最優化基礎(1)：單目標優化問題求解基礎(Formulation of single objective optimization problem)

CH1:Formulation of single objective optimization problem

1. Motivations

The aim of the first class is to understand the definition,familiar with the formula and know how to generate the expression.

1.1 Optimization

Optimization is the way to find using an algorithmic approach the “best possible” solutions from a given set of feasible(applicable or acceptable) solutions.

1.2 Design variables

Design variables (they are also called decision variables or optimizations variables) are the variables on which the designer can act to improve the product or the system while improving the criterion. These variables are represented as a vector$x$ of dimension n.
$x \in R^n,x = [x_1,x_2,x_3,x_4...x_i...x_n]^T$

1.3 Fixed parameters

Fixed parameters are given by a previous decision or are input from another disciplinary of a previous stage, or non modifiable data for a given problem. These fixed parameters are represented as a vector $p$ of dimension n.
$p \in R^n,x = [p_1,p_2,p_3,p_4...p_i...p_n]^T$

1.4 Constraints

Constraints are some conditions to be respected by the solution. The boundaries of the decision variables also to be considered as constraints.
These constraints are represented as :

1.5 Feasible solution

A point $x \in R^n$ is said to be feasible if it satisfies all the constraints(inequality and equality). And Feasible region(also called design space, admissible space or domain$\Omega$) is the set of all the feasible points.

We can use this figure to review our constraints are active or inactive.

1.6 Generic formulation

2. Local and global minimum

We can always find a local minimum value . And global minimum value is the smallest local minimum in the domain.

3. Classification of optimization problems

4. Example

All the examples are in this website.

Example	Remark
BEAM	solve multi-objective function problem :cost and performance
MANUFACTURING	find a better solution around (187.5,125)
Toys	Data Reformulation
ROBOT RR	objective function can be f(x)=norm(QP)
ANTENNAS	transform multi-objective function problem to a single one
KNAPSACK PROBLEMS	COMBINATORIAL OPTIMIZATION
Resources Allocation	binary variable,perfect formulation in this way

Resources Allocation

If the variable x can take only one of the discrete variables (d1, d2,…, dn) (integer or not) then it is possible to introduce n supplementary binary variables $y_i$ and 2 equality constraints

5. Summary

This chapter is an introduction about what is the optimization and how we do the optimization. It shows us the tool we could use for optimization. Optimization includes optimization problem, design variables, fixed parameters, objective function and constraints.Several examples are mentioned to make sure we know how to formulate the problem.