.
LECTURES
ON
MODERN CONVEX OPTIMIZATION
Aharon Ben-Tal and Arkadi Nemirovski
The William Davidson Faculty of Industrial Engineering & Management,
Technion Israel Institute of Technology, [email protected]
IE 8534
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Lecture 4. The Dual Cone and Dual Problem
Shuzhong Zhang
IE 8534
2
For a convex cone K, its dual cone is dened as
K = cfw_y | x, y 0, x K.
The inner-product can be replaced by xT y if the co
IE 8534
1
Lecture 3. Conic Optimization: What and Why?
Shuzhong Zhang
IE 8534
2
This course is about convex conic optimization models and the
solution methods for such models.
A few key words are wort
Convex Optimization Boyd & Vandenberghe
2. Convex sets
ane and convex sets
some important examples
operations that preserve convexity
generalized inequalities
separating and supporting hyperplane
IE 8534
1
IE 8534: Conic Optimization and Applications
Professor Shuzhong Zhang
Department of Industrial and Systems Engineering
College of Science and Engineering
University of Minnesota
Shuzhong Zha
IE 8534
1
Lecture 2. The KKT Condition and Exchange Economy
Shuzhong Zhang
IE 8534
2
The KKT optimality condition has wide applications. In this lecture,
we will just illustrate one application in eco
IE 8534
1
Preliminaries: Basic Facts
Sets and set operations S T , S T and S \ T .
Vector space X :
- If x X and R then x X .
- If x, y X then x + y X .
Set of all real numbers
R and the Euclidean spa
IE 8534
1
Supplement: Homans Error Bounds
Shuzhong Zhang
IE 8534
2
In Lecture 1 we learned that linear program
(P ) min
s.t.
cT x
Ax = b
x 0,
and its dual problem
max
bT y
s.t.
(D)
AT y + s = c
s0
und
IE 8534
1
Supplement: Error Bounds for Conic Optimization
Shuzhong Zhang
IE 8534
2
Let us now discuss the projection onto a convex cone. First note the
following important result.
Theorem 1 (Moreau) S
IE 8534
1
Lecture 5. Theorems of Alternatives and Self-Dual Embedding
Shuzhong Zhang
IE 8534
2
A system of linear equations may not have a solution.
It is well known that either
Ax = c
has a solution,
IE 8534
1
Lecture 6. Some Graph Problems and Their SDP Relaxation
Shuzhong Zhang
IE 8534
2
For a given graph G = (V, E ; W ), where the weight on the edge (i, j ) is
wij , the max-cut problem can be s
IE 8534
1
Supplement: Universal Self-Concordant Barrier Functions
Shuzhong Zhang
IE 8534
2
Recall that a self-concordant barrier function for K is a barrier
function satisfying
|3 F (x)[h, h, h]|
2(2
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Lecture 12. The Self-Concordant Barrier and Path Following
Shuzhong Zhang
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2
A natural method for solving constrained optimization, which has
been around for more than a half century,
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1
Supplement: Duality and the Minimax Theorem
Shuzhong Zhang
IE 8534
2
Duality plays an important role in convex analysis and optimization.
We have seen dual objects such as polar sets, dual c
IE 8534
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Supplement: The Lwner-John Ellipsoids
o
Shuzhong Zhang
IE 8534
2
An important result in convex analysis is the so-called Lwner-John
o
theorem, which says the following. For any compact conve
IE 8534
1
Lecture 10. Primal-Dual Interior Point Method for LP
Shuzhong Zhang
IE 8534
2
Consider a linear program
cT x
(P ) minimize
subject to
Ax = b
x0
and its dual
maximize
bT y
subject to
(D )
AT
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1
Lecture 8. SDP and the Matrix Decomposition
Shuzhong Zhang
IE 8534
2
Let A be any n n matrix, the eld of values of A is given by
F (A) := cfw_z H Az | z H z = 1 C.
This set, like the spectra
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Lecture 7. SDP and Non-Convex Quadratic Optimization
Shuzhong Zhang
IE 8534
2
The idea of applying SDP relaxation can be used to solve broader
classes of non-convex quadratic models. In this
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Lecture 7: Nesterovs First Order Method
Shuzhong Zhang
IE 8534
2
Optimization is always a matter of weighing over dierent concerns.
There is no optimal method in optimization!
Even for conve
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Lecture 6: Optimality Conditions for Nonlinear Programming
Shuzhong Zhang
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2
In this lecture we will discuss a very important topic: how to
characterize and solve a nonlinear program
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Lecture 5: Constrained Optimization
Shuzhong Zhang
IE 8534
2
Let us consider a general form of a constrained problem:
(P )
minimize
f (x)
subject to x X.
Certainly the most popular type of c
IE 8534
1
Lecture 6. Some Graph Problems and Their SDP Relaxation
Shuzhong Zhang
IE 8534
2
For a given graph G = (V, E ; W ), where the weight on the edge (i, j ) is
wij , the max-cut problem can be s
IE 8534
1
Lecture 5. Theorems of Alternatives and Self-Dual Embedding
Shuzhong Zhang
IE 8534
2
A system of linear equations may not have a solution.
It is well known that either
Ax = c
has a solution,
IE 8534
1
Lecture 4. The Dual Cone and Dual Problem
Shuzhong Zhang
IE 8534
2
For a convex cone K, its dual cone is dened as
K = cfw_y | x, y 0, x K.
The inner-product can be replaced by xT y if the co
IE 8534
1
Lecture 3. Conic Optimization: What and Why?
Shuzhong Zhang
IE 8534
2
This course is about convex conic optimization models and the
solution methods for such models.
A few key words are wort
IE 8534
1
Lecture 2. The KKT Condition and Exchange Economy
Shuzhong Zhang
IE 8534
2
The KKT optimality condition has wide applications. In this lecture,
we will just illustrate one application to the
IE 8534
1
IE 8534: Conic Optimization and Applications
Professor Shuzhong Zhang
Department of Industrial and Systems Engineering
College of Science and Engineering
University of Minnesota
Shuzhong Zha
Convex Optimization Boyd & Vandenberghe
2. Convex sets
ane and convex sets
some important examples
operations that preserve convexity
generalized inequalities
separating and supporting hyperplane
IE 8534
1
Review: Some Basic Facts
Sets and set operations S T , S T and S \ T .
Vector space X :
- If x X and R then x X .
- If x, y X then x + y X .
Set of all real numbers
R and the Euclidean space