Constraint Optimization - Definition

Definition

A constraint optimization problem can be defined as a regular constraint satisfaction problem in which constraints are weighted and the goal is to find a solution maximizing the weight of satisfied constraints.

Alternatively, a constraint optimization problem can be defined as a regular constraint satisfaction problem augmented with a number of "local" cost functions. The aim of constraint optimization is to find a solution to the problem whose cost, evaluated as the sum of the cost functions, is maximized or minimized. The regular constraints are called hard constraints, while the cost functions are called soft constraints. These names illustrate that hard constraints are to be necessarily satisfied, while soft constraints only express a preference of some solutions (those having a high or low cost) over other ones (those having lower/higher cost).

A general constrained optimization problem may be written as follows:


\begin{array}{rcll}
\max &~& f(\mathbf{x}) & \\
\mathrm{subject~to} &~& g_i(\mathbf{x}) = c_i &\mathrm{for~} i=1,\cdots,n \quad \rm{Equality~constraints} \\ &~& h_j(\mathbf{x}) \le d_j &\mathrm{for~} j=1,\cdots,m \quad \rm{Inequality~constraints}
\end{array}

Where is a vector residing in a n-dimensional space, is a scalar valued objective function, and are constraint functions that need to be satisfied.

Read more about this topic:  Constraint Optimization

Famous quotes containing the word definition:

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)

    The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.
    Ralph Waldo Emerson (1803–1882)

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)