terça-feira, 18 de julho de 2017

Mathematical optimization




Minimum and maximum value of a function[edit]

Consider the following notation:
This denotes the minimum value of the objective function , when choosing x from the set of real numbers . The minimum value in this case is , occurring at .
Similarly, the notation
asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

Optimal input arguments[edit]

Consider the following notation:
or equivalently
This represents the value (or values) of the argument x in the interval  that minimizes (or minimize) the objective function x2 + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answer is x = –1, since x = 0 is infeasible, i.e. does not belong to the feasible set.
or equivalently
represents the  pair (or pairs) that maximizes (or maximize) the value of the objective function , with the added constraint that x lie in the interval  (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form (5, 2kπ) and (−5,(2k+1)π), where k ranges over all integers.
arg min and arg max are sometimes also written argmin and argmax, and stand for argument of the minimum and argument of the maximum.

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