update background

Author: andrewrosemberg

class01/background_materials/optimization_homework.jl

@@ -0,0 +1,225 @@
+### A Pluto.jl notebook ###
+# v0.20.13
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 881eed45-e7f0-4785-bde8-530e378d7050
+begin
+using Pkg; Pkg.activate("..")
+end
+
+# ╔═╡ 9f5675a3-07df-4fb1-b683-4c5fd2a85002
+begin
+	using PlutoUI
+	using Random
+	using LinearAlgebra
+	using HypertextLiteral
+	using PlutoTeachingTools
+	using ShortCodes, MarkdownLiteral
+	using Random
+	using Plots
+	Random.seed!(8803)
+end
+
+# ╔═╡ 9ce52307-bc22-4f66-a4af-a4e4ac382212
+begin
+    using JuMP
+    using HiGHS        # Solver for LPs and MILPs
+    using Ipopt        # Solver for NLPs
+    using Test         # For quick validation checks
+    # const MOI = JuMP.MathOptInterface
+end
+
+# ╔═╡ 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
+difficulty:
+
+1. **Linear program (LP)**
+2. **Mixed‑integer linear program (MILP)**
+3. **Non‑linear program (NLP)** – a taste of what shows up constantly in **optimal‑control** and simulaing **non‑linear systems**.
+
+For every task you will:
+
+* Write down the mathematical formulation.
+* Translate it into a JuMP model.
+* Solve it.
+* Run the provided `@testset` to make sure your implementation is correct.  
+  When the tests are green ✅ you can be confident that your model is producing the expected answer.
+"""
+
+# ╔═╡ 6f67ca7c-1391-4cb9-b692-cd818e037587
+md"""
+---
+
+## 1. Linear program – Production planning
+
+A workshop makes **widgets** \(w\) and **gadgets** \(g\).
+
+|            | Machine‑hours | Labour‑hours | Profit (\$) |
+|------------|---------------|--------------|--------------|
+| Widget (\(w\)) | 2             | 3            | 3            |
+| Gadget (\(g\)) | 4             | 2            | 5            |
+
+Resources available this week: **100 machine‑hours** and **90 labour‑hours**.
+
+### 1.1  Your tasks
+1. Write the mathematical model *(maximization)*.
+2. Fill in the JuMP code in the next cell.
+3. Run the tests.
+"""
+
+# ╔═╡ 49042d6c-cf78-46d3-bfee-a8fd7ddf3aa0
+begin
+# === Your LP model goes below ===
+# Replace the contents of this cell with your own model.
+model_lp = Model(HiGHS.Optimizer)
+
+# Suggested variable names
+# @variable(model_lp, w >= 0)
+# @variable(model_lp, g >= 0)
+
+# --- YOUR CODE HERE ---
+
+# optimize!(model_lp)
+end
+
+# ╔═╡ 6fb672d0-5a18-4ccc-b7b3-184839c2401b
+begin
+# === Quick check ===
+@testset "LP check" begin
+    @test termination_status(model_lp) == MOI.OPTIMAL
+    @test isapprox(objective_value(model_lp), 135.0; atol = 1e-3)
+end
+end
+
+# ╔═╡ 808c505d-e10d-42e3-9fb1-9c6f384b2c3c
+md"""
+---
+
+## 2. MILP – 0‑1 Knapsack
+
+You have a backpack that can carry at most **10 kg**.  
+There are three items:
+
+| Item | Value | Weight |
+|------|-------|--------|
+| 1    | 10    | 4      |
+| 2    | 7     | 3      |
+| 3    | 5     | 2      |
+
+### 2.1  Your tasks
+1. Write the mathematical model with **binary** decision variables \(x_i \in \{0,1\}\).
+2. Complete the JuMP model and solve it.
+3. Pass the tests.
+"""
+
+# ╔═╡ 39617561-bbbf-4ef6-91e2-358dfe76581c
+begin
+# === Your MILP model goes below ===
+# Replace the contents of this cell with your own model.
+model_milp = Model(HiGHS.Optimizer)
+
+# Example:
+# @variable(model_milp, x[1:3], Bin)
+
+# --- YOUR CODE HERE ---
+
+# optimize!(model_milp)
+end
+
+# ╔═╡ 01367096-3971-4e79-ace2-83600672fbde
+begin
+# === Quick check ===
+@testset "MILP check" begin
+    @test termination_status(model_milp) == MOI.OPTIMAL
+    @test isapprox(objective_value(model_milp), 22.0; atol = 1e-3)
+end
+end
+
+# ╔═╡ 5e3444d0-8333-4f51-9146-d3d9625fe2e9
+md"""
+---
+
+## 3. Non‑linear program – Rosenbrock valley
+
+Non‑linear models dominate **optimal control** because discretising the differential equations that
+describe a physical system almost always yields a **non‑linear program (NLP)**.
+
+A classic (and benign) test problem is the **Rosenbrock** function
+
+\[
+\min_{x,\,y\in\mathbb R}  \; f(x,y)= (1-x)^{2} + 100\,(y - x^{2})^{2}.
+\]
+
+It has a single global optimum at \((x^{\star},y^{\star}) = (1,1)\) with \(f^{\star}=0\).
+
+### 3.1  Your tasks
+1. Build and solve the model with **Ipopt**.
+2. Inspect the solution and objective.
+3. Check your work below.
+"""
+
+# ╔═╡ 00728de8-3c36-48c7-8520-4c9f408a7c5f
+begin
+# === Your NLP model goes below ===
+# Replace the contents of this cell with your own model.
+model_nlp = Model(Ipopt.Optimizer)
+
+# --- YOUR CODE HERE ---
+
+# optimize!(model_nlp)
+end
+
+# ╔═╡ 254b9a87-17f9-4fea-8b28-0e3873b58fe2
+begin
+# === Quick check ===
+@testset "NLP check" begin
+    @test termination_status(model_nlp) == MOI.LOCALLY_SOLVED || termination_status(model_nlp) == MOI.OPTIMAL
+    @test isapprox(objective_value(model_nlp), 0.0; atol = 1e-6)
+end
+end
+
+# ╔═╡ 147fe732-fe65-4226-af43-956b33a75bff
+md"""
+---
+
+## Why non‑linear models matter in optimal control 🚀
+
+When you discretise a continuous‑time optimal‑control problem (for example with **direct collocation**)
+you obtain an optimisation problem whose variables are the states, controls, and possibly parameters
+at many discrete time points:
+
+```math
+\begin{aligned}
+&\min_{x_{k},u_{k}} && \sum_{k=0}^{N-1} \; \ell(x_{k},u_{k}) \\
+&\text{s.t.} && x_{k+1} = x_{k} + h\,f(x_{k},u_{k}), \qquad k=0,\dots,N-1, \\
+& && g(x_{k},u_{k}) \le 0, \\
+& && x_{0}=x_{\text{init}}, \; x_{N}=x_{\text{goal}}.
+\end{aligned}
+```
+
+Even when \(f\) and \(g\) are **polynomial** the resulting constraints are *non‑linear* in the decision variables.
+Hence your optimisation solver must tackle *general NLPs*.  
+Getting comfortable with modelling and debugging small nonlinear examples like Rosenbrock will pay off
+when you step up to thousands of variables in real control problems!
+"""
+
+# ╔═╡ Cell order:
+# ╟─881eed45-e7f0-4785-bde8-530e378d7050
+# ╟─9f5675a3-07df-4fb1-b683-4c5fd2a85002
+# ╟─0df8b65a-0527-4545-bf11-00e9912bced0
+# ╠═9ce52307-bc22-4f66-a4af-a4e4ac382212
+# ╟─6f67ca7c-1391-4cb9-b692-cd818e037587
+# ╠═49042d6c-cf78-46d3-bfee-a8fd7ddf3aa0
+# ╠═6fb672d0-5a18-4ccc-b7b3-184839c2401b
+# ╠═808c505d-e10d-42e3-9fb1-9c6f384b2c3c
+# ╠═39617561-bbbf-4ef6-91e2-358dfe76581c
+# ╠═01367096-3971-4e79-ace2-83600672fbde
+# ╠═5e3444d0-8333-4f51-9146-d3d9625fe2e9
+# ╠═00728de8-3c36-48c7-8520-4c9f408a7c5f
+# ╠═254b9a87-17f9-4fea-8b28-0e3873b58fe2
+# ╟─147fe732-fe65-4226-af43-956b33a75bff

class01/class01.md

@@ -26,6 +26,7 @@ The course will use Julia for programming assignments and projects. If you are n
 
 - [Julia for Beginners](https://juliaacademy.com/p/julia-for-beginners)
 - [Parallel Computing and Scientific Machine Learning (SciML): Methods and Applications](https://book.sciml.ai/)
+- [JuMP Julia Tutorial](https://jump.dev/JuMP.jl/stable/tutorials/getting_started/getting_started_with_julia/)
 - [Julia ML Course](https://adrianhill.de/julia-ml-course/)
 
 Julia is a high-level, general-purpose dynamic programming language, designed to be fast and productive, for e.g. data science, artificial intelligence, machine learning, modeling and simulation, most commonly used for numerical analysis and computational science.
@@ -68,3 +69,7 @@ To run a local notebook file that you have not opened before, then you need to e
 ### **Linear Algebra**: 
 We have prepared a basic (Pluto) [Linear Algebra Primer](./background_materials/basics_math.jl) to help you brush up on essential concepts. This primer covers key topics such as matrix operations, eigenvalues, and eigenvectors besides other fundamental calculus concepts. It is recommended to review this primer before the first class.
 
+### **Optimization**:
+We will use JuMP for some optimization tasks. If you are new to JuMP, please review the [JuMP Tutorial](https://jump.dev/JuMP.jl/stable/tutorials/getting_started/getting_started_with_JuMP/) to familiarize yourself with its syntax and capabilities.
+
+Test your knowledge with this (Pluto) [Modeling Exercise](./background_materials/optimization_homework.jl).