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In order to define a cost function to minimize, when points and are in the same cluster we add their positive distance , and subtract it when they are in different clusters. This way, an optimal solution tends to place points which are far apart into different clusters, and points that are close into the same cluster.
One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible.
Intuitively, the cost function encourages facilities with high flows between each other to be placed close together. The problem statement resembles that of the assignment problem , except that the cost function is expressed in terms of quadratic inequalities, hence the name.
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
where f(y) is the value/cost of the solution y for the instance x. Clearly, the performance guarantee is greater than or equal to 1 and equal to 1 if and only if y is an optimal solution. If an algorithm A guarantees to return solutions with a performance guarantee of at most r ( n ), then A is said to be an r ( n )-approximation algorithm and ...
Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function = at any = based on the value and slope of the function at =, given that () is differentiable on [,] (or [,]) and that is close to .
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The function f is variously called an objective function, criterion function, loss function, cost function (minimization), [8] utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution.