Lectures on Modern Convex Optimization,9780898714913

Lectures on Modern Convex Optimization

by ;
ISBN13: 9780898714913
ISBN10: 0898714915
Format: Paperback
Pub. Date: 2001-08-01
Publisher(s): Society for Industrial & Applied
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Summary

Here is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.

Author Biography

Aharon Ben-Tal is a professor at the Technion-Israel Institute of Technology and head of the MINERVA Optimization Center Arkadi Nemirovski is a professor at the Technion-Israel Institute of Technology

Table of Contents

Preface xi
Linear Programming
1(42)
Linear Programming: Basic notions
1(1)
Example: Tschebyshev approximation and its applications
2(9)
Best uniform approximation
2(1)
Application: Synthesis of filters
3(2)
Filter synthesis revisited
5(2)
Synthesis of arrays of antennae
7(4)
Duality in linear programming
11(19)
Certificates for solvability and insolvability
12(4)
Dual to a linear programming program: Origin
16(1)
Linear programming duality theorem
17(2)
Illustration: Problem dual to the Tschebyshev approximation problem
19(2)
Application: Truss topology design
21(9)
Exercises to Lecture 1
30(13)
Uniform approximation
30(4)
Theorem on the alternative
34(1)
Proof of the homogeneous Farkas lemma
35(3)
Helley theorem
38(2)
How many bars are needed in an optimal truss?
40(3)
From Linear Programming to Conic Programming
43(36)
Orderings of Rm and convex cones
44(2)
What is conic programming?
46(4)
Conic duality
50(7)
Geometry of the primal and dual problems
53(4)
Conic duality theorem
57(10)
Is something wrong with conic duality?
60(1)
Consequences of the conic duality theorem
61(3)
Robust solvability status
64(3)
Conic duality revisited
67(6)
Exercises to Lecture 2
73(6)
Cones
73(3)
Conic problems
76(1)
Feasible and level sets of conic problems
77(2)
Conic Quadratic Programming
79(60)
Conic quadratic problems: Preliminaries
79(2)
Examples of conic quadratic problems
81(4)
Best linear approximation of complex-valued functions
81(1)
Contact problems with static friction
82(3)
What can be expressed via conic quadratic constraints?
85(24)
More examples of conic quadratic-representable functions and sets
104(5)
More applications
109(22)
Tschebyshev approximation in relative scale
109(1)
Robust linear programming
110(10)
Truss topology design
120(11)
Exercises to Lecture 3
131(8)
Optimal control in discrete time linear dynamic system
131(1)
Conic quadratic representations
132(5)
Does conic quadratic programming exist?
137(2)
Semidefinite Programming
139(196)
Semidefinite cone and semidefinite programs
139(5)
Preliminaries
139(5)
What can be expressed via linear matrix inequalities?
144(15)
Applications I: Combinatorics
159(19)
Shor's semidefinite relaxation scheme
161(3)
Stability number, Shannon capacity, and Lovasz capacity of a graph
164(6)
MAXCUT problem
170(2)
Extensions
172(3)
S-lemma
175(3)
Applications II: Stability analysis
178(24)
Dynamic stability in mechanics
178(2)
Lyapunov stability analysis and synthesis
180(9)
Interval stability analysis and synthesis
189(13)
Applications III: Robust quadratic programming
202(8)
Applications IV: Synthesis of filters and antennae arrays
210(9)
Applications V: Design of chips
219(8)
Building the model
220(6)
Wire sizing
226(1)
Applications VI: Structural design
227(30)
Building a model
228(5)
Standard case
233(3)
Semidefinite reformulation of the standard SSD problem
236(7)
From primal to dual
243(5)
Back to primal
248(4)
Explicit forms of the standard truss and shape problems
252(5)
Applications VII: Extremal ellipsoids
257(19)
Ellipsoidal approximations of unions and intersections of ellipsoids
262(2)
Approximating sums of ellipsoids
264(12)
Exercises to Lecture 4
276(59)
Around positive semidefiniteness, eigenvalues, and ≥ ordering
276(15)
Semidefinite representations of epigraphs of convex polynomials
291(2)
Lovasz capacity number and semidefinite relaxations of combinatorial problems
293(6)
Lyapunov stability analysis
299(1)
S-lemma
300(23)
Antenna synthesis
323(3)
Ellipsoidal approximations
326(9)
Computational Tractability of Convex Programs
335(42)
Numerical solution of optimization programs --- preliminaries
335(7)
Mathematical programming programs
335(1)
Solution methods and efficiency
336(6)
Black box-represented convex programs
342(10)
Polynomial solvability of convex programming
352(11)
Polynomial solvability of convex programming
359(4)
Difficult problems and NP-completeness
363(14)
CCT --- a quick introduction
363(4)
From the real arithmetic complexity theory to the CCT and back
367(10)
Interior Point Polynominal Time Methods for Linear Programming, Conic Quadratic Programming, and Semidefinite Programming
377(66)
Motivation
377(2)
Interior point methods
378(1)
Newton method and the interior penalty scheme
379(5)
Unconstrained minimization and the Newton method
379(1)
Classical interior penalty scheme: Construction
380(2)
Classical interior penalty scheme: Drawbacks
382(1)
But
382(2)
Interior point methods for linear programming, conic quadratic programming, and semidefinite programming: Building blocks
384(5)
Canonical cones and canonical barriers
384(3)
Elementary properties of canonical barriers
387(2)
Primal-dual pair of problems and the primal-dual central path
389(8)
Problem(s)
389(1)
Central path(s)
390(7)
Tracing the central path
397(24)
Path-following scheme
397(1)
Speed of path-tracing
398(4)
Primal and dual path-following methods
402(3)
Semidefinite programming case
405(16)
Complexity bounds for linear programming, conic quadratic programming, and semidefinite programming
421(3)
Complexity of linear programming
422(1)
Complexity of conic quadratic programming
423(1)
Semidefinite programming
423(1)
Concluding remarks
424(2)
Exercises to Lecture 6
426(17)
Canonical barriers
426(1)
Scalings of canonical cones
427(2)
Dikin ellipsoid
429(2)
More on canonical barriers
431(1)
Primal path-following method
432(3)
Infeasible start path-following method
435(8)
Solutions to Selected Exercises 443(42)
Exercises to Lecture 1
443(3)
Exercises to Lecture 2
446(3)
Exercises to Lecture 3
449(10)
Exercises to Lecture 4
459(16)
Exercises to Lecture 6
475(10)
Index 485

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