LEADER 05218nam 2200613Ia 450 001 9910786967903321 005 20200520144314.0 010 $a981-4412-52-X 035 $a(CKB)2670000000361827 035 $a(EBL)1193426 035 $a(SSID)ssj0000950915 035 $a(PQKBManifestationID)11529078 035 $a(PQKBTitleCode)TC0000950915 035 $a(PQKBWorkID)10881425 035 $a(PQKB)11603865 035 $a(MiAaPQ)EBC1193426 035 $a(Au-PeEL)EBL1193426 035 $a(CaPaEBR)ebr10700616 035 $a(CaONFJC)MIL486883 035 $a(OCoLC)843871633 035 $a(PPN)189428279 035 $a(EXLCZ)992670000000361827 100 $a20130419n2013 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aUndergraduate convexity$b[electronic resource] $efrom Fourier and Motzkin to Kuhn and Tucker /$fNiels Lauritzen 210 $aSingapore $cWorld Scientific$d2013 215 $a1 online resource (300 p.) 300 $aDescription based upon print version of record. 311 $a981-4412-51-1 320 $aIncludes bibliographical references and index. 327 $aPreface; Acknowledgments; Contents; 1. Fourier-Motzkin elimination; 1.1 Linear inequalities; 1.2 Linear optimization using elimination; 1.3 Polyhedra; 1.4 Exercises; 2. Affine subspaces; 2.1 Definition and basics; 2.2 The affine hull; 2.3 Affine subspaces and subspaces; 2.4 Affine independence and the dimension of a subset; 2.5 Exercises; 3. Convex subsets; 3.1 Basics; Minkowski sum, dilation and the polar of a subset; 3.2 The convex hull; 3.3 Faces of convex subsets; Interlude: Integral points in convex subsets; 3.4 Convex cones; The recession cone; Finitely generated cones 327 $a3.5 Carathe?odory's theorem3.6 The convex hull, simplicial subsets and Bland's rule; Non-cycling; 3.7 Exercises; 4. Polyhedra; 4.1 Faces of polyhedra; 4.2 Extreme points and linear optimization; 4.3 Weyl's theorem; 4.4 Farkas's lemma; 4.5 Three applications of Farkas's lemma; 4.5.1 Markov chains and steady states; 4.5.2 Gordan's theorem; 4.5.3 Duality in linear programming; 4.6 Minkowski's theorem; 4.7 Parametrization of polyhedra; 4.8 Doubly stochastic matrices: The Birkhoff polytope; 4.8.1 Perfect pairings and doubly stochastic matrices; 4.9 Exercises; 5. Computations with polyhedra 327 $a5.1 Extreme rays and minimal generators in convex cones5.2 Minimal generators of a polyhedral cone; 5.3 The double description method; 5.3.1 Converting from half space to vertex representation; 5.3.2 Converting from vertex to half space representation; 5.3.3 Computing the convex hull; 5.4 Linear programming and the simplex algorithm; 5.4.1 Two examples of linear programs; 5.4.2 The simplex algorithm in a special case; 5.4.3 The simplex algorithm for polyhedra in general form; 5.4.4 The simplicial hack; 5.4.5 The computational miracle of the simplex tableau; The simplex algorithm 327 $aExplaining the steps5.4.6 Computing a vertex in a polyhedron; 5.5 Exercises; 6. Closed convex subsets and separating hyperplanes; 6.1 Closed convex subsets; 6.2 Supporting hyperplanes; 6.3 Separation by hyperplanes; 6.4 Exercises; 7. Convex functions; 7.1 Basics; 7.2 Jensen's inequality; 7.3 Minima of convex functions; 7.4 Convex functions of one variable; 7.5 Differentiable functions of one variable; 7.5.1 The Newton-Raphson method for finding roots; 7.5.2 Critical points and extrema; 7.6 Taylor polynomials; 7.7 Differentiable convex functions; 7.8 Exercises 327 $a8. Differentiable functions of several variables8.1 Differentiability; 8.1.1 The Newton-Raphson method for several variables; 8.1.2 Local extrema for functions of several variables; 8.2 The chain rule; 8.3 Lagrange multipliers; The two variable case; The general case and the Lagrangian; 8.4 The arithmetic-geometric inequality revisited; 8.5 Exercises; 9. Convex functions of several variables; 9.1 Subgradients; 9.2 Convexity and the Hessian; 9.3 Positive definite and positive semidefinite matrices; 9.4 Principal minors and definite matrices; 9.5 The positive semidefinite cone 327 $a9.6 Reduction of symmetric matrices 330 $aBased on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples.Starting from linear inequalities and Fourier-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point algori 606 $aConvex domains 606 $aAlgebras, linear 615 0$aConvex domains. 615 0$aAlgebras, linear. 676 $a515.88 700 $aLauritzen$b Niels$f1964-$01529547 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910786967903321 996 $aUndergraduate convexity$93773877 997 $aUNINA