OR 6310: Mathematical Programming II. Spring 2011.
Homework Set 1. Due: Tuesday February 15.
1. (Symmetric quadratic programming duality)
a) Consider the unconstrained quadratic minimization problem
(P ) :
1
min f (v ) := dT v + v T Bv,
v
2
where B = B T

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Bangrui Chen
Lecture 24
April 24, 2014
The story of the Pessimist vs. Optimist (Robust optimization)
Well consider linear programming problems where some or all of the data are uncertain.
We want a

OR 6310: Mathematical Programming II. Spring 2014.
Homework Set 2. Due: Tuesday March 11.
1. Let Pi I di be a nonempty polyhedron dened by ni inequalities, i = 1, 2, and let
R
P := P1 P2 := cfw_(x1 ; x2 ) : x1 P1 , x2 P2 .
a) Show that P is a polyhedron i

OR 6310: Mathematical Programming II. Spring 2014.
Homework Set 1. Due: Thursday February 20.
1. (KKT solutions and minimizers)
a) Find all KKT solutions (s which with suitable multipliers satisfy the KKT conditions) for
x
mincfw_x1 x2 + x2 : 1 x1 1, and

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Jake Feldman
Lecture 3
February 1st, 2011
In this lecture we focus on bimatrix games (two-person, non-zero sum) and how they
relate to the LCPs that we have been looking at. As a reminder, a zero-sum

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Daniel Fleischman
Lecture 26
April 28, 2011
Data Classication
Suppose we have a training set x1 , . . . , xn I d with associated labels
R
y1 , . . . , yn cfw_1, 1. We want to nd a rule that assigns l

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Rolf Waeber
1
Lecture 25
April 26, 2011
Robust Optimization (Optimist vs. Pessimist)
We consider uncertainty in the input to the LP, i.e., some or all of the data A, b and c is not
known exactly. The

OR 631: Mathematical Programming II. Spring 2014.
Homework Set 4. Due: Tuesday April 29.
1. Suppose we wish to solve the problem
hi (x) 0, i = 1, . . . , m,
min f (x),
x B(0, R),
and suppose that f is convex and has range at most 1 on B(0, R), that each h

OR 631: Mathematical Programming II. Spring 2014.
Homework Set 3. Due: Tuesday April 15.
1. This question and the next are concerned with central cuts. Suppose we have an
ellipsoid E := E(B, y), and we add two cuts symmetrically placed with respect to the

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Kecheng Xu
Lecture 19
April 8, 2014
Equivalence of separation and optimization
G Rn , convex body: B(0, r) G B(0, R) (well-rounded).
(i) Strong separation weak optimization (proved last time).
Appl

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Ibrahim Issa
Lecture 18
March 27, 2014
Another Interpretation/Implementation of the Ellipsoid
Method
This section is based on the paper The ellipsoid method generates dual variables (found on
the c

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Jian Wu
Lecture 23
April 23, 2014
Last topic: Interpretable duals.
Regression: want to t a vector b Rm using as explanatory variables the columns of a
matrix A Rmn . Want x Rn with b Ax small.
Deni

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Nanjing Jian
Lecture 22
April 17, 2014
Let x0 Rn , and f FLR := cfw_f : Rn R : f convex, C 1,1 , with Lipschitz constant L,
and with a minimizer x X = cfw_x : f (x) = min f (Rn ) and x0 x R.
Recall

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: Jiayi Guo
Lecture 21
April 15, 2014
Recall the algorithm for minimizing a convex function f over B(0, R), where we assume f
has range at most one on this ball. Let x minimize f over the ball.
Start

Mathematical Programming II
ORIE 6310 Spring 2014
Scribe: James Dong
1
Lecture 20
April 10, 2014
Subgradient Method
Let f : Rn R be a convex function. Recall that the subdierential of f is the set of
subgradients:
f (x) := g Rn : f (y) f (x) + g T (y x) ,

OR 631: Mathematical Programming II. Spring 2014.
Homework 3 solutions.
1. This question and the next are concerned with central cuts. Suppose we have an
ellipsoid E := E(B, y), and we add two cuts symmetrically placed with respect to the center
y. Consid

OR 631: Mathematical Programming II. Spring 2014.
Homework Set 4. Due: Tuesday April 29.
1. Suppose we wish to solve the problem
hi (x) 0, i = 1, . . . , m,
min f (x),
x B(0, R),
and suppose that f is convex and has range at most 1 on B(0, R), that each h

OR 6310: Mathematical Programming II. Spring 2014.
Homework Set 1 Solutions. Due: Thursday February 20.
1. (KKT solutions and minimizers)
a) Find all KKT solutions (s which with suitable multipliers satisfy the KKT conditions) for
x
mincfw_x1 x2 + x2 : 1

OR 6310: Mathematical Programming II. Spring 2014.
Homework 2 Solutions.
1. Let Pi I di be a nonempty polyhedron dened by ni inequalities, i = 1, 2, and let
R
P := P1 P2 := cfw_(x1 ; x2 ) : x1 P1 , x2 P2 .
a) Show that P is a polyhedron in I d dened by n

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Kenneth Chong
Lecture 24
April 21, 2011
Dual Interpretability
By the above, we refer to a class of problems that can be formulated as conic optimization
problems whose duals have a nice interpretatio

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Nick James
Lecture 22
April 14, 2011
Complexity for Large n, and Moderate
When we say large n we mean that n 104 , and moderate means 102 . Since n is large
we would like our algorithm to have no dep

Mathematical Programming II
OR 6310 Spring 2011
Scribe: James Davis
Lecture 9
February 22, 2011
Continuation of Kalais Theorem
Theorem (Kalai). H (d, n) 2nlog2 d+1 .
We continue the proof of this theorem with a series of small claims.
3. The bottom of P c

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Shanshan Zhang
Lecture 7
February 15, 2011
The examples from last time, like those for several variants of the simplex method, required
an exponential number of pivots. In the next few lectures, we a

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Tia Sondjaja
Lecture 6
February 10, 2011
Recall Lemkes theorem from last time:
Theorem 1 If Lemkes algorithm is applied to an LCP(M, q ) with M monotone, it processes
the LCP. That is, either it nds

Mathematical Programming II
OR 6310 Spring 2011
Scribe: Chaoxu Tong
Lecture 5
February 8, 2011
Now we consider the third approach to nd a complementary solution to LCPs:
w = dz0 + M z + q,
w 0, z0 0, z 0,
wi zi = 0 (i = 1, . . . , n).
(1 )
(2 )
(3)
We cho