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Methods of optimization

• Linear algebra
• Constrained optimization
• Effective implementation

Methods of optimization are tool for finding minima of multivariable functions. These functions are called objective functions or goal functions. The objective functions are assumed to be continuous and differentiable or almost everywhere differentiable. They are often minimized with respect to equality or inequality constraints. Optimization problems arise in many applications from diverse areas such as allocation of resources in logistics or design of technical systems.

Because of the difficulty of practical optimization problems, it is necessary to use any numerical method for their solution. The choice of proper optimization algorithm should be done according to the nature of the objective function and constraints.

Linear algebra

We use a language of linear algebra to describe most of practical problems. One of them is the transportation problem. It is an example of a problem described as minimization of linear function with linear constraints. We use so-called linear programming for solving of this problem. The solution of the transportation problem played an important role in planning the invasion of Normandy.

The great number of problems arises in mechanics. One of them is the minimization of potential energy of a system. This leads to the quadratic programming problem, i.e. the problem of minimization of a quadratic function.

Constrained optimization

Constrained optimization problems arise from models that include explicit constraints on the variables. These constraints often describe some non-penetration condition in contact problems.

Effective implementation

We can use plenty of methods for the numerical solution of the optimization problem. It is important to focus on the selection of optimization algorithm and its implementation in the case of really huge optimization problem.

Areas of use

• Contact problems
• Shape optimization
• Material optimization
• Topological optimization