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

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