Global optimization : deterministic approaches /
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| Author / Creator: | Horst, Reiner |
|---|---|
| Edition: | 2nd rev. ed. |
| Imprint: | Berlin ; New York : Springer-Verlag, c1993. |
| Description: | xvi, 698 p. : ill. ; 25 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/1497916 |
Table of Contents:
- Pt. A. Introduction and Basic Techniques. Ch. I. Some Important Classes of Global Optimization Problems. 1. Global Optimization. 2. Concave Minimization. 2.1. Definition and Basic Properties. 2.2. Brief Survey of Direct Applications. 2.3. Integer Programming and Concave Minimization. 2.4. Bilinear Programming and Concave Minimization. 2.5. Complementarity Problems and Concave Minimization. 2.6. Max-Min Problems and Concave Minimization. 3. D.C. Programming and Reverse Convex Constraints. 3.1. D.C. Programming: Basic Properties. 3.2. D.C. Programming: Applications. 3.3. Reverse Convex Constraints. 3.4. Canonical D.C. Programming Problems. 4. Lipschitzian Optimization and Systems of Equations and Inequalities. 4.1. Lipschitzian Optimization. 4.2. Systems of Equations and Inequalities. Ch. II. Outer Approximation. 1. Basic Outer Approximation Method. 2. Outer Approximation by Convex Polyhedral Sets. 3. Constraint Dropping Strategies. 4. On Solving the Subproblems (Q[subscript k]). 4.1. Finding an Initial Polytope D[subscript 1] and its Vertex Set V[subscript 1]. 4.2. Computing New Vertices and New Extreme Directions. 4.3. Identifying Redundant Constraints. Ch. III. Concavity Cuts. 1. Concept of a Valid Cut. 2. Valid Cuts in the Degenerate Case. 3. Convergence of Cutting Procedures. 4. Concavity Cuts for Handling Reverse Convex Constraints. 5. A Class of Generalized Concavity Cuts. 6. Cuts Using Negative Edge Extensions. Ch. IV. Branch and Bound. 1. A Prototype Branch and Bound Method. 2. Finiteness and Convergence Conditions. 3. Typical Partition Sets and their Refinement. 3.1. Simplices. 3.2. Rectangles and Polyhedral Cones. 4. Lower Bounds. 4.1. Lipschitzian Optimization. 4.2. Vertex Minima. 4.3. Convex Subfunctionals. 4.4. Duality. 4.5. Consistency. 5. Deletion by Infeasibility. 6. Restart Branch and Bound Algorithm
- Pt. B. Concave Minimization. Ch. V. Cutting Methods. 1. A Pure Cutting Algorithm. 1.1. Valid Cuts and a Sufficient Condition for Global Optimality. 1.2. Outline of the Method. 2. Facial Cut Algorithm. 2.1. The Basic Idea. 2.2. Finding an Extreme Face of D Relative to M. 2.3. Facial Valid Cuts. 2.4. A Finite Cutting Algorithm. 3. Cut and Split Algorithm. 3.1. Partition of a Cone. 3.2. Outline of the Method. 3.3. Remarks. 4. Generating Deep Cuts: The Case of Concave Quadratic Functionals. 4.1. A Hierarchy of Valid Cuts. 4.2. Konno's Cutting Method for Concave Quadratic Programming. 4.3. Bilinear Programming Cuts. Ch. VI. Successive Approximation Methods. 1. Outer Approximation Algorithms. 1.1. Linearly Constrained Problem. 1.2. Problems with Convex Constraints. 1.3. Reducing the Sizes of the Relaxed Problems. 2. Inner Approximation. 2.1. The (DG) Problem. 2.2. The Concept of Polyhedral Annexation. 2.3. Computing the Facets of a Polytope. 2.4. A Polyhedral Annexation Algorithm. 2.5. Relations to Other Methods. 2.6. Extensions. 3. Convex Underestimation. 3.1. Relaxation and Successive Underestimation. 3.2. The Falk and Hoffman Algorithm. 3.3. Rosen's Algorithm. 4. Concave Polyhedral Underestimation. 4.1. Outline of the Method. 4.2. Computation of the Concave Underestimators. 4.3. Computation of the Nonvertical Facets. 4.4. Polyhedral Underestimation Algorithm. 4.5. Alternative Interpretation. 4.6. Separable Problems. Ch. VII. Successive Partition Methods. 1. Conical Algorithms. 1.1. The Normal Conical Subdivision Process. 1.2. The Main Subroutine. 1.3. Construction of Normal Subdivision Processes. 1.4. The Basic NCS Process. 1.5. The Normal Conical Algorithm. 1.6. Remarks Concerning Implementation. 1.7. Example. 1.8. Alternative Variants. 1.9. Concave Minimization with Convex Constraints. 1.10. Unbounded Feasible Domain. 1.11. A Class of Exhaustive Subdivision Processes. 1.12. Exhaustive Nondegenerate Subdivision Processes. 2. Simplicial Algorithms. 2.1. Normal Simplicial Subdivision Processes. 2.2. Normal Simplicial Algorithm. 2.3. Construction of an NSS Process. 2.4. The Basic NSS Process. 2.5. Normal Simplicial Algorithm for Problems with Convex Constraints. 3. An Exact Simplicial Algorithm. 3.1. Simplicial Subdivision of a Polytope. 3.2. A Finite Branch and Bound Procedure. 3.3. A Modified ES Algorithm. 3.4. Unbounded Feasible Set. 4. Rectangular Algorithms. 4.1. Normal Rectangular Algorithm. 4.2. Construction of an NRS Process. 4.3. Specialization to Concave Quadratic Programming. 4.4. Example. Ch. VIII. Decomposition of Large Scale Problems. 1. Decomposition Framework. 2. Branch and Bound Approach. 2.1. Normal Simplicial Algorithm. 2.2. Normal Rectangular Algorithm. 2.3. Normal Conical Algorithm. 3. Polyhedral Underestimation Method. 3.1. Nonseparable Problems. 3.2. Separable Problems. 4. Decomposition by Outer Approximation. 4.1. Basic Idea. 4.2. Decomposition Algorithm. 4.3. An Extension. 4.4. Outer Approximation Versus Successive Partition. 4.5. Outer Approximation Combined with Branch and Bound. 5. Decomposition of Concave Minimization Problems over Networks. 5.1. The Minimum Concave Cost Flow Problem. 5.2. The Single Source Uncapacitated Minimum Concave Cost Flow Problem (SUCF). 5.3. Decomposition Method for (SUCF). 5.4. Extension. Ch. IX. Special Problems of Concave Minimization. 1. Bilinear Programming. 1.1. Basic Properties. 1.2. Cutting Plane Method. 1.3. Polyhedral Annexation. 1.4. Conical Algorithm. 1.5. Outer Approximation Method. 2. Complementarity Problems. 2.1. Basic Properties. 2.2. Polyhedral Annexation Method for the Linear Complementarity Problem (LCP). 2.3. Conical Algorithm for the (LCP). 2.4. Other Global Optimization Approaches to (LCP). 2.5. The Concave Complementarity Problem. 3. Parametric Concave Programming. 3.1. Basic Properties. 3.2. Outer Approximation Method for (LRCP). 3.3. Methods Based on the Edge Property. 3.4. Conical Algorithms for (LRCP)
- Pt. C. General Nonlinear Problems. Ch. X. D.C. Programming. 1. Outer Approximation Methods for Solving the Canonical D.C. Programming Problem. 1.1. Duality between the Objective and the Constraints. 1.2. Outer Approximation Algorithms for Canonical D.C. Problems. 1.3. Outer Approximation for Solving Noncanonical D.C. Problems. 2. Branch and Bound Methods. 3. Solving D.C. Problems by a Sequence of Linear Programs and Line Searches. 4. Some Special D.C. Problems and Applications. 4.1. The Design Centering Problem. 4.2. The Diamond Cutting Problem. 4.3. Biconvex Programming and Related Problems. Ch. XI. Lipschitz and Continuous Optimization. 1. Brief Introduction into the Global Minimization of Univariate Lipschitz Functions. 1.1. Saw-Tooth Covers. 1.2. Algorithms for Solving the Univariate Lipschitz-Problem. 2. Branch and Bound Algorithms. 2.1. Branch and Bound Interpretation of Piyavskii's Univariate Algorithm. 2.2. Branch and Bound Methods for Minimizing a Lipschitz Function over an n-dimensional Rectangle. 2.3. Branch and Bound Methods for Solving Lipschitz Optimization Problems with General Constraints. 2.4. Global Optimization of Concave Functions Subject to Separable Quadratic Constraints. 3. Outer Approximation. 4. The Relief Indicator Method. 4.1. Separators for f on D. 4.2. A Global Optimality Criterion. 4.3. The Relief Indicator Method.