Optimization

See also slides

model-based

• conclusions from the model of the system

Components:

• decision variables
• constraints
• objectives
• functions: mathematical function that determines the objective as a function of decision variable

$$\begin{array}{rlrl}\underset{x}{\min }\varphi =f(x)& & \leftarrow & \text{Objective function}\\ \text{s.t}& & \leftarrow & \text{Constraints}\\ h(x)=0& & \leftarrow & \text{Equality constraints}\\ g(x)\le 0& & \leftarrow & \text{Inequality constraints}\\ x_{lb}\le x\le x_{ub}& & \leftarrow & \text{Bounds}\end{array}$$

decision variables

discrete.

limited to a fixed or countable set of values

$$x_{\mathcal{D}}\mid a\in \mathcal{I}=\{1,2,3,4,5\}$$

continuous.

can take any value within a range

$$x_{\mathcal{C}}\subset \mathcal{R}$$

constraints

• physical limitations: cannot purchase negative raw materials

• model assumptions: assumptions about the system

domain of a definition

a decision upper and lower bounds ($x^{\mathcal{U}}$ and $x^{\mathcal{L}}$)

Properties

• Active/binding: $\exists x^{\ast }\mid g(x^{\ast })=0$

• Inactive: $\exists x^{\ast }\mid g(x^{\ast })<0$

graphing models

feasible set of an optimization model

The collection of decision variables that satisfy all constraints

$$\mathcal{S}\triangleq \{x:g(x)\le 0,h(x)=0,x^L\le x\le x^U\}$$

outcomes

optimal value

the optimal value ${\varphi }^{\ast }$ is the value of the objective at the optimum(s)

$${\varphi }^{\ast }\triangleq \varphi (x^{\ast })$$

Constraints satisfy, but it is not binding

Linear optimization problems

$$\begin{array}{rl}\underset{x_1,x_2}{\min }\varphi & =50x_1+37.5x_2\\ & \text{s.t}\\ 0.3x_1+0.4x_2& \ge 2000\\ 0.4x_1+0.15x_2& \ge 1500\\ 0.2x_1+0.35x_2& \le 1000,\\ x_1& \le 9000\\ x_2& \le 6000\\ x_i& \ge 0\end{array}$$

See also Linear Optimization