Lecture 1
Robust Linear Optimization:
Motivation, Concepts, Tractability
In this lecture, we introduce the concept of the uncertain Linear Optimization problem and its
Robust Counterpart, and study the computational issues associated with the emerging opt

Globalized Robust Counterparts
We are about to reconsider two of the basic assumptions on the decision environment we
made, namely, that
A.2. The decision maker is fully responsible for
consequences of the decisions to be made when,
and only when, the ac

Uncertain Conic Quadratic and Semidenite
Optimization
Canonical Conic problem:
A1x + b1 K1
min cT x + d :
x
AI x + bI KI
(CP)
x: decision vector
Ki: simple cone: nonnegative orthant Rm, or Lorentz cone
+
2
2
Lm = cfw_y Rm : ym y1 + . + ym1, or Semiden

Robust Linear Optimization
and
Chance Constraints
T [x; 1] aT x + b 0 , : uncertain (ULC)
T [x; 1] 0 U
(RC)
Question: How to specify an uncertainty set?
Answer: This is a modeling, heavily applicationdependent, issue and as such it is beyond the scope

Linear Optimization Problem, its Data and
Structure
Linear Optimization problem:
min cT x + d : Ax b
x
(LO)
x Rn: vector of decision variables,
c Rn and d R form the objective,
A: an m n constraint matrix,
b Rm: right hand side.
Problems structure: its

Appendix B
Solutions to Exercises
B.1
Exercises from Lecture 1
Exercise 1.1. We should prove that x is robust feasible if and only if it can be extended, by
properly chosen u, v 0 such that u v = x, to a feasible solution to (1.6.1). First, let x be
robus

Linear Optimization Models
An LO program. A Linear Optimization
problem, or program (LO), called also Linear
Programming problem/program, is the problem of optimizing a linear function cT x of an
n-dimensional vector x under nitely many
linear equality a

Lecture 5
Adjustable Robust Multistage
Optimization
In this lecture we intend to investigate robust multi-stage linear and conic optimization.
5.1
Adjustable Robust Optimization: Motivation
Consider a general-type uncertain optimization problem a collecti

Lecture 4
Globalized Robust Counterparts of
Uncertain Linear and Conic
Problems
In this lecture we extend the concept of Robust Counterpart in order to gain certain control on
what happens when the actual data perturbations run out of the postulated pertu

Lecture 3
Robust Conic Quadratic and
Semidenite Optimization
In this lecture, we extend the RO methodology onto non-linear convex optimization problems,
specically, conic ones.
3.1
Uncertain Conic Optimization: Preliminaries
3.1.1
Conic Programs
A conic o

Lecture 2
Robust Linear Optimization and
Chance Constraints
2.1
How to Specify an Uncertainty Set
The question posed in the title of this section goes beyond general-type theoretical considerations
this is mainly a modeling issue that should be resolved

Intermediate Summary
So far,
We have dened the notion of uncertain conic problem a family P of instances
A1[ ]x + b1[ ] K1
.
min cT x :
,Z
x
Am[ ]x + bm[ ] Km
with the data A1, b1, ., Am, bm anely parameterized
by the perturbation running through a giv