update

Author: andrewrosemberg

class01/background_materials/optimization_basics.jl

@@ -30,14 +30,29 @@ begin
     using Ipopt        # Solver for NLPs
 end
 
-# ╔═╡ 0df8b65a-0527-4545-bf11-00e9912bced0
+# ╔═╡ b879c4bf-8292-45a0-87d0-b1f0fa456585
 md"""
-
 | | | |
 |-----------:|:--|:------------------|
 |  Lecturer   | : | Rosemberg, Andrew |
 |  Date   | : | 28 of July, 2025 |
 
+"""
+
+# ╔═╡ eeceb82e-abfb-4502-bcfb-6c9f76a0879d
+md"""
+## Prelude
+
+Research and make sure you understand the following concepts/questions:
+ - What is a convex maximization optimziation problem? and a minimization one? Why are convex problems desirable (compared to non-convex ones)?
+ - What is a linear program?
+ - What are Lagrangian multipliers?
+ - What are Integer problems? and Mixed-Integer problems?
+ - What is duality? Can you construct the dual of at least a linear program?
+"""
+
+# ╔═╡ 0df8b65a-0527-4545-bf11-00e9912bced0
+md"""
 # Background – Modeling Optimization Problems in JuMP 🏗️
 
 This short Pluto notebook walks you through three small optimisation models of increasing
@@ -253,9 +268,60 @@ replace the quadratic constraint by a sequence of cutting planes
 3. **Add Cut**
     * Add cut & repeat.
 
-2
+At any point in the solution process, the current LP solution looks like (for a given set of cuts indexed by $j$):
+```math
+\begin{aligned}
+\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.
+\end{aligned}
+```
+
+Your tasks
+
+1. **Implement** the iterative procedure in JuMP + HiGHS:  
+   - master model with variables `m`, `c`;  
+   - a loop that solves, checks feasibility, and adds cuts when needed.  
+
+2. **Record** and (optionally) plot the objective value after each iteration.  
+
 """
 
+# ╔═╡ 7d855c60-41dd-40af-b00a-60b3e779ad13
+question_box(md"How can you check your answer?")
+
+# ╔═╡ ae263aac-1668-4f18-8104-ac25953a4503
+question_box(md"""
+### How do you solve two-stage problems?		 
+
+How can you use cutting planes to solve a linear Two--Stage problem
+
+```math
+\begin{aligned}
+\min_{x}\; & c^{\top}x 
+            \;+\;
+            \mathbb{E}_{\xi}\!\Bigl[
+              \min_{y(\xi)} \; q^{\top}y(\xi)
+            \Bigr] \\[6pt]
+\text{s.t.}\; & A\,x = b, \quad x \ge 0
+               \qquad\qquad\text{(first-stage decisions)}
+\end{aligned}
+```
+			 
+For each realisation $\xi = \bigl(h(\xi),\,T(\xi)\bigr)$ of the uncertain data,
+the **second-stage (recourse) problem** is
+			 
+```math
+\begin{aligned}
+\min_{y(\xi)}\; & q^{\top}y(\xi) \\[4pt]
+\text{s.t.}\; & T(\xi)\,y(\xi) = h(\xi) - W\,x, \\
+              & y(\xi) \ge 0
+              \qquad\qquad\text{(second-stage decisions)}
+\end{aligned}
+```
+""")
+
 # ╔═╡ 808c505d-e10d-42e3-9fb1-9c6f384b2c3c
 md"""
 ---
@@ -709,6 +775,8 @@ end
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