Unit Representation	412	(2)
Reactor Network Superstructure	414	(1)
Mathematical Formulation of Reactor Superstructure	415	(7)
Synthesis of Reactor-Separator-Recycle Systems	422	(13)
Introduction	422	(2)
Problem Statement	424	(1)
Basic Idea	424	(1)
Reactor-Separator-Recycle Superstructure	425	(3)
Mathematical Formulation	428	(7)
Bibliography	435	(18)
Index	453