Constrained optimization

In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function which is to be minimized, or a reward function or utility function, which is to be maximized. Constraints can be either hard constraints which set conditions for the variables that are required to be satisfied, or soft constraints which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.

A general constrained minimization problem may be written as follows:

where ${\displaystyle g_{i}(\mathbf {x} )=c_{i}~\mathrm {for~} i=1,\ldots ,n}$ and ${\displaystyle h_{j}(\mathbf {x} )\geq d_{j}~\mathrm {for~} j=1,\ldots ,m}$ are constraints that are required to be satisfied; these are called hard constraints.

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