Optimization Theory - (1) Introduction 1 Models and Examples

1. Formation and development

• In the discussion of architectural aesthetics in 500 BC, ancient Greece had discovered that the optimal ratio of the length and width of a rectangle was 1.618, which became the environmental division ratio:
  satisfy all: major part = major part: minor part
• Before the advent of calculus, many scholars began to study optimization problems with mathematical methods.
  For example, Archimedes proved that: given the circumference, the area enclosed by a circle is the largest. This is why almost all ancient castles in Europe were built in circles.
• The classical optimization method
  was proposed by Newton and Leibniz in the calculus they created in the 17th century.extreme value of functionquestion.
• Modern optimization methods
  Before and after the Second World War, modern optimization methods were formed: represented by the Soviet Union Л.В. Kontorovich and the American GB Danzik.linear programming;Represented by American Kuhn and Tuckernonlinear programming; Represented by R. Bellman of the United Statesdynamic programming; Represented by Л.С. Pontryakin of the Soviet UnionMaximum principlewait.

2. An example of the classic extreme value problem

A solid metal ball with a radius of 1 is melted and cast into a solid cylinder. What size should the cylinder take to minimize its surface area?

3. Model and classification of optimization problems

1. The general form of the real vector-valued function optimization problem with vectors as variables

2. According to the different requirements of the actual problem, the optimization model has different forms, but it can be converted into the above general form after appropriate transformation.

3. Classification of constrained optimization problems

4. Unconstrained optimization problems

5. Unconstrained optimization problems are the basis of optimization

• Many practical optimization problems are themselves unconstrained optimization problems.
• Many constrained optimization methods convert the constrained optimization problem into an unconstrained optimization problem through transformation, and then use appropriate unconstrained optimization methods to solve it.

6. Optimized classification

• According to the characteristics of the included equations
  • Linear programming: an optimization problem in which both the objective function and the constraints are linear functions , that is, they are both linear functions;
  • Nonlinear programming: Optimization problems with one or more nonlinear functions in the objective function and constraints ;
• According to the number of objective functions
  • Single-objective optimization problem: an optimization problem with only one objective function ;
  • Multi-objective optimization problem: an optimization problem containing multiple objective functions ;
• According to the value of the decision variable
  • If the feasible set where the decision variable is located is continuous , such as a plane, an interval, etc., it is called continuous optimization ;
  • If the decision variables take values ​​on a discrete set , then the corresponding optimization problem is called discrete optimization . The most common discrete optimization problem is integer programming , whose decision variables take values ​​on the integer set. Solving discrete optimization problems is more difficult than solving continuous optimization problems . This book only introduces the theory and methods of continuous optimization.;

4. Examples of optimization problems

1. Transportation issues

2. Facility issues

3. Assignment issues

5. General steps for solving problems using optimization methods

• Raise questions that require optimization and start collecting relevant information and data;
• Establish relevant mathematical models for solving optimization problems, determine variables, and list objective functions and relevant constraints;
• Analyze the model and select appropriate optimization methods;
• Solve the equation. Generally, the optimal solution is obtained on an electronic computer by programming;
• Verification and implementation of optimal solutions.