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Lectures
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Date
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Subjects Covered
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Lecture Contents/Supplementary Materials
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Lecture 24
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Thurs. Dec. 3
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Review
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projection for multiple sets
review e.g.:
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basic concepts of convex sets, functions
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subdifferentials, normal cones, tangent cones, directional
derivatives
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sandwich and separation theorems
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Fenchel conjugate, Fenchel dual
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Lecture 23
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Tues. Dec. 1
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Convex Feasibility cont...
(see
Techniques of Variational Analysis, available
online at
UofW library)
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Projection as a minimization problem
Attracting mappings and Fejer sequences
Convergence
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Lecture 22
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Thurs. Nov. 26
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KKT Conditions and Convex Feasibility Problems
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KKT:
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(text Sect. 7.2) Karush-Kuhn-Tucker Theorem for equality and
inequality general NLP.
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Mangasarian-Fromovitz CQ (MFCQ)
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Proof of KKT using the weakest CQ (tangent cone equals linearizing
cone)
Convex Feas. Probs:
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Existence/uniqueness of nearest points to a closed convex set; normal
cone characterization
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projection map and properties (monotone operator)
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Lecture 21
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Tues. Nov. 24
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Fenchel Duality of convex programs, KKT
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Fenchel duality of convex nonlinear program (NLP)
KKT conditions for a general nonlinear program
tangent cones, linearizing cones
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Lectures 18-20
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Thurs. Nov. 12-19
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Generalized cone programs
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SDP
Fenchel duality and Lagrangian duality
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Lecture 17
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Tues. Nov. 10
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optimization problems with cone constraints
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applications to general linear programming
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Lecture 16
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Thurs. Nov. 5
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KKT
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Karush-Kuhn-Tucker conditions for the convex case
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Lecture 15
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Tues. Nov. 3
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further characterizations of optimality
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relations: normal cone, subgradient cone, linearizing cone
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Lecture 14
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Thurs. Oct. 29
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Normal Cones characterization of optimality
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Examples
Theorems of the alternative
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Lecture 13
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Tues. Oct. 27
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Set constrained optimization
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sandwich theorem with proof using hyperplane separation
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Lecture 12
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Thurs. Oct. 22
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Subgradients algebra; Fenchel duality for minimization
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locally Lip. continuity; f' closed; subgradient convex, compact
Fenchel duality and sandwich theorems
(For a Lagrange multiplier approach see e.g.
MAA Lester Ford prize article by Pourciau, i.e.
Bruce H. Pourciau, Modern multiplier rules, Amer. Math. Monthly 87 (1980), 433-452)
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Lecture 11
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Tues. Oct. 20
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Fenchel-Legendre conjugate/duality;
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support functions; directional derivatives; subgradients
subgradients/derivatives of eigenvalue functions
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Lecture 10
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Thurs. Oct. 15
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Fenchel-Legendre transform; conjugate/duality;
optimality conditions
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More on support functions
f**=f if and only if f is closed, convex (with proof).
optimality conditions of simple optimization problems
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Lecture 9
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Tues. Oct. 13
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Fenchel-Legendre conjugate/duality;
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Fenchel Conjugate f* Properties: relation to affine functions; f* is
closed and convex; when is f* proper and when is the domain of f* nonempty;
f majorizes g implies g* majorizes f*; Fenchel-Young inequality with
equality relationship to subdifferential.
support, positively homogeneous, sublinear, subadditive functions.
Sf is the set supported by f
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Lecture 8
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Thurs. Oct. 8
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Fenchel-Legendre transform; conjugate/duality;
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Summary of Notation and Basic Results
FYI:
symbolic Fenchel conjugation/convex analysis maple packages
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Lecture 7
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Tues. Oct. 6
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Homework 2, with
Supplementary Problems;
Fenchel-Legendre conjugate/duality;
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f is a closed convex function (iff f is the
sup of all majorized affine functions)
conjugate of a function, f*
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Lecture 6
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Thurs. Oct. 1
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Applications of (Fenchel/conjugate) duality;
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What shape is your conjugate? A survey of computational convex analysis and its applications, MR2496900, by
Lucet, Yves, SIAM J. Optim. 20 (2009), no. 1, 216--250.
basic hyperplane separation
So, polar of a set and its properties
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Lecture 5
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Tues. Sept. 29
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CVX outline, examples; Convex Hulls and Epigraphs
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(CVX outline/examples)
compositions of functions that yield convexity
convex combinations, convex hulls of sets and functions
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Lecture 4
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Thurs. Sept. 24
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Properties of Convex Sets and Convex Functions
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sublevel sets, coercive functions, attainment of minimization problems
quasi-convex functions, indicator function
operations that preserve convexity (for convex sets and functions)
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Lecture 3
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Tues. Sept. 22
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basic concepts of Euclidean spaces;
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Supplementary Problems
Homework 1 definition of domain added; GPs discussed;
difference between
Cartesian products and
Direct sums;
basic concepts of Euclidean spaces;
norms, open sets, closed sets, relative interior, affine manifolds,
affine hull
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Lecture 2
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Thurs. Sept. 17
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Homework 1, with
Supplementary Problems;
Convex Optimization (CO) Problems; Euclidean (vector) spaces
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video lectures at Stanford 1;
video lectures at Stanford 2;
Convex Optimization (CO) problems: LPs, SDPs, GPs
(and info).
basic concepts of Euclidean spaces; constructing Euclidean spaces, e.g.
using e.g.
Cartesian products;
norms, open sets, closed sets
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Lecture 1
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Tues. Sept. 15
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Introduction/Administrivia; texts and supplementary references;
CVX software
(examples); Basic properties (
convex functions and
convex sets)
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"The great watershed in optimization is not
between linearity and nonlinearity, but convexity and nonconvexity."
(
Rockafellar, 1993.)
NEOS Wiki and the
Optimization Tree;
(graphic)
References: on reserve text is
Convex Analysis and Nonlinear Optimization, by Borwein & Lewis.
on reserve supplementary reference is
Convex Optimization Theory, by D.P. Bertsekas.
CVX, Matlab Software for
Disciplined Convex Programming (download/install/example);
using the online book
Convex Optimization, by
Boyd & Vandenberghe.
Lecture covered:
convex functions: affine, exponential,
entropy, quadratic, max, norms; convex sets: polyhedral, ellipsoids,
cones
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