Lecture 18 Notes: Duality
1
Overview
The Practical Importance of Duality
Denition of the Dual Problem
Steps in the Construction of the Dual Problem
Examples of Dual Constructions
The Column Geome
Lecture 19 Notes: Necessary Optimality Conditions
1
Introduction
Recall that a constrained optimization problem is a problem of the form
(P) minx f (x)
s.t.
g(x) 0
h(x) = 0
x X,
where X is an open set
Lecture 6 Notes: Constrained Optimization Part 1
1
Introduction
Recall that a constrained optimization problem is a problem of the form
(P) minx f (x)
s.t.
g(x) 0
h(x) = 0
x X,
where X is an open set
Lecture 1 Notes: Unconstrained Optimization
1
Preliminaries
1.1
Types of optimization problems
Unconstrained Optimization Problem:
f (x)
(P) minx
s.t.
x X,
where x = (x1 , . . . , xn ) n , f (x) : n ,
Lecture 9 Notes: Projection Methods
1
Review of Steepest Descent
Suppose we want to solve
P : minimize f (x)
s.t.
x n ,
where f (x) is dierentiable. At the point x = x, f (x) can be
approximated by it
Lecture 3 Notes: Newmans Methods
1
Newtons Method
Suppose we want to solve:
(P:)
min f (x)
x n .
At x = x, f (x) can be approximated by:
f (x) h(x) := f (x) +
1
f (x)T (x ) + (x )t H()(x x),
which is
Lecture 4 Notes: Quadratic Forms
1
Quadratic Optimization
A quadratic optimization problem is an optimization problem of the form:
(QP) : minimize f (x) := 2 xT Qx + cT x
s.t.
x n.
Problems of the for
Lecture 5 Notes: Steepest Descent Method
1
The Algorithm
The problem we are interested in solving is:
P : minimize f (x)
s.t.
x n,
where f (x) is dierentiable. If x = x is a given point, f (x) can be
Lecture 10 Notes: Projections
1
Introduction
Consider the constrained optimization problem P :
P : minimize f (x)
x
s.t.
gi (x) 0, i = 1, . . . , m
hi (x) = 0, i = 1, . . . , k
x n ,
whose feasible re
Lecture 14 Notes: Interior-Point Method
1
The Problem
The logarithmic barrier approach to solving a linear program dates back to
the work of Fiacco and McCormick in 1967 in their book Sequential Uncon
Lecture 12 Notes: Barrier Method
1
The Conditional-Gradient Method for Constrained
Optimization (Frank-Wolfe Method)
We now consider the following optimization problem:
P : minimizex
s.t.
f (x)
xC .
W
Exploratory Data Analysis
Lesson 5 Notes
Explore Many Variables
Multivariate Data
In the last lesson, you learned how to examine the relationship between two
quantitative variables. In this lesson, yo