AskSia

Plus

1. Let

a^{i}, i=1, \ldots, m

be given vectors in

\mathbb{R}^{n}

, let $b_{i}...

Sep 8, 2024

Solution by Steps

step 1

We start with the function defined as

f(z) = \max \{0, |z| - d\}

. This function is piecewise defined, where

f(z) = |z| - d

if |z| > d and

f(z) = 0

|z| \leq d

step 2

The nonlinear program we need to analyze is given by

\min_{x \in \mathbb{R}^{n}} \sum_{i=1}^{m} f(b_{i} - a^{i^{T}} x)

. We will substitute the piecewise definition of

f(z)

into this expression

step 3

For each

i

, we can express the objective function as:

\sum_{i=1}^{m} f(b_{i} - a^{i^{T}} x) = \sum_{i=1}^{m} \max \{0, |b_{i} - a^{i^{T}} x| - d\}

This indicates that the objective function is a sum of maximum functions, which can be transformed into linear constraints

step 4

To convert this into a linear programming problem, we introduce auxiliary variables

t_i

such that:

t_i \geq b_{i} - a^{i^{T}} x - d \quad \text{and} \quad t_i \geq - (b_{i} - a^{i^{T}} x) - d

This allows us to express the nonlinear constraints in a linear form. The new linear program can be formulated as:

\min_{x, t} \sum_{i=1}^{m} t_i \quad \text{subject to the linear constraints above.}

Answer

The nonlinear program is equivalent to a linear programming problem by introducing auxiliary variables and transforming the objective function and constraints into linear forms.

Key Concept

Nonlinear programs can sometimes be transformed into linear programs by introducing auxiliary variables and reformulating the objective and constraints.

Explanation

The transformation allows us to handle the maximum function in a linear programming framework, making it easier to solve using linear programming techniques.