lectures/Optimization_MOC.md

---
title: "Optimization Map of Content (MOC)"
course: "[[Optimization]]"
type: "moc"
updated: "2026-08-20"
tags:
  - moc
  - course/Optimization
---

# 🗺️ Optimization: Map of Content (MOC)

> [!NOTE] Master Course Knowledge Hub
> This hub connects all lecture notes, derivations, and exam warnings for **Optimization**.

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## 📅 1. Chronological Lecture Syllabus

| Date | Topic | Note Link |
| :--- | :--- | :--- |
| `2026-08-19` | KKT Conditions | [[2026-08-19_Optimization_KKT_Conditions\|KKT Conditions Note]] |
| `2026-08-19` | KKT Conditions 2026 08 19 | [[2026-08-19_Optimization_KKT_Conditions_2026_08_19\|KKT Conditions 2026 08 19 Note]] |
| `2026-10-15` | KKT Conditions | [[2026-10-15_Optimization_KKT_Conditions\|KKT Conditions Note]] |

---

## 📐 2. Key Derivations & Theorems Index

- **[[2026-08-19_Optimization_KKT_Conditions_2026_08_19#Theorem: Global Optimality under Convexity|Theorem: Global Optimality under Convexity]]** _(Topic: KKT Conditions 2026 08 19, Date: `2026-08-19`)_
- **[[2026-08-19_Optimization_KKT_Conditions_2026_08_19#Proof:|Proof:]]** _(Topic: KKT Conditions 2026 08 19, Date: `2026-08-19`)_
- **[[2026-10-15_Optimization_KKT_Conditions#Lagrangian Formulation|Lagrangian Formulation]]** _(Topic: KKT Conditions, Date: `2026-10-15`)_

---

## ⚠️ 3. High-Yield Exam Pitfalls Aggregator

### From [[2026-08-19_Optimization_KKT_Conditions|KKT Conditions (2026-08-19)]]:
> - **Midterm Exam Alert**: Proving **strong duality under Slater's condition** will be explicitly tested on the midterm examination. Master the proof steps connecting strict feasibility to dual multiplier existence.
> - **Sign Convention Traps**: Dual multipliers for inequality constraints expressed as $g_i(x) \le 0$ must satisfy $\lambda_i \ge 0$. Reversing the constraint inequality to $g_i(x) \ge 0$ changes the required sign of the multiplier or Lagrangian term.
> - **Necessity vs. Sufficiency**: Remember that KKT conditions are *necessary and sufficient* for global optimality **only** when $f(x)$ and $g_i(x)$ are convex functions. For non-convex problems, KKT conditions are merely necessary first-order conditions (under constraint qualifications) and only yield stationary points.

### From [[2026-08-19_Optimization_KKT_Conditions_2026_08_19|KKT Conditions 2026 08 19 (2026-08-19)]]:
> - **Midterm Exam Mandate**: The professor explicitly warned that **proving strong duality under Slater's condition** will be directly tested on the midterm exam.
> - **Slater's Condition Requirement**: Remember that Slater's condition requires strictly feasible primal points—i.e., there exists at least one $x \in \text{relint}(\mathcal{D})$ such that $g_i(x) < 0$ for all non-affine inequality constraints.
> - **Sign Errors**: A common trap on exams is incorrectly writing dual feasibility as $\lambda_i \le 0$ or placing the wrong sign in the Lagrangian stationarity equation. Ensure $\lambda_i \ge 0$ when formulation uses $g_i(x) \le 0$.

### From [[2026-10-15_Optimization_KKT_Conditions|KKT Conditions (2026-10-15)]]:
> - **Explicit Midterm Exam Topic**: You will be required to write a formal proof of **strong duality under Slater's condition** on the midterm examination.
> - **Multiplier Sign Conventions**: Inequality constraints expressed as $g_i(x) \le 0$ dictate $\lambda_i \ge 0$. If written as $g_i(x) \ge 0$, the corresponding multiplier sign flips.
> - **Complementary Slackness Application**: $\lambda_i^* g_i(x^*) = 0$ implies a strict trade-off:
>   - If constraint $i$ is inactive ($g_i(x^*) < 0$), then $\lambda_i^* = 0$.
>   - If $\lambda_i^* > 0$, the constraint must be active ($g_i(x^*) = 0$).