Lecture Notes for the Course
CONIC AND ROBUST
OPTIMIZATION
Aharon Ben-Tal
[email protected], http:/iew3.technion.ac.il/Home/Users/morbt.phtml
Minerva Optimization Center
Technion Israel Institute of Technology
University of Rome La Sapienza
July 2
Technion Israel Institute of Technology
Faculty of Industrial Engineering and Management
Technion-City, Haifa 32000, Israel
32000 ,-
CONVEX OPTIMIZATION IN ENGINEERING:
Modeling, Analysis, Algorithms
Aharon Ben-Tal and Arkadi Nemirovski
Copyright 1
ISE 6514 Adv. Topics in Math Programming Exercise #11
Abstract
In the last lectures, we defined chance constrained problems and studied some structural
properties of its feasible set X. In particular, we gave general sufficient conditions under which
X is
A course on Robust
Optimization
Aharon Ben-Tal
Minerva Optimization center
Technion Israel Institute Of Technology
Lecture 1
Conic Optimization
01-15
From Linear to Conic Programming
When passing from a generic LP problem
T
min
c x : Ax b 0
x
[A : m n]
(
ISE 6514 Adv. Topics in Math Programming Exercise #9
1
Two-stage linear stochastic programming
Consider the following Linear two-stage stochastic programming problem:
mincfw_cT x + E(Q(x, ) : Ax = b, x 0
(2SLSP),
where Q(x, ) is the optimal value of the p
Topics in Management Mathematics/
Conic Optimization
20102011
Example class 1 model solutions
Given a cone K along with its dual K , we define a complementary pair every
pair x K, y K such that y T x = 0.
Recall some basic fact on trace of a square matrix
ISE 6514 Adv. Topics in Math Programming Exercise #8
1. (Does Two-stage stochastic programming exists?) Consider the following Linear two-stage
stochastic programming problem:
Z := mincfw_cT x + E(Q(x, ) : Ax = b, x 0
(),
where Q(x, ) is the optimal value
ISE 6514 Adv. Topics in Math Programming Exercise #7
1. (LP application example)
(a) Find in the literature (paper/report/book/magazine) a recent (no more than 15 years
ago) application of linear programming. Give a precise (and accesible) reference.
(b)
ISE 6514 Adv. Topics in Math Programming Exercise #10
1. Read Section 3.1.3 Scenario Trees and Section 3.1.4 Algebraic Formulation of Nonanticipativity Constraints in the Lectures on Stochastic Programming textbook (http:/www2.
isye.gatech.edu/people/facu
ISE 6514 Adv. Topics in Math Programming Exercise #5
1. (Conic quadratic representability in terms of projection) Recall the definition of conic
quadratic representability :
Definition 1. A set X Rn is conic quadratic representable if
o
n
X = x Rn : u Rp
ISE 6514 Adv. Topics in Math Programming Exercise #3
1. (Second Order Conic Programming (SOCP) in R3 )
q
(a) Show that the Lorentz cone L3 = cfw_x R3 : x12 + x22 x3 is a regular cone (i.e. is a
pointed closed convex cone with nonempty interior).
(b) Show
ISE 6514 Adv. Topics in Math Programming Exercise #6
1. Read Section 3.7 Extremal ellipsoids in the Lectures on Modern Convex Optimization textbook (http:/www2.isye.gatech.edu/nemirovs/Lect_ModConvOpt.pdf) from page 208
to page 211.
(a) Write the optimiza
ISE 6514 Adv. Topics in Math Programming Exercise #4
1. The min-max Steiner problem (Page 74 in the textbook). Consider the problem as follows: Given N points b1 , . . . , bN Rn , find a point x Rn which minimizes the maximum
(Euclidean) distance from its
ISE 6514 Adv. Topics in Math Programming Exercise #12
1. Give a numerical/applied example of a problem of the form:
min f := E(F(x, )
xX
(T)
and write the corresponding sample average approximation problem:
N
1X
min fN :=
F(x, j )
xX
N
(SAAN ).
j=1
If it
Chapter 1
Stochastic Linear and Nonlinear Programming
1.1 Optimal land usage under stochastic uncertainties
1.1.1 Extensive form of the stochastic decision program
We consider a farmer who has a total of 500 acres of land available
for growing wheat, corn
ISE 6514 Adv. Topics in Math Programming Exercise #2
1. Let K Rm be a regular cone (that is, K is a pointed closed convex cone with nonempty
interior) and let A be a m n matrix and b Rm . Show that the feasible region of a the
general conic programming pr