lectures/Optimization_MOC.md

metadata

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

🗺️ Optimization: Map of Content (MOC)

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]]

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📐 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)

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⚠️ 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 ($g_i(x^*) = 0$).