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− Applications • New problem paradigms with rich applications • Dualitybased decomposition
− Largescale resource allocation
− Lagrangian relaxation, discrete optimization
− Stochastic programming
• Conic programming
− Robust optimization
− Semideﬁnite programming • Machine learning
− Support vector machines
− l1 regularization/Robust regression/Compressed
sensing
15 METHODOLOGICAL TRENDS
• New methods, renewed interest in old methods.
− Interior point methods
− Subgradient/incremental methods
− Polyhedral approximation/cutting plane methods
− Regularization/proximal methods
− Incremental methods
• Renewed emphasis on complexity analysis
− Nesterov, Nemirovski, and others ...
− “Optimal algorithms” (e.g., extrapolated gradient methods)
• Emphasis on interesting (often dualityrelated)
largescale special structures 16 COURSE OUTLINE
• We will follow closely the textbook
− Bertsekas, “Convex Optimization Theory,”
Athena Scientiﬁc, 2009, including the online
Chapter 6 and supplementary material at
http://www.athenasc.com/convexduality.html
• Additional book references:
− Rockafellar, “Convex Analysis,” 1970.
− Boyd and Vanderbergue, “Convex Optimization,” Cambridge U. Press, 2004. (Online at http://www.stanford.edu/~boyd/cvxbook/) − Bertsekas, Nedic, and Ozdaglar, “Convex Analysis and Optimization,” Ath. Scientiﬁc, 2003.
• Topics (the text’s design is modular, and the
following sequence involves no loss of continuity):
− Basic Convexity Concepts: Sect. 1.11.4.
− Convexity and Optimization: Ch. 3.
− Hyperplanes & Conjugacy: Sect. 1.5, 1.6.
− Polyhedral Convexity: Ch. 2.
− Geometric Duality Framework: Ch. 4.
− Duality Theory: Sect. 5.15.3.
− Subgradients: Sect. 5.4.
− Algorithms: Ch. 6.
17 WHAT TO EXPECT FROM THIS COURSE
• Requirements: Homework (25%), midterm (25%),
and a term paper (50%)
• We aim:
− To develop insight and deep understanding
of a fundamental optimization topic
− To treat with mathematical rigor an important branch of methodological research, and
to provide an account of the state of the art
in the ﬁeld
− To get an understanding of the merits, limitations, and characteristics of the rich set of
available algorithms
• Mathematical level:
− Prerequisites are linear algebra (preferably
abstract) and real analysis (a course in each)
− Proofs will matter ... but the rich geometry
of the subject helps guide the mathematics
• Applications:
− They are many and pervasive ... but don’t
expect much in this course. The book by
Boyd and Vandenberghe describes a lot of
practical convex optimization models
− You can do your term paper on an application area
18 A NOTE ON THESE SLIDES
• These slides are a teaching aid, not a text • Don’t expect a rigorous mathematical development
• The statements of theorems are fairly precise,
but the proofs are not
• Many proofs have been omitted or greatly abbreviated
• Figures are meant to convey and enhance understanding of ideas, not to express them precisely
• The omitted proofs and a fuller discussion can
be found in the “Convex Optimization Theory”
textbook and its supplementary material 19...
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This document was uploaded on 03/19/2014 for the course EECS 6.253 at MIT.
 Spring '12
 DimitrisBertsekas
 Optimization

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