Optimization is one of the most important branches in applied mathematics that focuses on finding the best solutions of an alternative set of groups. This could mean maximizing or minimizing objective functions based on certain boundaries. Optimization has extensive applications in different fields, such as economics, engineering, management and computer science. The two most common optimizations are linear programming (linear programming) and Non-linear programming (non-linear programming).

Linear Programming

Linear programming (PL) is an optimizing method in which objective functions and all the boundaries involved are linear functions. This means that variable-variable in the PL problem only appears in the first power and no non-linear interactions between variables. The classic example of linear programming is a problem maximizing profit In terms of production, where companies want to determine how many products have to be made to maximize profit by limit, such as the cost of raw materials and labor.

Linear Programming Formulation

Linear programming problems are generally defined as: maximize (or Minimize) c1x1 + c2x2 + cnxnntext

Here, the objective function of c1x1 + c2x2 + x2 + xn + cnxnc _ 1x _ 1 + c _ 2x + cdots + c _ nc1 + c2 x2 Border Variable

One of the most famous algorithms to solve linear programming problems is Simplex Method It was found by George Dantzig in 1947. This method is efficient and can solve big problems quite quickly. In modern times, Simplex's methods and variations were used in software optimizations such CPLEX and Gurobi.

Linear Programming Application

Linear programming has many applications, including:

• Production planning: Optimize the combination of products produced to maximize profit while considering resource constraints.
• Supply chain management: Minimize distribution costs while fulfilling customer needs.
• Budget location: Funding for various projects for profit is optimal.

Non-Linear Programming

Unlike linear programming, nonlinear programming (PNL) used when objective functions or constraints have nonlinear forms, such as involving squared or interacting between variables. In many real-world cases, intervariable relationships aren't always linear, so nonlinear programming is more realistic and relevant to various applications.

Non- Linear Programming Formulation

Non-linear programming problems are usually defined as: maximized (or Minimize) f (x1, x2, xn, x, xn, x, x2, x, x, x, x, x, x, xx, xx, xx, xx, xx, x@@

In PNL, objective functions (x1, x2,..., xn) f (x _ 1, x _ 2, ldots, x _ n) f (x@@

Non- Linear Programming Methods

No single method can be used to solve all non-linear problems. Some methods are generally used among others:

• Gradient Method: This method seeks the derivative of objective functions and moves towards minimum (or maximum) until solutions are reached.
• Lagrange MethodThis method works when there's a limit to problems. By using Lagrange multiplier, we can turn limited problems into infinite problems.
• Interior Point Method: This is an iterative method often used in big and complex non-linear problems.

Non-Linear Programming Application

Non-linear programming also has extensive applications, including:

• Portolio optimization: Determines optimal asset allocation to maximize return at minimal risk, using squared functions in risk models.
• Energy planning: Optimize limited use of energy resources while minimizing cost and environmental impact.
• Engineering design: Designing the structure or the optimal mechanical system, as in the design of airplanes or cars.

Linear vs. programming. Non- Linear

The main difference between linear and nonlinear programming lies in complexity and the form of objective functions and its limits. Linear programming is easier to solve because it has a simple structure and a more efficient algorithm. Instead, nonlinear programming requires a more sophisticated approach because of more complex function forms.

However, despite the more complex, nonlinear programming is often more accurate in representing real world problems involving nonlinear intervariable relationships.

Conclusion

Optimization, both in linear and nonlinear programming, is an important tool in solving problems in the real world. While linear programming is easier to understand and apply, nonlinear programming allows more complex and realistic models. With technology growing and the method of optimization, we're more and more able to solve problems that used to be difficult to overcome, to make a huge contribution to economics, to engineering, to computer science.

source: Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). Nonlinear Programming: John Wiley & Sons.