MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2012 Spring 6.253 Midterm Exam
Instructor: Prof. Dimitri Bertsekas
Problem 1. (60 p oints) In the following, X is a nonempty convex subset of n , A is a matrix of appropriate
dimension, b is a vector of appropriate di
6.253: Convex Analysis and Optimization
Midterm
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
State which of the following statements are true and which are false. You dont have to justify your
answers:
1. If X1 , X2 are convex sets that can be
LECTURE 6
LECTURE OUTLINE
Nonemptiness of closed set intersections
Simple version
More complex version
Existence of optimal solutions
Preservation of closure under linear transformation
Hyperplanes
All figures are courtesy of Athena Scientific, and
LECTURE 5
LECTURE OUTLINE
Recession cones and lineality space
Directions of recession of convex functions
Local and global minima
Existence of optimal solutions
Reading: Section 1.4, 3.1, 3.2
All figures are courtesy of Athena Scientific, and are used
LECTURE 4
LECTURE OUTLINE
Relative interior and closure
Algebra of relative interiors and closures
Continuity of convex functions
Closures of functions
Reading: Section 1.3
All figures are courtesy of Athena Scientific, and are used with permission.
1
LECTURE 3
LECTURE OUTLINE
Dierentiable Convex Functions
Convex and Ane Hulls
Caratheodorys Theorem
Reading: Sections 1.1, 1.2
All figures are courtesy of Athena Scientific, and are used with permission.
1
DIFFERENTIABLE CONVEX FUNCTIONS
f (z )
f (x) +
LECTURE 2
LECTURE OUTLINE
Convex sets and functions
Epigraphs
Closed convex functions
Recognizing convex functions
Reading: Section 1.1
All figures are courtesy of Athena Scientific, and are used with permission.
1
SOME MATH CONVENTIONS
All of our wo
LECTURE SLIDES ON
CONVEX ANALYSIS AND OPTIMIZATION
BASED ON 6.253 CLASS LECTURES AT THE
MASS. INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASS
SPRING 2012
BY DIMITRI P. BERTSEKAS
http:/web.mit.edu/dimitrib/www/home.html
Based on the book
Convex Optimization Theory
6.253: Convex Analysis and Optimization
Homework 5
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
Consider the convex programming problem
minimize
f (x)
x
subject to x X,
g (x) 0,
of Section 5.3, and assume that the set X is described by equalit
6.253: Convex Analysis and Optimization
Homework 4
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
Let f : Rn R be the function
n
f (x ) =
1
|xi |p
p
i=1
where 1 < p. Show that the conjugate is
n
f (y ) =
1
|yi |q ,
q
i=1
where q is dened by the
6.253: Convex Analysis and Optimization
Homework 3
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
(a) Show that a nonpolyhedral closed convex cone need not be retractive, by using as an example
the cone C = cfw_(u, v, w) | (u, v ) w, the recessi
6.253: Convex Analysis and Optimization
Homework 2
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
(a) Let C be a nonempty convex cone. Show that cl(C ) and ri(C ) is also a convex cone.
(b) Let C = cone(cfw_x1 , . . . , xm ). Show that
m
ri(C )
6.253: Convex Analysis and Optimization
Homework 1
Prof. Dimitri P. Bertsekas
Spring 2010, M.I.T.
Problem 1
(a) Let C be a nonempty subset of Rn , and let 1 and 2 be positive scalars. Show that if C is
convex, then (1 + 2 )C = 1 C + 2 C . Show by example