Nonlinear programming is one of the components of mathematical programming

Nonlinear programming is part of mathematical programming, in which a non-linear function is represented by certain constraints or a goal function. The main task of non-linear programming is to find the optimal value of a given objective function with a certain number of parameters and constraints.

The problems of non-linear programming differ from the problems of a linear content of the optimal result not only within an area that has certain limitations, but also beyond its limits. These types of tasks include those tasks of mathematical programming that can be represented by either equalities or inequalities.

Nonlinear programming is classified depending on the variety of the function F (x), the constraint function and the dimension of the solution vector x. So, the name of the task depends on the number of variables. With a single variable, nonlinear programming can be performed using unconditional one-parameter optimization. With a number of variables greater than one, unconditional multiparametric optimization can be used.

Linearity problems are solved using standard linear programming methods (for example, simplex method). But for a nonlinear general method of solution does not exist, it is chosen in each individual case and it also depends on the function F (x).

Nonlinear programming is common in everyday life quite often. For example, this is a disproportionate increase in costs to the number of goods produced or purchased.

Sometimes, in order to find the optimal solution in problems of nonlinear programming, we try to approximate linear problems. An example is quadratic programming, in which the function F (x) is represented by a polynomial of the second degree with respect to variables, while linearity of constraints is observed. A second example is the use of the method of penalty functions, the application of which, under certain restrictions, reduces the task of finding an extremum to a similar procedure without such limitations, which can be solved much easier.

However, if we analyze in general, then nonlinear programming is a solution to the problems of increased computational difficulty. Very often during their decision, it is necessary to use approximate optimization methods. Another powerful tool that can be proposed to solve this type of problem is numerical methods that allow finding the right solution with a given accuracy.

As already mentioned above, non-linear programming requires an individual special approach, which must take into account its specificity.

There are the following non-linear programming methods:

- Gradient methods based on the property of a functional gradient at a point. In other words, it is a vector of partial derivatives computed at the point taken as the sign of the direction of the greatest increase in the function in the vicinity of this point.

- The Monte Carlo method, in which a parallelepiped of the n-th dimension, including a set of plans, is determined for the subsequent modeling of random N-points with uniform distribution in a given parallelepiped.

- The method of dynamic programming reduces to a multidimensional task of optimizing tasks to a smaller dimension.

- The convex programming method is implemented in the search for the minimum value of a convex function or the maximum value of a set of plans concave on a convex part. In the case where a set of plans is a convex polyhedron, then a simplex method can be applied .