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The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions. Without the constraints, the solution would be ...
Set constraints obtained from abstract interpretation of the Collatz algorithm. In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of equations between numbers, methods are studied for
The only translation-invariant measure on = with domain ℘ that is finite on every compact subset of is the trivial set function ℘ [,] that is identically equal to (that is, it sends every to ) [6] However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover ...
For very simple problems, say a function of two variables subject to a single equality constraint, it is most practical to apply the method of substitution. [4] The idea is to substitute the constraint into the objective function to create a composite function that incorporates the effect of the
Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem. Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
f : ℝ n → ℝ is the objective function to be minimized over the n-variable vector x, g i (x) ≤ 0 are called inequality constraints; h j (x) = 0 are called equality constraints, and; m ≥ 0 and p ≥ 0. If m = p = 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem.
The equality constraint functions :, =, …,, are affine transformations, that is, of the form: () =, where is a vector and is a scalar. The feasible set C {\displaystyle C} of the optimization problem consists of all points x ∈ D {\displaystyle \mathbf {x} \in {\mathcal {D}}} satisfying the inequality and the equality constraints.