Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

2.7. THE PROBLEM OF POSING PROBLEMS 29
Linear programs
A very important situation occurs when both the objective and the constraint functions
are linear and, moreover, the constraints are defined either by equalities or
inequalities, that is, the constraints are given by linear equations and linear inequalities.
As we have remarked before, the solution set is a solid polyhedron
of the euclidean space. Optimization problems of this nature are called linear
programs and their study constitute a sub-area of optimization. In fact, several
practical problems can be posed as linear programs, and there exists a huge number
of techniques that solve them, exploiting its characteristics: the linearity of
the objective function and the piecewise linear structure of the solution set (polyhedra).
2.7 The problem of posing problems
Most of the problems in general, and in particular the problems from computer
graphics, can be posed using the concept of operator between two spaces. 1 These
problems can be classified in two categories: direct and inverse problems.
Direct problem. Given two spaces O 1 and O 2 of graphical objects, an operator
T : O 1 → O 2 , and a graphical object x ∈ O 1 . Problem: compute the object
y = T (x) ∈ O 2 . That is, we are given the operator T and a point x in its domain,
and we must compute the image y = T (x)
Because T is an operator, it is not multi-valued, therefore the direct problem
always has a unique solution.
Inverse problems.
There are two types of inverse problems:
Inverse problem of first kind. Given two spaces O 1 e O 2 of graphic objects,
an operator T : O 1 → O 2 , and an element y ∈ O 2 , determine and object x ∈ O 1
tal que T (x) = y. That is, we are given an operator T and a point y, belonging to
1 In some books the use of the term operator implies that it is linear. Here we use the term to
mean a continuous function.