Economic Optimization See also
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
min x ϕ = f ( x ) ← Objective function s.t ← Constraints h ( x ) = 0 ← Equality constraints g ( x ) ≤ 0 ← Inequality constraints x l b ≤ x ≤ x u b ← Bounds \begin{align*}
\min_{x} \phi = f(x) & &\leftarrow &\space \text{Objective function} \\\
\text{s.t} & &\leftarrow &\space \text{Constraints} \\\
h(x) = 0 & &\leftarrow &\space \text{Equality constraints} \\\
g(x) \leq 0 & &\leftarrow &\space \text{Inequality constraints} \\\
x_{lb} \leq x \leq x_{ub} & &\leftarrow &\space \text{Bounds}
\end{align*} x min ϕ = f ( x ) s.t h ( x ) = 0 g ( x ) ≤ 0 x l b ≤ x ≤ x u b ← ← ← ← ← Objective function Constraints Equality constraints Inequality constraints Bounds decision variables discrete.
limited to a fixed or countable set of values
x D ∣ a ∈ I = { 1 , 2 , 3 , 4 , 5 } x_{\mathcal{D}} \mid a \in \mathcal{I} = \lbrace 1, 2, 3, 4, 5 \rbrace x D ∣ a ∈ I = { 1 , 2 , 3 , 4 , 5 } continuous.
can take any value within a range
x C ⊂ R x_{\mathcal{C}} \subset \mathcal{R} x C ⊂ R constraints
a decision upper and lower bounds (x U x^{\mathcal{U}} x U x L x^{\mathcal{L}} x L
Active/binding : ∃ x ∗ ∣ g ( x ∗ ) = 0 \exists \space x^{*} \mid g(x^{*}) = 0 ∃ x ∗ ∣ g ( x ∗ ) = 0 Inactive : ∃ x ∗ ∣ g ( x ∗ ) < 0 \exists \space x^{*} \mid g(x^{*}) < 0 ∃ x ∗ ∣ g ( x ∗ ) < 0
graphing models
feasible set of an optimization model
The collection of decision variables that satisfy all constraints
S ≜ { x : g ( x ) ≤ 0 , h ( x ) = 0 , x L ≤ x ≤ x U } \mathcal{S} \triangleq \lbrace x : g(x) \leq 0, h(x) = 0, x^L \leq x \leq x^U \rbrace S ≜ { x : g ( x ) ≤ 0 , h ( x ) = 0 , x L ≤ x ≤ x U }
outcomes
the optimal value ϕ ∗ \phi^{*} ϕ ∗
ϕ ∗ ≜ ϕ ( x ∗ ) \phi^{*} \triangleq \phi(x^{*}) ϕ ∗ ≜ ϕ ( x ∗ )
Constraints satisfy, but it is not binding
Linear optimization problems
min x 1 , x 2 ϕ = 50 x 1 + 37.5 x 2 s.t 0.3 x 1 + 0.4 x 2 ≥ 2000 0.4 x 1 + 0.15 x 2 ≥ 1500 0.2 x 1 + 0.35 x 2 ≤ 1000 , x 1 ≤ 9000 x 2 ≤ 6000 x i ≥ 0 \begin{aligned}
\underset{x_1,x_2}{\min} \space \phi &= 50x_1 + 37.5x_2 \\
&\text{s.t} \\\
0.3x_1 + 0.4x_2 &\geq 2000 \\\
0.4x_1 + 0.15x_2 &\geq 1500 \\\
0.2x_1 + 0.35x_2 &\leq 1000, \\\
x_1 &\leq 9000 \\\
x_2 &\leq 6000 \\\
x_i &\geq 0
\end{aligned} x 1 , x 2 min ϕ 0.3 x 1 + 0.4 x 2 0.4 x 1 + 0.15 x 2 0.2 x 1 + 0.35 x 2 x 1 x 2 x i = 50 x 1 + 37.5 x 2 s.t ≥ 2000 ≥ 1500 ≤ 1000 , ≤ 9000 ≤ 6000 ≥ 0 See also