diff --git a/class02/class02_constrained_min.jl b/class02/class02_constrained_min.jl
new file mode 100644
index 0000000..23a99a2
--- /dev/null
+++ b/class02/class02_constrained_min.jl
@@ -0,0 +1,264 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ a509f9d8-9c6d-11f0-3db9-cb5fe2e85d64
+md"""# Constrained Optimization (Equality & Inequality KKT)
+ 
+[⬅ Back to Class 02 Overview](class02_overview.jl) 
+
+[⬅ Previous: Unconstrained Minimization](class02_unconstrained_min.jl) 
+
+[➡ Next: Methods (Penalty/ALM/IPM)](class02_methods_barrier_alm.jl)
+
+**In this section you will:**
+
+* Build the **geometry** of equality constraints and the **KKT** conditions.
+* See the **Newton-on-KKT** linear system (saddle point) and when it’s well-posed.
+* Contrast **full Newton** vs. **Gauss–Newton** on the KKT system.
+* Extend to **inequality constraints** and understand **complementarity**.
+ 
+"""
+
+# ╔═╡ 57cd6d88-ea7e-4f5c-bc17-7fc65fb78e95
+md"""## Equality-constrained minimization: geometry and conditions
+
+**Problem:**
+[
+\min_{x\in\mathbb{R}^n} f(x) \quad \text{s.t.}\quad C(x)=0,\ \ C:\mathbb{R}^n\to\mathbb{R}^m.
+]
+
+**Geometric picture.** At an optimum on the manifold (C(x)=0), the negative gradient must lie in the tangent space:
+[
+\nabla f(x^\star)\ \perp\ \mathcal{T}_{x^\star}={p:\ J_C(x^\star)p=0}.
+]
+Equivalently, the gradient is a linear combination of the constraint normals:
+[
+\nabla f(x^\star)+J_C(x^\star)^{!T}\lambda^\star=0,\qquad C(x^\star)=0\quad(\lambda^\star\in\mathbb{R}^m).
+]
+
+**Lagrangian.** (L(x,\lambda)=f(x)+\lambda^{!T}C(x)).
+"""
+
+# ╔═╡ b7763163-fc68-4c56-bac9-c37de527858f
+md"""
+## Equality constraints: picture first
+
+**Goal.** Minimize (f(x)) while staying on the surface (C(x)=0).
+
+* **Feasible set as a surface.** Think of (C(x)=0) as a smooth surface embedded in (\mathbb{R}^n) (a manifold).
+* **Move without breaking the constraint.** Tangent directions are the “along-the-surface” moves keeping (C(x)) unchanged to first order.
+* **What must be true at the best point.** At (x^\star), there’s no downhill direction within the tangent space.
+* **Normals enter the story.** If the gradient can’t point along the surface, it must be balanced by the normals ({J_C(x^\star)_{i:}^{!T}}), producing multipliers (\lambda^\star). 
+"""
+
+# ╔═╡ 8a08b045-3a1b-4601-a081-27a6a22d05e6
+md"""
+## From the picture to KKT (equality only)
+
+For a regular local minimum:
+
+1. **Feasibility:** (C(x^\star)=0).
+2. **Stationarity:** (\nabla f(x^\star) + J_C(x^\star)^{!T}\lambda^\star = 0).
+
+**Lagrangian viewpoint.** Define (L(x,\lambda)=f(x)+\lambda^{!T}C(x)). At a solution, (x^\star) is stationary for (L) w.r.t. (x), while (C(x^\star)=0) ensures feasibility.
+
+**Interpreting (\lambda^\star).** Each (\lambda_i^\star) reflects how strongly the (i)-th constraint “pushes back”; it’s also a sensitivity of the optimal value to perturbations in (C_i).
+"""
+
+# ╔═╡ 907459d1-4a09-441c-a989-71ff687da873
+md"""
+## KKT system for equalities (first order) & Newton on KKT
+
+**KKT (FOC):**
+[
+\nabla_x L(x,\lambda)=\nabla f(x)+J_C(x)^{!T}\lambda=0,\qquad C(x)=0.
+]
+
+**Newton on KKT (linearize both blocks):**
+[
+\begin{bmatrix}
+\nabla^2 f(x) + \sum_{i=1}^{m}\lambda_i,\nabla^2 C_i(x) & ; J_C(x)^{!T}[2pt]
+J_C(x) & ; 0
+\end{bmatrix}
+\begin{bmatrix}\Delta x\ \Delta\lambda\end{bmatrix}
+=-
+\begin{bmatrix}
+\nabla f(x)+J_C(x)^{!T}\lambda[2pt] C(x)
+\end{bmatrix}.
+]
+
+**Notes.** This is a symmetric **saddle-point** system. Practical solves use block elimination (Schur complement) and sparse factorizations.
+ """
+
+# ╔═╡ aec59d16-c254-43b6-aee2-6261823fb7c3
+md"""
+## Newton on KKT: practice & safeguards
+
+**Works best when:**
+
+* (J_C(x^\star)) has **full row rank** (regularity).
+* The **reduced Hessian** is **positive definite**.
+* A **globalization** (e.g., merit/penalty line search) and mild **regularization** are present.
+
+**Common safeguards:**
+
+* **Regularize** the ((1,1)) block (e.g., (+\beta I)) to ensure a good search direction.
+* **Merit/penalty line search** balancing feasibility vs. optimality.
+* **Scaling** constraints to improve conditioning of the KKT system.
+"""
+
+# ╔═╡ a14a100c-5c92-42cc-899d-8c6eb0619368
+md"""
+## Gauss–Newton vs. full Newton (equality case)
+
+* **Full Newton Lagrangian Hessian:**
+  [
+  \nabla_{xx}^2 L(x,\lambda)=\nabla^2 f(x)+\sum_{i=1}^m \lambda_i,\nabla^2 C_i(x).
+  ]
+* **Gauss–Newton approximation:** drop the constraint-curvature term:
+  [
+  H_{\text{GN}}(x)\approx \nabla^2 f(x).
+  ]
+
+**Trade-offs.**
+
+* **Full Newton:** fewer iterations near the solution; costlier steps; less robust far away.
+* **Gauss–Newton:** cheaper per step and often more stable; may need more iterations but competitive in wall-clock on many problems.
+"""
+
+# ╔═╡ 300c917e-e61c-4881-82d0-bc79ace66795
+md"""
+## Solving the KKT system: Schur complement (intuition)
+
+Given
+[
+\begin{bmatrix} H & A^{!T}\ A & 0\end{bmatrix}
+\begin{bmatrix}\Delta x\ \Delta\lambda\end{bmatrix}
+=-
+\begin{bmatrix} g\ c\end{bmatrix},
+]
+with (H\approx \nabla_{xx}^2 L), (A=J_C(x)), (g=\nabla f+J_C^{!T}\lambda), (c=C(x)).
+
+* Eliminate (\Delta x): (\Delta x = -H^{-1}(g + A^{!T}\Delta\lambda)).
+* Schur system in (\Delta\lambda):
+  [
+  (A H^{-1} A^{!T}),\Delta\lambda = c + A H^{-1} g.
+  ]
+* Then recover (\Delta x).
+  Exploit **sparsity**: factor (H) once per iteration; reuse structure across iterations.
+
+"""
+
+# ╔═╡ 28a43bca-618a-4fec-a7eb-ad7a734ceac5
+md"""
+## Inequality-constrained minimization and KKT
+
+**Problem:** (\min f(x)\ \text{s.t.}\ c(x)\ge 0,\ \ c:\mathbb{R}^n\to\mathbb{R}^p).
+
+**KKT (FOC):**
+[
+\begin{aligned}
+&\text{Stationarity:} && \nabla f(x)-J_c(x)^{!T}\lambda=0,\
+&\text{Primal feasibility:} && c(x)\ge 0,\
+&\text{Dual feasibility:} && \lambda\ge 0,\
+&\text{Complementarity:} && \lambda^{!T}c(x)=0\quad(\lambda_i c_i(x)=0,\ \forall i).
+\end{aligned}
+]
+
+**Interpretation.**
+
+* **Active** constraints: (c_i(x)=0\Rightarrow \lambda_i) can be nonzero (acts like an equality).
+* **Inactive** constraints: (c_i(x)>0\Rightarrow \lambda_i=0) (no influence on stationarity).
+"""
+
+# ╔═╡ 3b6300c0-d69f-4004-9b91-41116d0ce832
+md"""
+## Complementarity: intuition & Newton’s challenge
+
+**What (\lambda_i c_i(x)=0) means.**
+
+* Tight constraint ((c_i=0)) → can press back ((\lambda_i\ge 0)).
+* Loose constraint ((c_i>0)) → no force ((\lambda_i=0)).
+
+**Why naïve Newton struggles.**
+
+* Complementarity brings **nonsmoothness** and **inequalities** ((\lambda\ge 0), (c(x)\ge 0)).
+* Equality-style Newton can violate nonnegativity or bounce across the boundary.
+
+**Two main strategies (preview).**
+
+* **Active-set:** guess actives → solve equality-constrained subproblem → update the set.
+* **Barrier / PDIP / ALM:** smooth or relax complementarity, use damped Newton, and drive the relaxation to zero.
+ 
+"""
+
+# ╔═╡ ab782dbb-5f3c-404f-87b7-091b4382b0aa
+md"""
+## Globalization with constraints: merit functions
+
+To balance feasibility and optimality during updates ((x,\lambda)\to(x+\alpha\Delta x,\lambda+\alpha\Delta\lambda)), use a **merit/penalty** function, e.g.
+[
+\Phi_\mu(x) = f(x) + \mu,|C(x)|*1 \quad \text{(equality case)},
+]
+or for inequalities, a penalty on **violation** (v(x)=\sum_i \max(0,-c_i(x))).
+Do a **backtracking line search** on (\Phi*\mu) to ensure robust progress.
+
+*(You’ll see barrier and ALM variants in the next section.)*
+ 
+"""
+
+# ╔═╡ 0f722888-d66c-47ae-a1ee-1e2b7c9b4a58
+md"""
+## Conditioning & scaling
+
+* **Scale constraints** so rows of (J_C) have comparable norms → better KKT conditioning.
+* **Regularize** (H) when indefinite/ill-conditioned (modified Cholesky or (+\beta I)).
+* **Exploit structure:** block-banded, sparse patterns common in trajectory problems.
+* **Warm-starts** from previous solves (e.g., along continuation or time steps) improve robustness.
+"""
+
+# ╔═╡ f4d85409-9095-4bc0-b515-ae283e43f344
+md"""
+## Where to next
+
+* Proceed to **Methods: Penalty vs. Augmented Lagrangian vs. Interior-Point** to see practical algorithms that *enforce* the KKT conditions reliably, including complementarity handling for inequalities.
+* Later, we’ll assemble these pieces into **SQP**.
+
+[➡ Methods (Penalty/ALM/IPM) (next)](class02_methods_barrier_alm.jl) · [⬅ Back to overview](class02_overview.jl)
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
+[deps]
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000002
+PLUTO_MANIFEST_TOML_CONTENTS = """
+# This file is machine-generated - editing it directly is not advised
+
+julia_version = "1.10.0"
+manifest_format = "2.0"
+project_hash = "da39a3ee5e6b4b0d3255bfef95601890afd80709"
+
+[deps]
+"""
+
+# ╔═╡ Cell order:
+# ╠═a509f9d8-9c6d-11f0-3db9-cb5fe2e85d64
+# ╠═57cd6d88-ea7e-4f5c-bc17-7fc65fb78e95
+# ╠═b7763163-fc68-4c56-bac9-c37de527858f
+# ╠═8a08b045-3a1b-4601-a081-27a6a22d05e6
+# ╠═907459d1-4a09-441c-a989-71ff687da873
+# ╠═aec59d16-c254-43b6-aee2-6261823fb7c3
+# ╠═a14a100c-5c92-42cc-899d-8c6eb0619368
+# ╠═300c917e-e61c-4881-82d0-bc79ace66795
+# ╠═28a43bca-618a-4fec-a7eb-ad7a734ceac5
+# ╠═3b6300c0-d69f-4004-9b91-41116d0ce832
+# ╠═ab782dbb-5f3c-404f-87b7-091b4382b0aa
+# ╠═0f722888-d66c-47ae-a1ee-1e2b7c9b4a58
+# ╠═f4d85409-9095-4bc0-b515-ae283e43f344
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/class02/class02_methods_barrier.jl b/class02/class02_methods_barrier.jl
new file mode 100644
index 0000000..973c921
--- /dev/null
+++ b/class02/class02_methods_barrier.jl
@@ -0,0 +1,308 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 8f4c1994-9c6a-11f0-2d1d-296c30c0d0ed
+md"""
+# Unconstrained Minimization (Newton + Globalization)
+
+*(Class 02 — interactive chapter section)*
+
+[⬅ Back to Class 02 Overview](class02_overview.jl) · [⬅ Previous: Root-Finding](class02_root_finding.jl)
+
+**In this section you will:**
+
+* Review first/second-order optimality conditions.
+* Derive **Newton’s method for minimization** from the quadratic model.
+* Make Newton **robust** with **line search** (Armijo/Wolfe) and **trust regions**.
+* See when/why to use **quasi-Newton (BFGS/L-BFGS)**.
+* Practice on the **Rosenbrock** test problem (path & convergence diagnostics).
+
+"""
+
+# ╔═╡ db452cd5-1c97-40b5-9836-ac74ec96e950
+md"""
+## Big picture
+
+We often recast tasks as
+[
+\min_{x\in\mathbb{R}^n} f(x),
+]
+then solve for a stationary point. Compared to generic root-finding (f(x)=0), minimization gives us **objective structure** (descent directions, merit functions, curvature) that we can exploit for **globalization** and **diagnostics**.
+
+"""
+
+
+# ╔═╡ fa119b30-970f-41a3-8ac6-2d132123d8c7
+md"""
+
+## First- and second-order conditions
+
+Let (f:\mathbb{R}^n\to\mathbb{R}) be twice continuously differentiable.
+
+* **First order (necessary):** at a local minimizer (x^\star),
+  [
+  \nabla f(x^\star)=0.
+  ]
+* **Second order:**
+
+  * **Necessary:** (\nabla^2 f(x^\star)\succeq 0) (positive semidefinite).
+  * **Sufficient (strict local min):** (\nabla^2 f(x^\star)\succ 0) (positive definite).
+
+These are **local** conditions; nonconvex problems can have multiple minima/saddles.
+
+"""
+
+# ╔═╡ 44366de6-1152-45bb-b300-c766d64e85f8
+md"""
+## Quadratic Taylor model at (x)
+
+The second-order Taylor expansion around (x) is
+[
+m_x(d) ;=; f(x) + \nabla f(x)^{!T}d + \tfrac12 d^{!T}\nabla^2 f(x),d.
+]
+Newton-type methods “approximately” minimize this model to choose the step (d).
+
+* If (\nabla^2 f(x)\succ 0), (m_x) is strictly convex and has a unique minimizer.
+
+"""
+
+# ╔═╡ fe5dfe3c-f4d4-426f-a840-dc950c816222
+md"""
+## Derivation of the Newton step
+
+Set (\nabla_d m_x(d)=0):
+[
+\nabla f(x) + \nabla^2 f(x),d ;=; 0
+\quad\Longrightarrow\quad
+d_{\text{N}} ;=; -\big(\nabla^2 f(x)\big)^{-1},\nabla f(x).
+]
+Update (x \leftarrow x + d_{\text{N}}) and repeat.
+
+* **Quadratic convergence** near a nondegenerate minimizer ((\nabla^2 f(x^\star)) nonsingular).
+* Each iteration requires solving a linear system with (\nabla^2 f(x)).
+
+"""
+
+# ╔═╡ 39bbcd3b-7ab7-45e3-ab27-eb94a25f507f
+md"""
+## Indefiniteness, overshooting, and saddles
+
+* If (\nabla^2 f(x)) is **indefinite**, (d_{\text{N}}) may not be a **descent direction** ((\nabla f(x)^{!T} d < 0)).
+* Even with (\nabla^2 f(x)\succ 0), a full step can **overshoot**; objective can increase.
+* Newton may converge to a **saddle** if started in the wrong basin.
+
+We fix this with **regularization** and **globalization**.
+
+"""
+
+# ╔═╡ 16b72035-3c8b-4c46-a1e7-35b062851422
+md"""
+## Positive-definite Hessian via regularization
+
+If (\nabla^2 f(x)) is not PD, make it so:
+[
+H(x) \leftarrow \nabla^2 f(x) + \lambda I,\quad \lambda>0,
+]
+or use a **modified Cholesky** factorization to obtain (H(x)\succ 0). Then solve
+[
+H(x),d ;=; -\nabla f(x).
+]
+This ensures the **Newton direction is descent**: (\nabla f(x)^{!T} d < 0).
+
+> In practice, (\lambda) is reduced toward (0) as we approach a minimizer.
+
+"""
+
+# ╔═╡ 50a7ea9f-72ad-4050-89c6-b3c7d02d07d8
+md"""
+Globalization I: line search
+
+## Backtracking with Armijo / Wolfe
+
+Given a descent direction (d), pick a step length (\alpha\in(0,1]).
+
+* **Armijo (sufficient decrease):** find smallest (\alpha=c^j) (e.g., (c=\frac12)) such that
+  [
+  f(x+\alpha d) ;\le; f(x) + c_1,\alpha,\nabla f(x)^{!T} d, \quad 0<c_1<1.
+  ]
+* **(Strong) Wolfe:** also enforce curvature
+  [
+  \big|\nabla f(x+\alpha d)^{!T} d\big| ;\le; c_2,\big|\nabla f(x)^{!T} d\big|,\quad c_1 < c_2 < 1.
+  ]
+
+**Takeaway:** Armijo prevents overshooting; Wolfe also controls the slope, improving quasi-Newton updates.
+
+"""
+
+# ╔═╡ b4a652af-1fda-455c-b6d2-7e818120fdf2
+md"""
+Globalization II: trust regions
+
+## Trust-region viewpoint
+
+Instead of choosing (\alpha), **trust** the model only in a ball:
+[
+\min_{d}; m_x(d)
+\quad \text{s.t.}\quad |d| \le \Delta.
+]
+
+* Solve approximately (e.g., **Cauchy point**, **dogleg**, truncated CG).
+* If the actual decrease matches predicted decrease, **enlarge** (\Delta); else **shrink** and retry.
+
+**Pros:** robust near indefiniteness and in poor scaling; integrates nicely with Hessian-free solvers.
+
+"""
+
+# ╔═╡ 49f460a5-e70e-472f-8149-f7dd6cf9e481
+md"""
+Quasi-Newton (BFGS & L-BFGS)
+
+## When Hessians are pricey
+
+Quasi-Newton builds a PD approximation (B_k \approx \nabla^2 f(x_k)) from gradients:
+
+* **BFGS update** (ensures (B_{k+1}\succ 0) if (s_k^{!T}y_k>0))
+  [
+  s_k = x_{k+1}-x_k,\quad y_k=\nabla f(x_{k+1})-\nabla f(x_k),
+  ]
+  [
+  B_{k+1} = B_k - \frac{B_k s_k s_k^{!T} B_k}{s_k^{!T} B_k s_k}
+  + \frac{y_k y_k^{!T}}{y_k^{!T} s_k}.
+  ]
+* **L-BFGS** stores only a few ( (s_k, y_k) ) pairs → great for large-scale problems.
+
+Use a **Wolfe line search** to satisfy the curvature condition (s_k^{!T}y_k>0).
+
+"""
+
+# ╔═╡ 1f90d0e6-8d3e-4aed-9cbb-748243e3a25a
+md"""
+Stopping criteria
+
+## When to stop
+
+Typical checks (use more than one):
+
+* **Gradient norm:** (|\nabla f(x)| \le \varepsilon_g\cdot \max(1,|\nabla f(x_0)|)).
+* **Step norm:** (|x_{k+1}-x_k| \le \varepsilon_x\cdot \max(1,|x_k|)).
+* **Objective change:** (|f(x_{k+1})-f(x_k)| \le \varepsilon_f\cdot \max(1,|f(x_k)|)).
+* **Iteration/time** limits.
+
+Report **status** (converged, maxiter, failed-descent) and basic **history** for debugging.
+
+"""
+
+# ╔═╡ 5e586027-baa1-463b-a032-1b25f87fe270
+md"""
+
+Example: Rosenbrock function
+
+## The Rosenbrock “banana”
+
+[
+f(x,y) = (1-x)^2 + 100,(y-x^2)^2.
+]
+
+* Narrow, curved valley; poor conditioning away from the minimizer (x^\star=(1,1)).
+* Great to illustrate **line search**, **Hessian PD-repair**, and **path plots**.
+
+**What to try (when you add code):**
+
+* Start at ((-1.2,1.0)). Compare **pure Newton**, **regularized Newton**, **BFGS**.
+* Plot contour lines and the **iterate path**. Track (|\nabla f|) on a **log scale**.
+
+
+"""
+
+# ╔═╡ ffe2800b-add1-4817-adeb-efc7660ead61
+md"""
+Conditioning & scaling
+
+## Make life easier with scaling
+
+Poor scaling slows convergence and breaks line searches.
+
+* **Variable scaling:** change variables (x=S z) to make typical curvature similar.
+* **Preconditioning:** solve (\tilde H d = -\tilde g) with (\tilde H = P^{-1/2} H P^{-1/2}).
+* **Hessian-vector products:** Newton-CG/trust-region methods avoid forming (H) explicitly.
+* **Automatic differentiation** gives accurate gradients/Hessians (or Hessian-vector products) cheaply.
+
+"""
+
+# ╔═╡ 0bb5977b-889d-408a-a073-c9a9b4e1eabc
+md"""
+Practical recipe (putting it together)
+
+## A robust Newton-type solver (sketch)
+
+1. Compute (g=\nabla f(x)). If (|g|) small → **stop**.
+2. Compute/approximate (H=\nabla^2 f(x)) or (B\approx H).
+3. **Make PD** (modified Cholesky or (+\lambda I)) to get a descent direction (d) from (H d=-g).
+4. **Line search** (Armijo/Wolfe) or use a **trust region** to pick the step length/size.
+5. Update (x\leftarrow x+\alpha d).
+6. If quasi-Newton, update (B) via **BFGS**.
+7. Repeat until stopping criteria met; record diagnostics.
+
+"""
+
+# ╔═╡ 2ef96814-04d3-466a-9d91-a43a3ae1f069
+md"""
+Check your understanding
+
+1. **Newton direction is descent when (H\succ 0).** Show that if (H\succ 0) and (H d = -g\neq 0), then (g^{!T} d < 0).
+2. **Rosenbrock Hessian.** Derive (\nabla f) and (\nabla^2 f) for Rosenbrock. Where is (H) poorly conditioned?
+3. **Armijo parameter sensitivity.** Experiment (mentally or later in code) with (c_1\in[10^{-4},10^{-1}]) and backtracking factor (c\in{1/2,2/3}). How do step counts and robustness trade-off?
+4. **BFGS curvature condition.** Explain why enforcing Wolfe guarantees (s^{!T}y>0) for smooth (f).
+
+"""
+
+# ╔═╡ 1dece802-261f-4f0e-a681-dbc6fcde5bbc
+md"""
+## Where to next
+
+* Proceed to **Constrained Minimization & KKT** to extend Newton-type ideas to equality/inequality constraints and saddle-point systems.
+* Keep the Rosenbrock notebook handy to test different globalization and quasi-Newton settings.
+
+[⬅ Back to overview](class02_overview.jl)
+
+[➡ Constrained minimization & KKT (next)](class02_constrained_min.jl) 
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
+[deps]
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000002
+PLUTO_MANIFEST_TOML_CONTENTS = """
+# This file is machine-generated - editing it directly is not advised
+
+julia_version = "1.10.0"
+manifest_format = "2.0"
+project_hash = "da39a3ee5e6b4b0d3255bfef95601890afd80709"
+
+[deps]
+"""
+
+# ╔═╡ Cell order:
+# ╠═8f4c1994-9c6a-11f0-2d1d-296c30c0d0ed
+# ╠═db452cd5-1c97-40b5-9836-ac74ec96e950
+# ╠═fa119b30-970f-41a3-8ac6-2d132123d8c7
+# ╠═44366de6-1152-45bb-b300-c766d64e85f8
+# ╠═fe5dfe3c-f4d4-426f-a840-dc950c816222
+# ╠═39bbcd3b-7ab7-45e3-ab27-eb94a25f507f
+# ╠═16b72035-3c8b-4c46-a1e7-35b062851422
+# ╠═50a7ea9f-72ad-4050-89c6-b3c7d02d07d8
+# ╠═b4a652af-1fda-455c-b6d2-7e818120fdf2
+# ╠═49f460a5-e70e-472f-8149-f7dd6cf9e481
+# ╠═1f90d0e6-8d3e-4aed-9cbb-748243e3a25a
+# ╠═5e586027-baa1-463b-a032-1b25f87fe270
+# ╠═ffe2800b-add1-4817-adeb-efc7660ead61
+# ╠═0bb5977b-889d-408a-a073-c9a9b4e1eabc
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diff --git a/class02/class02_overview.jl b/class02/class02_overview.jl
new file mode 100644
index 0000000..4f8c880
--- /dev/null
+++ b/class02/class02_overview.jl
@@ -0,0 +1,420 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ f656fd22-6dce-40da-b67a-febcee9bff08
+using PlutoUI
+
+# ╔═╡ a89a5e16-9c64-11f0-3c6f-772fd0a68437
+md"""
+## Sections
+- [Root finding](class02_root_finding.jl)
+- [Unconstrained minimization](class02_unconstrained_min.jl)
+- [Constrained minimization](class02_constrained_min.jl)
+- [Barrier vs. ALM](class02_methods_barrier_alm.jl)
+- [Sequential Quadratic Programming](class02_sqp.jl)
+"""
+
+
+# ╔═╡ 46558377-74bc-4453-b2c2-273ccaf3d2d7
+md"""
+### How to read / run
+
+1. Skim the **learning goals** below.
+2. Open each subsection (links above). Read the short prose, then **run the cells top-to-bottom**.
+3. Use the **sliders and toggles** to see how stepsizes, conditioning, and feasibility affect convergence.
+4. Try at least **one “Check your understanding”** box per subsection.
+"""
+
+# ╔═╡ 61fc1519-10c9-4b57-ac8f-4b660c08b39d
+md"""## Learning goals (what you’ll be able to do)
+
+* **Pick and configure** an optimizer for small control problems (unconstrained & constrained).
+* **Derive KKT conditions** and form the **QP/SQP subproblem** for a nonlinear program.
+* **Explain differences** between **penalty**, **augmented Lagrangian (ALM)**, and **interior-point** methods, and when to prefer each.
+
+**Why this matters.** It lets us map classic control tasks—LQR, MPC, trajectory optimization—into QPs/NLPs, exploit sparsity/warm starts, and choose a robust globalization strategy.
+"""
+
+# ╔═╡ 82c39021-20d1-4605-b10a-b1c6a29029e5
+md"""## Big picture: why optimization for control?
+
+* **Controller synthesis as optimization.** Many controllers are designed or run by solving optimization problems repeatedly.
+* **MPC** solves a QP/NLP online each step—**warm-starts** and **sparsity** drive speed.
+* **Trajectory optimization** uses nonlinear programming (often with collocation); robust **globalization** prevents solver stalls.
+* **Learning-based control** wraps optimizers inside training loops (differentiable programming), so smoothness and convergence matter.
+
+**Mental model (any time step (k))**
+State (x_k) → **optimizer** solves a small problem → control (u_k) → plant evolves (x_{k+1}=f(x_k,u_k)).
+The solver is part of the loop; reliability and speed translate directly to control performance.
+"""
+
+# ╔═╡ 515f280d-70e8-4b98-ba81-3861433594b3
+md"""
+## Notation I: derivatives & Jacobians
+
+We keep **dimensions explicit** so shapes are clear.
+
+* **Scalar-valued** (f:\mathbb{R}^n\to\mathbb{R})
+
+  * Row-derivative (row gradient): (\dfrac{\partial f}{\partial x}\in\mathbb{R}^{1\times n})
+  * Column gradient (our default): (\nabla f(x):=\left(\dfrac{\partial f}{\partial x}\right)^{!T}\in\mathbb{R}^{n})
+
+  **First-order model:**
+  [
+  f(x+\Delta x)\approx f(x)+\nabla f(x)^{!T}\Delta x.
+  ]
+
+* **Vector-valued** (g:\mathbb{R}^m\to\mathbb{R}^n)
+
+  * **Jacobian:** (\dfrac{\partial g}{\partial y}\in\mathbb{R}^{n\times m})
+
+  **First-order model:**
+  [
+  g(y+\Delta y)\approx g(y)+\frac{\partial g}{\partial y},\Delta y.
+  ]
+"""
+
+# ╔═╡ ad3d636c-571f-4671-a049-d7c4a9546058
+md"""
+## Notation II: gradient, Hessian & Taylor
+
+* **Gradient (column):** (\nabla f(x)\in\mathbb{R}^{n}).
+* **Hessian:** (\nabla^{2}f(x)\in\mathbb{R}^{n\times n}).
+* **Shape check:** (\Delta x^{!T}\nabla^{2}f(x)\Delta x\in\mathbb{R}).
+
+**Second-order Taylor expansion (near (x))**
+[
+f(x+\Delta x)\approx f(x)+\nabla f(x)^{!T}\Delta x+\tfrac{1}{2},\Delta x^{!T}\nabla^{2}f(x),\Delta x.
+]
+
+**Conventions in this chapter**
+
+* We use the **column-gradient** convention in derivations.
+* For constraints (g(x)=0) and (h(x)\le 0), Jacobians are stacked by rows so shapes align in KKT systems.
+"""
+
+# ╔═╡ 333765ac-6dc9-40fd-9b14-5905cd95cb30
+md"""
+KKT snapshot (for reference)
+
+For
+[
+\min_x f(x)\quad \text{s.t.}\quad g(x)=0,\ \ h(x)\le 0,
+]
+the (informal) KKT conditions at a candidate (x^\star) are:
+
+* **Stationarity:** (\nabla f(x^\star)+\nabla g(x^\star)^{!T}\lambda^\star+\nabla h(x^\star)^{!T}\mu^\star=0).
+* **Primal feasibility:** (g(x^\star)=0,\ \ h(x^\star)\le 0).
+* **Dual feasibility:** (\mu^\star\ge 0).
+* **Complementarity:** (\mu_i^\star,h_i(x^\star)=0) for each inequality (i).
+
+You’ll see how penalty/ALM/interior-point handle these conditions differently (explicitly, indirectly, or via barriers).
+"""
+
+# ╔═╡ a65e8de6-8f17-4adb-8359-cbe8a349eff8
+md"""
+
+"""
+
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+
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+
+# ╔═╡ Cell order:
+# ╠═f656fd22-6dce-40da-b67a-febcee9bff08
+# ╠═a89a5e16-9c64-11f0-3c6f-772fd0a68437
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+# ╠═515f280d-70e8-4b98-ba81-3861433594b3
+# ╠═ad3d636c-571f-4671-a049-d7c4a9546058
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+# ╠═a65e8de6-8f17-4adb-8359-cbe8a349eff8
+# ╠═12cbe2ab-9a12-4db6-a044-de0578bb49f3
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/class02/class02_root_finding.jl b/class02/class02_root_finding.jl
new file mode 100644
index 0000000..806b8aa
--- /dev/null
+++ b/class02/class02_root_finding.jl
@@ -0,0 +1,646 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ cc19c250-9c64-11f0-069e-a1b411d0212b
+md"[⬅ Back to Class 02 Overview](class02_overview.jl)"
+
+# ╔═╡ c268beac-c0b0-4be1-b34e-bc5294751df9
+md"# Root Finding and Fixed Points
+
+In the first subsection of Chapter 2, we will be covering root finding and fixed points. We want to build an intuition for root-finding and fixed-point iteration. We also want to understand when/why simple fixed-point iteration converges. We will also derive Newton’s method from first principles. Apply it to an implicit Backward Euler step. Learn practical globalization (damping / line search on residual)."
+
+# ╔═╡ 8a457e90-7526-4d5a-89d7-63588be4690d
+md"""
+## Root-Finding and Fixed Points (big picture)
+
+* **Root-finding:** given $f:\mathbb{R}^n!\to!\mathbb{R}^n$, find ($x^\star$) with ($f(x^\star)=0$).
+  Examples: steady states, solving nonlinear equations, implicit time-stepping.
+* **Fixed point:** ($x^\star$) is a fixed point of (g) if ($g(x^\star)=x^\star$).
+* **Bridge:** pick ($g(x)=x-\alpha f(x)$) with ($\alpha>0$). Then
+  $$f(x^\star)=0 \quad\Longleftrightarrow\quad g(x^\star)=x^\star$$ .
+* **Mindset:** start at ($x_0$) and iterate ($x_{k+1}=g(x_k)$) until nothing changes (within tolerance).
+"""
+
+# ╔═╡ 164e269d-21c7-4466-915f-95f5bcad08c9
+md"""
+Nuni - add a small julia visualization for 1d or 2d motion
+"""
+
+# ╔═╡ 30bb23ab-1ba1-4222-859d-bd095cc231f8
+md"""## When does fixed-point iteration converge?
+ 
+* Near $(x^\star)$, (g) behaves like its Jacobian ($J_g(x^\star)$) (linearization).
+* **Contraction test:**
+
+  * Scalar: ($|g'(x^\star)|<1$)
+  * Vector: spectral radius ($\rho!\left(J_g(x^\star)\right)<1$)
+* Smaller contraction ($\Rightarrow$) faster (linear) convergence.
+* If the contraction fails ($(\ge 1)$), expect divergence or oscillations.
+* Convergence is **local**: you need to start in the **basin of attraction**.
+"""
+
+# ╔═╡ 8336a88f-7a6b-473f-a3cf-711df97d2297
+md"""
+## Fixed-Point Iteration: minimal recipe
+
+1. Choose ($g$) (often ($g(x)=x-\alpha f(x))$) and an initial guess ($x_0$).
+2. Loop: ($x_{k+1}\leftarrow g(x_k)$).
+3. Stop when either
+
+   * **residual** ($|f(x_{k+1})|$) is small, or
+   * **step size** ($|x_{k+1}-x_k|$) is small, or
+   * **max iterations** hit.
+4. **Report both** residual and step size to diagnose false convergence.
+
+**Diagnostics tip.** Small step but large residual ($\Rightarrow$) stagnation (bad $(g)/(\alpha))$. Large step but small residual $(\Rightarrow)$ near solution but overshooting.
+"""
+
+# ╔═╡ 192e2d28-c502-4cc5-b365-21086d7e1bd4
+md"""
+## Tuning the simple iteration
+
+* **Step size ($\alpha$):** too small $(\Rightarrow)$ slow; too large $(\Rightarrow)$ divergence/oscillation. Start modest, adjust cautiously.
+* **Damping / relaxation:** $(x_{k+1}\gets (1-\beta)x_k+\beta,g(x_k))$, with $(0<\beta\le 1)$ stabilizes updates.
+* **Rescaling/preconditioning:** choose $(g(x)=x-P,f(x))$ with a good matrix (P) (approx $(J_f(x^\star)^{-1})$) to speed convergence.
+* **Better initial guesses** shrink the “distance to the basin”.
+
+**Optimization link.** Gradient descent is fixed-point iteration on (\nabla F):
+$(g(x)=x-\eta\nabla F(x))$ solves $(\nabla F(x^\star)=0)$.
+
+"""
+
+# ╔═╡ a8ab5084-6fb4-4f54-8a60-831c0a77e660
+md"""
+## Newton’s Method Derived (from linearization)
+
+**TL;DR.** Instead of solving $(f(x)=0)$ directly, at the current $(x)$ solve a **linearized** system for a correction $(\Delta x)$.
+
+* Linearize $(f)$ near (x):
+  $$f(x+\Delta x);\approx; f(x) + J_f(x),\Delta x$$ .
+  
+* Set the approximation to zero and solve for ($\Delta x$):
+  $$f(x) + J_f(x),\Delta x = 0
+  \quad\Longrightarrow\quad
+  \Delta x = -J_f(x)^{-1} f(x)$$
+* Update and repeat: $(x \leftarrow x + \Delta x)$.
+
+"""
+
+# ╔═╡ 8658738a-b448-4aa9-9a1c-5a913cd3ceb1
+md"""
+## Newton: local behavior & requirements
+
+* If (f) is smooth, (J_f(x^\star)) is **nonsingular**, and (x_0) is close enough to (x^\star), Newton converges **quadratically**:
+  (|x_{k+1}-x^\star| \approx C |x_k-x^\star|^2).
+* If (J_f(x^\star)) is singular or poorly conditioned, expect slow or erratic progress.
+* Good **initialization** matters; Newton is a **local** method.
+
+**Cost.** Each iteration solves a linear system with (J_f(x)): naive dense (O(n^3)), but real problems exploit **sparsity/structure** and reuse factorizations.
+"""
+
+
+# ╔═╡ 81832b4c-f010-41d9-8b62-3924bcb3669d
+md"""## Globalization for Newton (residual line search)
+
+Pure Newton can overshoot. A classic fix is a **line search on the residual norm**:
+
+* Define merit (\phi(x)=\tfrac12 |f(x)|_2^2).
+* Compute Newton step (\Delta x=-J_f(x)^{-1}f(x)).
+* **Backtrack** on (\alpha\in(0,1]) until
+  [
+  \phi(x+\alpha\Delta x) ;\le; \phi(x) + c,\alpha,\nabla\phi(x)^{!T}\Delta x,
+  \quad 0<c<1\ \text{(Armijo)}.
+  ]
+* Update (x\gets x+\alpha\Delta x).
+
+**Alternatives.** Trust-region methods (e.g., Levenberg–Marquardt for least-squares) control step size by solving a **restricted** subproblem.
+"""
+
+
+# ╔═╡ 24cd83da-feae-4da0-9604-83767615c017
+md"""
+## Example: Backward Euler implicit step
+
+Given dynamics (\dot{x}=f(x)) and step (h>0), **Backward Euler** seeks (x_{n+1}) via
+[
+x_{n+1} = x_n + h,f(x_{n+1}).
+]
+Define the residual in the unknown (x_{n+1}):
+[
+r(z) ;=; z - x_n - h,f(z) .
+]
+We solve the **root-finding** problem (r(z)=0) for (z=x_{n+1}).
+
+* Newton step: solve (J_r(z),\Delta = -r(z)) with
+  (J_r(z)=I - h,J_f(z)).
+* **Fast local convergence** (often quadratic) once close.
+* **Cost driver:** the linear solve; exploit **sparsity** in (J_f), reuse factorizations across timesteps, and **warm-start** with (z\approx x_n).
+
+**Try it (when you add code).**
+
+* Slider for (h) and initial guess (z_0).
+* Compare pure Newton vs. residual line search.
+* Plot (|r(z_k)|) per iteration (log scale).
+"""
+
+# ╔═╡ 060606a5-aca5-449e-a614-57b79b78a462
+md"""
+## Root-Finding via Minimization (and vice-versa)
+
+* **Solve (f(x)=0)** by minimizing the least-squares objective
+  (\min_x \tfrac12 |f(x)|_2^2).
+
+  * **Gauss–Newton** uses (J_f(x)^{!T}J_f(x)\Delta = -J_f(x)^{!T} f(x)).
+  * Good when (f) is a residual from data/physics (sum of squares).
+* **Solve (\nabla F(x)=0)** (unconstrained minimization) with **Newton for gradients**:
+  [
+  \Delta x = -(\nabla^2 F(x))^{-1}\nabla F(x),\qquad x\gets x+\Delta x.
+  ]
+  This is “Newton for optimization” and appears in the next section.
+"""
+
+# ╔═╡ cb3d5657-0ef6-44bf-8818-95c48ad5595e
+md"""
+## Diagnostics & Pitfalls
+
+* **Stagnation:** ($|x_{k+1}-x_k|\to 0$) but ($|f(x_k)|\not\to 0$).
+  *Fix:* better (g)/preconditioner (P), damping, or switch to Newton.
+* **Oscillation/divergence:** step too aggressive.
+  *Fix:* relaxation ($\beta<1$), backtracking on residual, smaller ($\alpha$).
+* **Singular/ill-conditioned Jacobian:** Newton step unstable.
+  *Fix:* regularize (e.g., add ($\gamma I$)), trust-region, or switch to Gauss–Newton if least-squares.
+* **Bad initial guess:** outside basin.
+  *Fix:* problem-specific heuristics, continuation (homotopy), or multiple restarts.
+
+"""
+
+# ╔═╡ 44d93a6b-6575-438a-911f-6913d0ca02b5
+md"""
+## Ensuring descent / stability (regularization)
+
+If the linear system for $(\Delta x)$ is unstable or poorly conditioned, **regularize**:
+ $$\big(J_f(x) + \gamma I\big),\Delta = -f(x),\qquad \gamma>0$$
+Then backtrack on ($\alpha$) and update ($x\gets x+\alpha\Delta$).
+
+* Often called **damped Newton** (shrinks steps).
+* Stabilizes solves and helps ensure decrease in the residual merit.
+* Choose ($\gamma$) adaptively (smaller as the residual shrinks).
+"""
+
+# ╔═╡ a7af3229-b2ee-4dc1-991b-2196273a38db
+md"""
+## Where to go next
+
+* Head to **Unconstrained Minimization (Newton + globalization)** to see how solving (\nabla F(x)=0) ties in and how to make Newton **reliably decrease** an objective.
+* Later, we’ll re-use these ideas inside **KKT systems** for constrained problems.
+
+[⬅ Back to overview](class02_overview.jl)
+
+[➡ Unconstrained minimization (next)](class02_unconstrained_min.jl) 
+"""
+
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diff --git a/class02/class02_sqp.jl b/class02/class02_sqp.jl
new file mode 100644
index 0000000..17c83fd
--- /dev/null
+++ b/class02/class02_sqp.jl
@@ -0,0 +1,269 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ cb245a30-f64e-4f2c-91f4-a50371b201a2
+md"""
+*(Class 02 — interactive chapter section)*
+
+[⬅ Back to Class 02 Overview](class02_overview.jl) · [⬅ Previous: Methods (Penalty/ALM/IPM)](class02_methods_barrier_alm.jl)
+
+**In this section you will:**
+
+* Build the SQP **local model** (quadratic Lagrangian + linearized constraints).
+* See the **QP subproblem** and how it relates to Newton on the KKT system.
+* Learn **globalization** (merit / filter / trust-region) and **Hessian updates** (BFGS).
+* Work through a small **toy example** and practical tips for control problems.
+
+"""
+
+# ╔═╡ 1adff7b2-d86b-4311-9626-22d52057f551
+md"""
+## Cell 2 — What is SQP?
+
+## What is SQP?
+
+**Idea.** Solve a nonlinear constrained problem by repeatedly solving a **quadratic program (QP)** built from local models.
+
+* Linearize constraints; build a quadratic model of the **Lagrangian/objective**.
+* Each iteration: solve a QP to get a step (d), then update (x \leftarrow x+\alpha d).
+* With good Hessian information, SQP enjoys **strong local convergence** (often **superlinear**).
+
+
+"""
+
+# ╔═╡ 3d685462-e495-4c33-a2de-470e109bff78
+md"""
+
+## Cell 3 — Target problem & KKT recap
+
+## Target NLP
+
+[
+\min_{x \in \mathbb{R}^n} \ f(x)
+\quad\text{s.t.}\quad
+g(x)=0,\quad h(x)\le 0,
+]
+with (g:\mathbb{R}^n!\to!\mathbb{R}^{m}) and (h:\mathbb{R}^n!\to!\mathbb{R}^{p}).
+
+**KKT at a candidate optimum (x^\star):** there exist (\lambda\in\mathbb{R}^m), (\mu\in\mathbb{R}^p_{\ge 0}) with
+[
+\nabla f(x^\star)+ \nabla g(x^\star)^{!T}\lambda + \nabla h(x^\star)^{!T}\mu = 0, \quad
+g(x^\star)=0, \quad
+h(x^\star)\le 0, \quad
+\mu\ge 0, \quad
+\mu \odot h(x^\star)=0.
+]
+
+"""
+
+# ╔═╡ 40e0acdf-f870-4ef6-9656-e909d5fb2e25
+md"""
+
+## Cell 4 — Local models at (x_k)
+
+## From NLP to local models
+
+At iterate (x_k) (with multipliers ((\lambda_k,\mu_k))):
+
+**Quadratic model of the Lagrangian**
+[
+m_k(d) ;=; \nabla f(x_k)^{!T} d ;+; \tfrac{1}{2}, d^{!T} B_k, d,
+]
+with (B_k \approx \nabla^2_{xx}\mathcal{L}(x_k,\lambda_k,\mu_k)).
+
+**Linearized constraints**
+[
+g(x_k)+\nabla g(x_k),d=0,
+\qquad
+h(x_k)+\nabla h(x_k),d \le 0.
+]
+
+"""
+
+# ╔═╡ b23e9ac3-192d-41e6-9411-99214b612ab9
+md"""
+
+## Cell 5 — The SQP QP subproblem
+
+## SQP subproblem (QP)
+
+[
+\begin{aligned}
+\min_{d \in \mathbb{R}^n}\quad & \nabla f(x_k)^{!T} d + \tfrac12 d^{!T} B_k d\
+\text{s.t.}\quad & \nabla g(x_k), d + g(x_k) = 0,\
+& \nabla h(x_k), d + h(x_k) \le 0.
+\end{aligned}
+]
+
+* Solve the QP (\Rightarrow) get step (d_k) and updated multipliers ((\lambda_{k+1},\mu_{k+1})).
+* Update (x_{k+1} = x_k + \alpha_k d_k) (line search, **filter**, or **trust-region**).
+
+"""
+
+# ╔═╡ 64d7aa02-a814-4d0d-b365-90d70814932a
+md"""
+
+## Cell 6 — Algorithm sketch
+
+## SQP: high-level algorithm
+
+1. Initialize (x_0), multipliers ((\lambda_0,\mu_0)), and (B_0 \succ 0).
+2. Form the QP at (x_k) using (B_k) and the linearized constraints.
+3. Solve the QP (\Rightarrow) obtain (d_k), ((\lambda_{k+1},\mu_{k+1})).
+4. **Globalize** (merit/filter/TR) to choose (\alpha_k) (and possibly restrict (d_k)).
+5. Set (x_{k+1}=x_k+\alpha_k d_k); update (B_{k+1}) (e.g., **BFGS**).
+6. Check convergence (stationarity, feasibility, complementarity) or continue.
+
+"""
+
+# ╔═╡ a725c268-9c6a-11f0-059f-fdcd96af48e3
+md"""
+
+## Cell 7 — Why this QP? (Newton–KKT connection)
+
+## Newton–KKT connection
+
+The QP KKT system is a **linearized** KKT system for the NLP.
+
+* If (B_k) equals the exact (\nabla_{xx}^2\mathcal{L}(x_k,\lambda_k,\mu_k)), the SQP step is a **Newton step** (on the KKT equations).
+* With **BFGS** (properly updated), SQP attains **superlinear** local convergence under standard assumptions (LICQ, strict complementarity, SOSC, Wolfe conditions on the merit).  
+"""
+
+# ╔═╡ ddb74901-0efe-4d4f-9c14-fc5098aaeced
+md"""
+## Cell 8 — Globalization I: (L_1) merit function
+
+## Merit function line search
+
+Define
+[
+\Phi_\mu(x)= f(x) + \mu\Big(,|g(x)|_1 + |h(x)^-|_1\Big),\quad
+h^-_i(x)=\max(0,,h_i(x)).
+]
+
+* Choose (\mu) large enough to ensure **descent coupling** of feasibility and optimality.
+* Backtrack on (\alpha\in(0,1]) to achieve an **Armijo** decrease in (\Phi_\mu).
+
+**Caveat.** Large (\mu) can cause the **Maratos effect** (over-penalizing curvature, step rejection). See SOC below.
+
+"""
+
+# ╔═╡ af20e5bb-dc3e-4d13-af71-f44e23c4c544
+md"""
+## Cell 9 — Globalization II: filter (alternative to a big penalty)
+
+## Filter method (accept if you improve *either* objective or feasibility)
+
+Track pairs ((f,\theta)) with (\theta=|g(x)|_1+|h(x)^-|_1).
+
+* Accept (x_{k+1}) if it **dominates** the filter: it reduces (f) for similar (\theta), or reduces (\theta) sufficiently.
+* If the step doesn’t pass, try a **feasibility restoration** phase (focus on decreasing (\theta)).
+  Filter avoids tuning a huge (\mu) and is widely used in practical SQP codes.
+
+"""
+
+# ╔═╡ 19da3f74-83da-4d6f-8bf0-55102050a58a
+md"""
+## Cell 10 — Trust-region SQP & step decomposition
+
+## Trust-region SQP (TRSQP)
+
+* Add (|d|\le \Delta) to the QP or use a **normal/tangential** split: first reduce constraint violation (normal step), then reduce (f) in the tangent space (tangential step).
+* Adjust (\Delta) based on the ratio of **actual vs. predicted** improvement (as in unconstrained TR methods).
+"""
+
+# ╔═╡ 6b101d4e-7eff-4b91-8721-bd7bb7fafa17
+md"""
+## Cell 11 — Hessian models & updates
+
+## Choosing (B_k): exact vs. quasi-Newton
+
+* **Exact Lagrangian Hessian:** (B_k=\nabla^2_{xx}\mathcal{L}(x_k,\lambda_k,\mu_k)) (best local rate, expensive/less robust far away).
+* **Gauss–Newton / curvature drop:** drop (\sum \mu_i \nabla^2 h_i) and (\sum \lambda_j \nabla^2 g_j) if residual-type; cheaper and often more stable.
+* **BFGS / damped BFGS:** update (B_k\succ 0) with curvature condition (s_k^{!T}y_k>0) (enforce with Wolfe or damping).
+* **Projected / null-space BFGS:** maintain positive definiteness on the **tangent space** to constraints.
+"""
+
+# ╔═╡ 7923cfd5-cd33-4001-9b71-ec8c17318ae8
+md"""
+## Cell 12 — The QP solver: structure matters
+
+## Solving the QP efficiently
+
+* **KKT structure:** sparse, block-saddle systems; use **Schur complements** or sparse symmetric indefinite factorizations.
+* **Active-set QP solvers** (great with warm-starts; common in fast MPC).
+* **Primal–dual IPM QP solvers** (robust for large QPs; excellent for dense inequality sets).
+* **Warm-starting** ((x,\lambda,\mu)) and reusing factorizations across SQP iterations (and across time steps in MPC) is key.
+
+"""
+
+# ╔═╡ c1711266-ecab-4a93-af4b-7c1b0691a6a9
+md"""
+## Cell 13 — Toy example (local models)
+
+## Toy example: local models
+
+[
+\min_{x\in\mathbb{R}^2}\ \tfrac12|x|^2
+\quad\text{s.t.}\quad
+g(x)=x_1^2+x_2-1=0,\quad
+h(x)=x_2-0.2\le 0.
+]
+
+At (x_k):
+[
+\nabla f(x_k)=x_k,\quad B_k=I,\quad
+\nabla g(x_k)=\begin{bmatrix}2x_{k,1} & 1\end{bmatrix},\quad
+\nabla h(x_k)=\begin{bmatrix}0 & 1\end{bmatrix}.
+]
+Solve the QP for (d_k), then set (x_{k+1}=x_k+\alpha_k d_k) using a merit/filter/trust-region globalization.
+
+*(When you add code: plot feasibility (|g|), violation (|h^-|), and objective vs. iteration.)*
+"""
+
+# ╔═╡ 03af9cf2-4ba7-4c31-9f36-88efe8e9eb37
+md"""
+## Where to next
+
+* You now have the full pipeline: **KKT modeling → methods (Penalty/ALM/IPM) → SQP assembly**.
+* In a control context (NMPC/trajectory optimization), plug in system sparsity and warm-starts to hit real-time targets.
+
+[⬅ Back to overview](class02_overview.jl)
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
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diff --git a/class02/class02_unconstrained_min.jl b/class02/class02_unconstrained_min.jl
new file mode 100644
index 0000000..6493a8d
--- /dev/null
+++ b/class02/class02_unconstrained_min.jl
@@ -0,0 +1,31 @@
+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 573d73d6-9c6a-11f0-28fb-83e9ff22e922
+md"""
+hgdks
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
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+[deps]
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000002
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+# This file is machine-generated - editing it directly is not advised
+
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