1.2.4

## Elimination of equality constraints

In many optimization problems especially with linear constraints one could obtain an un- constraint optimization problem by eliminating the equality constraints.

Example 9 Assume we need to solve the following problem

min s.t

x 2 1 + x 2 2

1

x_{1 }

x_{2 }= 0

It is easy to see that the equality constraint can be written as x_{1 }= x_{1 }and therefore the constraint optimization problem can be writtten as

min

2 x 2 1

1

1.2.5

## Change of variables

In many cases we are able to obtain equivalent problems by changing variables. We have to be careful that the map is one to one. That is

min f(x) = min f(ϕ(z)) if the map x = ϕ(z) has an inverse for all admisible x’s

Example 10 Using the exponent to replace strictly positive variables Consider the optimization problem

min (x_{1 }+ 1)^{2 }+ (x_{2 }+ 3)^{2 }

0.01(log(x_{1}) + log(x_{2}))

## By

setting

exp(t_{i}) = x_{i},

i = 1, 2

we

obtain

min

(exp(t_{1}) + 1)^{2 }+ (exp(t_{2}) + 3)^{2 }

0.01(t_{1 }+ t_{2})

# 1.3

# Problems

1. Choose any field of study and find an optimization problem from that field. Explain the objective function and the constraints.

2. Plot the function h(x) = max(0, x), comment on its differentiability. Design a smooth, continuously differentiable function that is similar to the function h(x) = max(0, x).

3. Reformulate the following problem as an unconstrained optimization problem

min

x 2 1 + x 2 2

+ x 2 3

s.t

1 1 1 1 2 3

x x 2 1 x 3

=

3 6

6