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【实例简介】
Nonlinear programming Bertsekas D. (2ed., Athena 学习优化理论与算法的经典教材，深入学习优化算法必备图书。
Contents 1. Unconstrained Optimization 1.I. Optimality conditions 4 1.1.1 Variational ideas p.4 1. 1.2. Main Optimality conditions 13 1. 2. Gradient Methods-Convergence 22 1. 2. 1. Descent Directions and Stepsize rules 22 1. 2.2. Convergence Results 43 1.3. Gradient Methods- Rate of Convergence 62 1.3. 1. The Local Analysis Approach 64 1.3.2. The rolc of the Condition number 65 1,3.3. Convergence Rate results p.75 1.4. Newton's method and variations p,88 1.5. Least Squares Problems p.102 1.5. 1. The Gauss-Ncwton Method p.107 1. 5.2. Incremental Gradient Methods* p.108 1.5.3. Incremental Forms of the Gauss-Newton Method* 119 1.6. Conjugate Direction Methods p.130 1.7. Quasi-Newton Methods p.148 1. 8. Nonderivative methods 158 1.8.1. Coordinate Descent p.160 1.8.2. Dircct Search Methods p.162 1. 9. Discrete- Tine Optinal Control ProbleIIIs* p 166 1.10. Some Practical Guidelines 183 L.ll. Notes and Sources p.187 2. Optimization Over a Convex Set ··p.191 2.l, Optimality Conditions p.192 2.2. Feasible Directions and the Conditional Gradient Method p.209 2.2.1. Descent Directions and Stepsize rules p.210 2.2.2. The Conditional Gradient Method p.215 2.3. Gradient Projection Methods p.223 2.3. 1. Feasible Directions and Stepsize Rules Based on Projection p, 223 2.3.2. Convergence Analysis p.234 2.4. Two-Metric Projection Methods p.244 2.5. Manifold Suboptimization Mcthods .250 2.6. Affine Scaling for Linear Programming p.259 2.7. Block Coordinate Descent Methods* p.267 2. 8. Notes and Sources p.272 3. Lagrange Multiplier Theory p275 3.1 Necessary Conditions for Equality Constraints p.277 3.1.1. The Penalty approach D.281 3. 2. The Elimination Approach p.283 3.1.3. The Lagrangian Function p.287 3.2. Sufficient Conditions and Sensitivity Analysis 295 3.2.1. The Augmented Lagrangian Approach p.297 3. 2.2. The Feasible Direction Approach p.300 3.2.3. Sensitivity p.301 3.3. Inequality constraints p.307 3.3. 1. Karush-Kuhn-Tucker Optimality Conditions 3.3. 2. Conversion to the Equality Case* onditions p.309 p.312 3.3.3. Second Order Sufficiency Conditions and Sensitivity p.314 3.3.4. Sufficiency Conditions and Lagrangian Minimization p.315 3.3.5. Fritz John Optimality Conditions p.317 3.3.6. Refinements* p.330 3.4. Linear Constraints and Duality* I.357 3.4.1. Convex Cost Functions and Linear Constraints 357 3.4.2. Duality Theory: A Simple Form for Linear Constraints p.359 3.5. Notes and sources p.367 4. Lagrange Multiplier Algorithms p.369 4.1. Barrier and Interior point methods p,370 4. 1. 1. Linear Programming and the logarithmic Barrier*K 373 4.2. Penalty and Augmented Lagrangian Methods 388 4.2. 1. The Quadratic Penalty Function Method I.390 4.2. 2. Multiplier methods- Main ideas 4.2.3. Convergence Analysis of Multiplier Methods* p.398 p.407 4.2. 4. Duality and Second Order Multiplier methods p.410 4.2.5. The Exponential Method of Multipliers* p.413 4. 3. Exact Penalties -Sequential Quadratic Programming p.421 4.3.1. Nondifferentiable exact penalty functions p.422 4.3.2. Differentiable Exact Penalty functions p.439 4.4. Lagrangian and Primal-Dual Interior Point Methods* p.446 4,4.1. First-Order methods p.446 4.4.2, Newton-Like Methods for Equality Constraints p.450 4.4.3. Global Convergence p.460 Contents VII 4.4.4. Primal-Dual Interior point methods p.463 4.4.5. Comparison of various met hods 471 4.5. Notes and sources 473 5. Duality and Convex Programming p、477 5.1. The Dual Problem p.479 5. 1.1, Lagrange Multipliers p.480 5. 1.2. The Weak Duality thcorem 485 5.1.3. Characterization of Primal and Dual Optimal solutions p.490 5. 1. The Case of an Infeasible or Unbounded Primal Problem. p, 4191 5. 1.5. Treatment of Equality Constraints p.493 5.1.6. Separable Problems and Their Geometry p.494 5.1.7. Additional Issues About Duality 5.2. Convex Cost- Linear Constraints* 498 .503 5.2. 1, Proofs of Duality Theorems p.505 5.3. Convex cost- Convox Constraints 511 5.4. Conjugate Functions and Fenchel Duality p.521 5.4.1. Monotropic Programming duality p.525 5.4.2. Network Optimization p.529 5.4.3. Games and the Minimax Theorem p.531 5. 4.4. The Primal function p.534 5.4.5. A Dual View of Penalty methods p.536 5.4.6. The Proximal and Entropy Minimization algorithms p.542 5.5. Discrctc Optimization and duality p.558 5.5.1. Examples of Discrete Optimization Problems 559 5.5.2. Branch-and -Bound p.567 5.5.3. Lagrangian Relaxation .p.576 6. Notes alld sources 587 6. Dual methods p.591 6.1. Dual Derivatives and Subgradients p.594 6.2. Dual Ascent Methods for Differentiable Dual Problems*... p 600 6.2.1. Coordinate Ascent for Quadratic Programining p.600 6.2.2. Decomposition and Primal Strict Convexity .p.603 6.2.3. Partitioning and Dual Strict Concavity 6.3. Nondifferentiable Optimization Methods* p.604 p.609 6.3. 1. Subgradient Mcthods p.610 6.3.2. Approximate and Incremental Subgradient Methods p.614 6.3.3. Cutting Plane methods p.618 6.3.4. Ascent and Approximate Ascent Methods ,625 6.4. Decomposition Methods* p.638 6.4.1. Lagrangian Relaxation of the Coupling Constraints p.639 61.4.2. Decomposition by Right-Hand Side Allocation p.642 6.5. Notes and sources p.645 Appendix A: Mathematical Background p.647 A 1. Vectors and matrices P.648 A. 2. Norms, Sequences, Limits, and Continuity p.649 A.3. Square Matrices and Eigenvalues ,656 A.4. Symmetric and Positive Definite Matrices .659 A,5. Derivatives 66 A 6. Contraction Mappings p.669 Appendix B: Convex Analysis p.671 B. 1. Convex Sets and Functions p.671 B 2. Separating Hyperplanes p.689 B 3. Cones and Polyhedral Convexity p.694 B 4. Extreme points p.701 B.5. Differentiability Issues p.707 Appendix C: Line Search Methods 723 1. Cubic Interpolation 723 C 2. Quadratic Interpolation p.724 C 3. The Golden section Method p.726 Appendix D: Implementation of Newton's Method p.729 D 1. Cholesky factorization 729 D 2. Application to a Modified Newton Method p.731 eferences p735 Index p.773 Preface Nonlinear programming is a mature field that has experienced major de velopments in the last ten years. The first such development is the merging of linear and nonlinear programming algorithms through the use of inte. rior point methods, This has resulted in a profound rethinking of how we solve lincar programming problems, and in a major reassessment of how we treat constraints in nonlinear programming. A second development, less visible but still important is the increased emphasis on large-scale problems, and the associated algorithms that take advantage of problem structure as well as parallel hardware. a third development has been the extensive use of iterative unconstrained optimization to solve the difficult east squares problems arising in the training of neural networks. As a result, simple gradient-like methods and stepsize rules have attained in- creased importance The purpose of this book is to provide an up-to-date, comprehe and rigorous account of nonlinear programming at the beginning graduate student levcl. In addition to the classical topics, such as descent rithIns, Lagrange inulliplier Theory, and duality, soine of Che inportant recent developments are covered: interior point methods for linear and nonlinear programs, major aspects of large-scale optimization, and least squares problems and neural network training A further noteworthy featurc of the book is that it treats lagrange multipliers and duality using two different and complementary approaches a variational approach based on the implicit function theorem, and a convex analysis approach based on geometrical arguments. The former approach applies to a broader class of problems, while the latter is more elegant and more powerful for the convex programs to which it applies he chapter-by-chapter description of the book follows Chapter 1: This chapter covers unconstrained optimization; main con- cepts, optimality conditions, and algorithms. The material is classic, but there are discussions of topics frequently left untreated, such as the be avior of algorithms for singular problcms, neural nctwork training, and discrete-time optimal control 工⊥cL Chapter 2: This chapter treats constrained optimization over a convex set without the use of Lagrange multipliers. i prefer to cover this ma- terial before dealing with the complex machinery of lagrange multipliers bccausc i have found that students absorb easily algorithms such as con ditional gradient, gradient projection, and coordinate descent, which can be viewed as natural extensions of unconstrained descent algorithms. This chapter contains also a treatment of the affine scaling method for linear progranming Chapter 3: This chapter gives a detailed treatment of Lagrange multi- pliers, the associated necessary and sufficient conditions, and sensitivity analysis, The first three sections deal with nonlinear equality and inequal- ity constraints. The last section deals with linear constraints and develops a simple form of duality theory for linearly constrained problems with dif ferentiable cost, including linear and quadratic programming. Chapter 4: This chapter treats constrained optimization algorithms that use penalties and Lagrange multipliers, including barrier, augmented La grangian, sequential quadratic programming, and primal-dual interior point methods for linear programming. The treatment is extensive, and borrows from my 1982 research monograph on Lagrange multiplier methods Chapter 5: This chapter provides an in-depth coverage of duality theory (Lagrange and Fenchel. The treatment is totally geometric, and every- thing is explained in terms of intuitive figures Chapter 6: This chapter deals with large-scale optimization methods based on duality. Some material is borrowed from my Parallel and dis tributcd Algorithms book(coauthorcd by John'Isitsiklis), but there is also an extensive treatment of nondifferentiable optimization, including subgra- dient, E-subgradient, and cutting plane methods. Decomposition methods such as dantzig Wolfe and Benders are also discussed Appendixes: Four appendixes are given, The first gives a summary of calculus, analysis, and linear algebra rcsults used in the text. The second is a fairly extensive account of convexity theory, including proofs of the basic polyhedral convexity results on extreme points and Farkas'lermma as well the basic facts about subgradients, The third appendix covers one-dimensional minimization methods, The last appendix discusses an implementation of Newton's mcthod for unconstrained optimization Inevitably, some coverage compromises had to be made. The subject of nonlinear optimization has grown so much that leaving out a number of important topics could not be avoided For example, a discussion of variational inequalities, a deeper treatment of optimality conditions, and a morc dctailed devclopment of Quasi-Newton mcthods arc not provided Also, a larger number of sample applications would have been desirable. I hope that instructors will supplement the book with the type of practical examples that their stidents are most familiar with The book was developed through a first-year graduate course that i taught at the Univ, of Illinois and at M.I.T. over a period of 20 years The mathematical prerequisites are matrix-vector algebra and advanced calculus, including a good understanding of convergence concepts, A course in analysis and or linear algebra should also bc very helpful, and would provide the mathematical maturity needed to follow and to appreciate the mathematical reasoning used in the book. Some of the sections in the book may be ommited at first reading without loss of continuity. These sections haye becn marked by a star. The rule followed here is that the material discussed in a starred section is not used in a non-starred section The book can be used to teach several different types of courses (a) a two-quarter course that covers most sections of every chapter (b) A onie-semester course that covers Clapter 1 except for Section 1. 9 Chapter 2 except for Sections 2, 4 and 2, 5, Chapter 3 except for Section 3, 4, Chapter 4 except for parts of Sections 4.2 and 4.3, the first three scctions of Chaptcr 5, and a selection from Section 5. 4 and Chapter 6. This is the course i usually teach at MiT (c a one-semester course that covers most of Chapters 1, 2, and 3, and selected algorithms from Chapter 4. i have taught this type of course several times. It is less demanding of the students because it does not require the machinery of convex analysis, yet it still provides a fairly powerful version of duality theory(Section 3. 4) (d)A one-quarter course that covers selected parts of Chapters 1, 2, 3 and 4. This is a less comprehensive version of (c) above (e) a one quarter course on convex analysis and optimization that starts with appendix b and covers Sections 1.1, 2.1, 3.4, and Chapter 5 There is a very extensive literature on nonlinear programming and to give a complete bibliography and a historical account of the research that led to the present form of the subject would have been impossible. I thus have not attempted to compile a comprehensive list of original con ributions to the field. I have cited sources that i have used extensively, that provide important extensions to the material of the book, that survey important topics, or that are particularly well suited for further reading I have also cited selectively a few sources that are historically significant but the reference list is far from exhaustive in this respect. Generally, to aid researchers in the field, i have preferred to cite surveys and text books for subjects that are relatively mature, and to give a larger number of references for rclatively rccent developments Finally i would like to express thanks to a number of individuals for their contributions to the book conceptual understanding of the subject was formed at Stanford University while I interacted with David XII Luenberger and i taught using his books. This experience had a lasting in- Auence on my thinking. My research collaboration with several colleagues particularly Joc Dunn, Eli Gaini, Paul Tseng, and John Tsitsiklis, were very useful and are refected in the book. I appreciate the suggestions and insights of a number of people, particularly David Castanon, Joe Dunn, ry Rockafellar, Paul Tseng, and John Tsitsiklis.I am thankful to the many students and collaborators whose comments led to corrections and clarifications, Steve Patek, Serap Savari, and Cynara Wu were particularly helpful in this respect. Davi id logan Steve patek and lakis polymenakos helped me to generate the graph of the cover, which depicts the cost func tion of a simple neural network training problem. My wife Joanna cheered me up with her presence and humor during the long hours of writing, as she has with her companionship of over 30 years. i dedicate this book to her with my loy Dimitri p bertsekas November、1995 【实例截图】
【核心代码】
Nonlinear programming Bertsekas D. (2ed., Athena 学习优化理论与算法的经典教材，深入学习优化算法必备图书。
Contents 1. Unconstrained Optimization 1.I. Optimality conditions 4 1.1.1 Variational ideas p.4 1. 1.2. Main Optimality conditions 13 1. 2. Gradient Methods-Convergence 22 1. 2. 1. Descent Directions and Stepsize rules 22 1. 2.2. Convergence Results 43 1.3. Gradient Methods- Rate of Convergence 62 1.3. 1. The Local Analysis Approach 64 1.3.2. The rolc of the Condition number 65 1,3.3. Convergence Rate results p.75 1.4. Newton's method and variations p,88 1.5. Least Squares Problems p.102 1.5. 1. The Gauss-Ncwton Method p.107 1. 5.2. Incremental Gradient Methods* p.108 1.5.3. Incremental Forms of the Gauss-Newton Method* 119 1.6. Conjugate Direction Methods p.130 1.7. Quasi-Newton Methods p.148 1. 8. Nonderivative methods 158 1.8.1. Coordinate Descent p.160 1.8.2. Dircct Search Methods p.162 1. 9. Discrete- Tine Optinal Control ProbleIIIs* p 166 1.10. Some Practical Guidelines 183 L.ll. Notes and Sources p.187 2. Optimization Over a Convex Set ··p.191 2.l, Optimality Conditions p.192 2.2. Feasible Directions and the Conditional Gradient Method p.209 2.2.1. Descent Directions and Stepsize rules p.210 2.2.2. The Conditional Gradient Method p.215 2.3. Gradient Projection Methods p.223 2.3. 1. Feasible Directions and Stepsize Rules Based on Projection p, 223 2.3.2. Convergence Analysis p.234 2.4. Two-Metric Projection Methods p.244 2.5. Manifold Suboptimization Mcthods .250 2.6. Affine Scaling for Linear Programming p.259 2.7. Block Coordinate Descent Methods* p.267 2. 8. Notes and Sources p.272 3. Lagrange Multiplier Theory p275 3.1 Necessary Conditions for Equality Constraints p.277 3.1.1. The Penalty approach D.281 3. 2. The Elimination Approach p.283 3.1.3. The Lagrangian Function p.287 3.2. Sufficient Conditions and Sensitivity Analysis 295 3.2.1. The Augmented Lagrangian Approach p.297 3. 2.2. The Feasible Direction Approach p.300 3.2.3. Sensitivity p.301 3.3. Inequality constraints p.307 3.3. 1. Karush-Kuhn-Tucker Optimality Conditions 3.3. 2. Conversion to the Equality Case* onditions p.309 p.312 3.3.3. Second Order Sufficiency Conditions and Sensitivity p.314 3.3.4. Sufficiency Conditions and Lagrangian Minimization p.315 3.3.5. Fritz John Optimality Conditions p.317 3.3.6. Refinements* p.330 3.4. Linear Constraints and Duality* I.357 3.4.1. Convex Cost Functions and Linear Constraints 357 3.4.2. Duality Theory: A Simple Form for Linear Constraints p.359 3.5. Notes and sources p.367 4. Lagrange Multiplier Algorithms p.369 4.1. Barrier and Interior point methods p,370 4. 1. 1. Linear Programming and the logarithmic Barrier*K 373 4.2. Penalty and Augmented Lagrangian Methods 388 4.2. 1. The Quadratic Penalty Function Method I.390 4.2. 2. Multiplier methods- Main ideas 4.2.3. Convergence Analysis of Multiplier Methods* p.398 p.407 4.2. 4. Duality and Second Order Multiplier methods p.410 4.2.5. The Exponential Method of Multipliers* p.413 4. 3. Exact Penalties -Sequential Quadratic Programming p.421 4.3.1. Nondifferentiable exact penalty functions p.422 4.3.2. Differentiable Exact Penalty functions p.439 4.4. Lagrangian and Primal-Dual Interior Point Methods* p.446 4,4.1. First-Order methods p.446 4.4.2, Newton-Like Methods for Equality Constraints p.450 4.4.3. Global Convergence p.460 Contents VII 4.4.4. Primal-Dual Interior point methods p.463 4.4.5. Comparison of various met hods 471 4.5. Notes and sources 473 5. Duality and Convex Programming p、477 5.1. The Dual Problem p.479 5. 1.1, Lagrange Multipliers p.480 5. 1.2. The Weak Duality thcorem 485 5.1.3. Characterization of Primal and Dual Optimal solutions p.490 5. 1. The Case of an Infeasible or Unbounded Primal Problem. p, 4191 5. 1.5. Treatment of Equality Constraints p.493 5.1.6. Separable Problems and Their Geometry p.494 5.1.7. Additional Issues About Duality 5.2. Convex Cost- Linear Constraints* 498 .503 5.2. 1, Proofs of Duality Theorems p.505 5.3. Convex cost- Convox Constraints 511 5.4. Conjugate Functions and Fenchel Duality p.521 5.4.1. Monotropic Programming duality p.525 5.4.2. Network Optimization p.529 5.4.3. Games and the Minimax Theorem p.531 5. 4.4. The Primal function p.534 5.4.5. A Dual View of Penalty methods p.536 5.4.6. The Proximal and Entropy Minimization algorithms p.542 5.5. Discrctc Optimization and duality p.558 5.5.1. Examples of Discrete Optimization Problems 559 5.5.2. Branch-and -Bound p.567 5.5.3. Lagrangian Relaxation .p.576 6. Notes alld sources 587 6. Dual methods p.591 6.1. Dual Derivatives and Subgradients p.594 6.2. Dual Ascent Methods for Differentiable Dual Problems*... p 600 6.2.1. Coordinate Ascent for Quadratic Programining p.600 6.2.2. Decomposition and Primal Strict Convexity .p.603 6.2.3. Partitioning and Dual Strict Concavity 6.3. Nondifferentiable Optimization Methods* p.604 p.609 6.3. 1. Subgradient Mcthods p.610 6.3.2. Approximate and Incremental Subgradient Methods p.614 6.3.3. Cutting Plane methods p.618 6.3.4. Ascent and Approximate Ascent Methods ,625 6.4. Decomposition Methods* p.638 6.4.1. Lagrangian Relaxation of the Coupling Constraints p.639 61.4.2. Decomposition by Right-Hand Side Allocation p.642 6.5. Notes and sources p.645 Appendix A: Mathematical Background p.647 A 1. Vectors and matrices P.648 A. 2. Norms, Sequences, Limits, and Continuity p.649 A.3. Square Matrices and Eigenvalues ,656 A.4. Symmetric and Positive Definite Matrices .659 A,5. Derivatives 66 A 6. Contraction Mappings p.669 Appendix B: Convex Analysis p.671 B. 1. Convex Sets and Functions p.671 B 2. Separating Hyperplanes p.689 B 3. Cones and Polyhedral Convexity p.694 B 4. Extreme points p.701 B.5. Differentiability Issues p.707 Appendix C: Line Search Methods 723 1. Cubic Interpolation 723 C 2. Quadratic Interpolation p.724 C 3. The Golden section Method p.726 Appendix D: Implementation of Newton's Method p.729 D 1. Cholesky factorization 729 D 2. Application to a Modified Newton Method p.731 eferences p735 Index p.773 Preface Nonlinear programming is a mature field that has experienced major de velopments in the last ten years. The first such development is the merging of linear and nonlinear programming algorithms through the use of inte. rior point methods, This has resulted in a profound rethinking of how we solve lincar programming problems, and in a major reassessment of how we treat constraints in nonlinear programming. A second development, less visible but still important is the increased emphasis on large-scale problems, and the associated algorithms that take advantage of problem structure as well as parallel hardware. a third development has been the extensive use of iterative unconstrained optimization to solve the difficult east squares problems arising in the training of neural networks. As a result, simple gradient-like methods and stepsize rules have attained in- creased importance The purpose of this book is to provide an up-to-date, comprehe and rigorous account of nonlinear programming at the beginning graduate student levcl. In addition to the classical topics, such as descent rithIns, Lagrange inulliplier Theory, and duality, soine of Che inportant recent developments are covered: interior point methods for linear and nonlinear programs, major aspects of large-scale optimization, and least squares problems and neural network training A further noteworthy featurc of the book is that it treats lagrange multipliers and duality using two different and complementary approaches a variational approach based on the implicit function theorem, and a convex analysis approach based on geometrical arguments. The former approach applies to a broader class of problems, while the latter is more elegant and more powerful for the convex programs to which it applies he chapter-by-chapter description of the book follows Chapter 1: This chapter covers unconstrained optimization; main con- cepts, optimality conditions, and algorithms. The material is classic, but there are discussions of topics frequently left untreated, such as the be avior of algorithms for singular problcms, neural nctwork training, and discrete-time optimal control 工⊥cL Chapter 2: This chapter treats constrained optimization over a convex set without the use of Lagrange multipliers. i prefer to cover this ma- terial before dealing with the complex machinery of lagrange multipliers bccausc i have found that students absorb easily algorithms such as con ditional gradient, gradient projection, and coordinate descent, which can be viewed as natural extensions of unconstrained descent algorithms. This chapter contains also a treatment of the affine scaling method for linear progranming Chapter 3: This chapter gives a detailed treatment of Lagrange multi- pliers, the associated necessary and sufficient conditions, and sensitivity analysis, The first three sections deal with nonlinear equality and inequal- ity constraints. The last section deals with linear constraints and develops a simple form of duality theory for linearly constrained problems with dif ferentiable cost, including linear and quadratic programming. Chapter 4: This chapter treats constrained optimization algorithms that use penalties and Lagrange multipliers, including barrier, augmented La grangian, sequential quadratic programming, and primal-dual interior point methods for linear programming. The treatment is extensive, and borrows from my 1982 research monograph on Lagrange multiplier methods Chapter 5: This chapter provides an in-depth coverage of duality theory (Lagrange and Fenchel. The treatment is totally geometric, and every- thing is explained in terms of intuitive figures Chapter 6: This chapter deals with large-scale optimization methods based on duality. Some material is borrowed from my Parallel and dis tributcd Algorithms book(coauthorcd by John'Isitsiklis), but there is also an extensive treatment of nondifferentiable optimization, including subgra- dient, E-subgradient, and cutting plane methods. Decomposition methods such as dantzig Wolfe and Benders are also discussed Appendixes: Four appendixes are given, The first gives a summary of calculus, analysis, and linear algebra rcsults used in the text. The second is a fairly extensive account of convexity theory, including proofs of the basic polyhedral convexity results on extreme points and Farkas'lermma as well the basic facts about subgradients, The third appendix covers one-dimensional minimization methods, The last appendix discusses an implementation of Newton's mcthod for unconstrained optimization Inevitably, some coverage compromises had to be made. The subject of nonlinear optimization has grown so much that leaving out a number of important topics could not be avoided For example, a discussion of variational inequalities, a deeper treatment of optimality conditions, and a morc dctailed devclopment of Quasi-Newton mcthods arc not provided Also, a larger number of sample applications would have been desirable. I hope that instructors will supplement the book with the type of practical examples that their stidents are most familiar with The book was developed through a first-year graduate course that i taught at the Univ, of Illinois and at M.I.T. over a period of 20 years The mathematical prerequisites are matrix-vector algebra and advanced calculus, including a good understanding of convergence concepts, A course in analysis and or linear algebra should also bc very helpful, and would provide the mathematical maturity needed to follow and to appreciate the mathematical reasoning used in the book. Some of the sections in the book may be ommited at first reading without loss of continuity. These sections haye becn marked by a star. The rule followed here is that the material discussed in a starred section is not used in a non-starred section The book can be used to teach several different types of courses (a) a two-quarter course that covers most sections of every chapter (b) A onie-semester course that covers Clapter 1 except for Section 1. 9 Chapter 2 except for Sections 2, 4 and 2, 5, Chapter 3 except for Section 3, 4, Chapter 4 except for parts of Sections 4.2 and 4.3, the first three scctions of Chaptcr 5, and a selection from Section 5. 4 and Chapter 6. This is the course i usually teach at MiT (c a one-semester course that covers most of Chapters 1, 2, and 3, and selected algorithms from Chapter 4. i have taught this type of course several times. It is less demanding of the students because it does not require the machinery of convex analysis, yet it still provides a fairly powerful version of duality theory(Section 3. 4) (d)A one-quarter course that covers selected parts of Chapters 1, 2, 3 and 4. This is a less comprehensive version of (c) above (e) a one quarter course on convex analysis and optimization that starts with appendix b and covers Sections 1.1, 2.1, 3.4, and Chapter 5 There is a very extensive literature on nonlinear programming and to give a complete bibliography and a historical account of the research that led to the present form of the subject would have been impossible. I thus have not attempted to compile a comprehensive list of original con ributions to the field. I have cited sources that i have used extensively, that provide important extensions to the material of the book, that survey important topics, or that are particularly well suited for further reading I have also cited selectively a few sources that are historically significant but the reference list is far from exhaustive in this respect. Generally, to aid researchers in the field, i have preferred to cite surveys and text books for subjects that are relatively mature, and to give a larger number of references for rclatively rccent developments Finally i would like to express thanks to a number of individuals for their contributions to the book conceptual understanding of the subject was formed at Stanford University while I interacted with David XII Luenberger and i taught using his books. This experience had a lasting in- Auence on my thinking. My research collaboration with several colleagues particularly Joc Dunn, Eli Gaini, Paul Tseng, and John Tsitsiklis, were very useful and are refected in the book. I appreciate the suggestions and insights of a number of people, particularly David Castanon, Joe Dunn, ry Rockafellar, Paul Tseng, and John Tsitsiklis.I am thankful to the many students and collaborators whose comments led to corrections and clarifications, Steve Patek, Serap Savari, and Cynara Wu were particularly helpful in this respect. Davi id logan Steve patek and lakis polymenakos helped me to generate the graph of the cover, which depicts the cost func tion of a simple neural network training problem. My wife Joanna cheered me up with her presence and humor during the long hours of writing, as she has with her companionship of over 30 years. i dedicate this book to her with my loy Dimitri p bertsekas November、1995 【实例截图】
【核心代码】
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