LP and QP in conic form

The LP

can always be represented in the conic form

for appropriate matrix and vector .

To prove this, we introduce three new sets of non-negative variables: , which represents the constraints, and , which contain the positive and negative parts of vector . We have , with and . The constraint then reads . Let us define the new variable / The relationship between can then be expressed as , with

The objective of the original problem can be written as , with

Putting this together, we express the original LP as claimed