Other Titles in Applied Mathematics最新文献

{"title":"Primal-Dual Interior-Point Methods","authors":"Stephen J. Wright","doi":"10.1137/1.9781611971453","DOIUrl":"https://doi.org/10.1137/1.9781611971453","url":null,"abstract":"Preface Notation 1. Introduction. Linear Programming Primal-Dual Methods The Central Path A Primal-Dual Framework Path-Following Methods Potential-Reduction Methods Infeasible Starting Points Superlinear Convergence Extensions Mehrotra's Predictor-Corrector Algorithm Linear Algebra Issues Karmarkar's Algorithm 2. Background. Linear Programming and Interior-Point Methods Standard Form Optimality Conditions, Duality, and Solution Sets The B {SYMBOL 200 f \"Symbol\"} N Partition and Strict Complementarity A Strictly Interior Point Rank of the Matrix A Bases and Vertices Farkas's Lemma and a Proof of the Goldman-Tucker Result The Central Path Background. Primal Method Primal-Dual Methods. Development of the Fundamental Ideas Notes and References 3. Complexity Theory. Polynomial Versus Exponential, Worst Case vs Average Case Storing the Problem Data. Dimension and Size The Turing Machine and Rational Arithmetic Primal-Dual Methods and Rational Arithmetic Linear Programming and Rational Numbers Moving to a Solution from an Interior Point Complexity of Simplex, Ellipsoid, and Interior-Point Methods Polynomial and Strongly Polynomial Algorithms Beyond the Turing Machine Model More on the Real-Number Model and Algebraic Complexity A General Complexity Theorem for Path-Following Methods Notes and References 4. Potential-Reduction Methods. A Primal-Dual Potential-Reduction Algorithm Reducing Forces Convergence A Quadratic Estimate of Phi _{rho } along a Feasible Direction Bounding the Coefficients in The Quadratic Approximation An Estimate of the Reduction in Phi _{rho } and Polynomial Complexity What About Centrality? Choosing {SYMBOL 114 f \"Symbol\"} and {SYMBOL 97 f \"Symbol\"} Notes and References 5. Path-Following Algorithms. The Short-Step Path-Following Algorithm Technical Results The Predictor-Corrector Method A Long-Step Path-Following Algorithm Limit Points of the Iteration Sequence Proof of Lemma 5.3 Notes and References 6. Infeasible-Interior-Point Algorithms. The Algorithm Convergence of Algorithm IPF Technical Results I. Bounds on nu _k delimiter \"026B30D (x^k,s^k) delimiter \"026B30D Technical Results II. Bounds on (D^k)^{-1} Delta x^k and D^k Delta s^k Technical Results III. A Uniform Lower Bound on {SYMBOL 97 f \"Symbol\"}k Proofs of Theorems 6.1 and 6.2 Limit Points of the Iteration Sequence 7. Superlinear Convergence and Finite Termination. Affine-Scaling Steps An Estimate of ({SYMBOL 68 f \"Symbol\"}x, {SYMBOL 68 f \"Symbol\"} s). The Feasible Case An Estimate of ({SYMBOL 68 f \"Symbol\"} x, {SYMBOL 68 f \"Symbol\"} s). The Infeasible Case Algorithm PC Is Superlinear Nearly Quadratic Methods Convergence of Algorithm LPF+ Convergence of the Iteration Sequence {SYMBOL 206 f \"Symbol\"}(A,b,c) and Finite Termination A Finite Termination Strategy Recovering an Optimal Basis More on {SYMBOL 206 f \"Symbol\"} (A,b,c) Notes and References 8. Extensions. The Monotone LCP Mixed and Horizontal LCP Strict Complementarity and LCP Convex QP Convex Programming Monotone N","PeriodicalId":101611,"journal":{"name":"Other Titles in Applied Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127507081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}