Nonlinear programming

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Nonlinear programming is a method by which an objective function is optimized, either by maximizing or minimizing. This, taking into account different restrictions given. It is characterized because the objective function, or some of the restrictions, can be non-linear.

Nonlinear programming is, then, a process where the function to be maximized, or any of the restrictions, is different from a linear or first degree equation, where the variables are raised to the power 1.

We must remember that a linear equation is a mathematical equality that can have one or more unknowns. Thus, it has the following basic form, where a and b are the constants, while x and y are the variables:

ax + b = y

It should be added that not all the elements that make up this type of programming will comply with this characteristic. For example, it may be that the objective function is an equation of the second degree and one of the variables is squared, fulfilling the following form:

y = ax2 + bx + c

Now, through non-linear programming, this function could be optimized, finding the maximum or minimum value of y. This, taking into account that x is subject to certain restrictions.

Elements of nonlinear programming

The main elements of nonlinear programming are the following:

  • Objective function: It is that function that is optimized, either by maximizing or minimizing its result.
  • Restrictions: These are the conditions that must be met when optimizing the objective function. It can be algebraic equations or inequalities.

Nonlinear programming exercise

Let's see, to finish, a nonlinear programming exercise.

Suppose we have the following function:

y = 25 + 10x-x2

We also have the following restriction:

y = 50-3x

As we can see in the graph, the objective function and the restriction intersect at two points, but where y is maximized is when x = 2.3, where y = 43 (decimals are approximate).

The cut-off points can be found by equating both equations:

25 + 10x-x2 = 50-3x

0 = x2-13x + 25

Then the quadratic equation above has two solutions or roots that can be found with the following formulas, where a = 1, b = -13, and c = 25.

Thus, we find that x1 = 2.3467 (y = 43) and x2 = 10.653 (y = 18).

We must warn that this type of programming is more complex than linear, and there are not so many tools available online to solve this type of optimization. The example shown is a very simplified case.

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