diff --git a/class01/background_materials/optimization_basics.html b/class01/background_materials/optimization_basics.html index 5b11fe6..ab359ed 100644 --- a/class01/background_materials/optimization_basics.html +++ b/class01/background_materials/optimization_basics.html @@ -1,19 +1,17 @@ - - + -
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\ No newline at end of file diff --git a/class01/background_materials/optimization_basics.jl b/class01/background_materials/optimization_basics.jl index 9413ed1..6ad17ce 100644 --- a/class01/background_materials/optimization_basics.jl +++ b/class01/background_materials/optimization_basics.jl @@ -1,5 +1,5 @@ ### A Pluto.jl notebook ### -# v0.20.15 +# v0.20.4 using Markdown using InteractiveUtils @@ -229,7 +229,7 @@ Because of fuselage geometry the two design variables must lie *inside* a circular envelope in the $(m,c)$-plane: ```math -m^{2} + c^{2} \; \le \; 100 . +(m-6)^{2} + (c-3)^{2} \; \le \; 30 . ``` The engineering goal is to minimise the total cost @@ -256,14 +256,14 @@ replace the quadratic constraint by a sequence of cutting planes * Solve for a tentative point $(\bar m,\bar c)$. 2. **Feasibility check** - * If $\bar m^{2}+\bar c^{2} \le 100$ the point is feasible $\Rightarrow$ **done**. + * If $\bar (m-6)^{2} + (\bar c-3)^{2} \le 30$ stop: feasible and optimal! * Otherwise generate a *supporting hyperplane* for the circle at $(\bar m,\bar c)$: ```math -2\,\bar m\,(m - \bar m) \;+\; 2\,\bar c\,(c - \bar c) \;\le\; 100 \;-\; \bar m^{2} \;-\; \bar c^{2}. +2\,(\bar m-6)\,(m - \bar m) + 2\,(\bar c-3)\,(c - \bar c) \le 30 - (\bar m-6)^{2} - (\bar c-3)^{2} . ``` -(Derived from the gradient of $g(m,c)=m^{2}+c^{2}-100$.) +(Derived from the gradient of $g(m,c)=(m-6)^{2} + (c-3)^{2}-30$.) 3. **Add Cut** * Add cut & repeat. @@ -274,7 +274,7 @@ At any point in the solution process, the current LP solution looks like (for a \min_{m,c} \quad & 200\,m + 80\,c \\ \text{s.t.}\quad & 0 \le m \le 12, \\ & 0 \le c \le 12, \\ - & 2\,\bar m_j\,(m - \bar m_j) + 2\,\bar c_j\,(c - \bar c_j) \le 100 - \bar m_j^{2} - \bar c_j^{2}, \quad j=1,\dots,J. + & 2\,\bar (m_j-6)\,(m - \bar m_j) + 2\,(\bar c_j-3)\,(c - \bar c_j) \le 30 - (\bar m_j-6)^{2} - (\bar c_j-3)^{2}, \quad j=1,2,\dots \end{aligned} ```