EX#7 Mathematical Optimization

 Assignment Structure 作業架構

md version 

Deadline:  Next Saturday at 23:59 (one more week) 期中考緣故，順延一周

Send all the share links to  me chang212@gmail.com by email with subject EX#7 [your id, your name]

Students choose from three progressively harder assignments. Each builds on the interactive "Optimization in Hyperspace" 3D visualizer, which demonstrates five algorithms navigating a fitness landscape.

同學從三份難度遞增的作業中逐一完成。每份皆以互動式「超空間最佳化」3D 視覺化工具為基礎，展示五種演算法在適應度地形上的行為。

Problem 1 — Conceptual Understanding 概念理解

• Task 任務： Written reflection (300–500 words) on how each algorithm behaves on the landscape — local vs. global optima, exploration vs. exploitation.
• 撰寫心得（300–500字），描述各演算法在地形上的行為差異——區域最佳解 vs. 全域最佳解、探索 vs. 開發的取捨。
• Deliverable 繳交物： md (1–2 pages)
• Difficulty 難度： ★☆☆

Problem 2 — Mathematical Analysis 數學分析

• Task 任務： Implement gradient descent from scratch on a multimodal 2D function f(x,y) = sin(x)·cos(y) + 0.1(x² + y²). Test 3 starting points × 3 learning rates. Plot the surface + convergence paths.
• 自行實作梯度下降法，應用於多峰 2D 函數。測試 3 個起點 × 3 種學習率 (α = 0.01, 0.1, 0.5)，繪製曲面與收斂路徑。
• Deliverable 繳交物： Code + 1-page report in artifact
• Difficulty 難度： ★★☆

Problem 3 — Comparative Algorithm Design 演算法比較實驗

• Task 任務： Compare Gradient Descent vs. Simulated Annealing (exponential cooling T(t) = T₀·γᵗ). Run 50 trials each from random starts; vary T₀ and γ; produce success-rate table + plot.
• 比較梯度下降 vs. 模擬退火法（指數冷卻排程）。各執行 50 次隨機起點實驗，調整 T₀ 與 γ，製作成功率表格與圖表。
• Deliverable 繳交物： Code + 1–2 page analysis with table & plot in artifact
• Difficulty 難度： ★★★

---

Five Algorithms in the Visualizer 視覺化工具中的五種演算法

Algorithm 演算法	Type 類型	Key Trait 特性
Gradient Descent 梯度下降	Local 區域	Follows steepest slope; gets trapped in local optima 沿最陡方向走，易陷入區域最佳解
Nelder-Mead 單純形法	Local 區域	Derivative-free simplex; can stall near local optima 無需導數，但仍可能停在區域解
A* Search A*搜尋	Heuristic 啟發式	Graph-based, uses heuristic to guide search 圖形搜尋，以啟發函數引導
Simulated Annealing 模擬退火	Global 全域	Accepts worse moves probabilistically; escapes traps 以機率接受較差解，可跳脫陷阱
Global Optimization 全域最佳化	Global 全域	Systematic global search 系統性全域搜尋

---

Grading Notes 評分備註

• problem 1 is reflection-based — check for genuine engagement with the visualizer, not just generic definitions. HW1 為心得型——確認同學確實操作過視覺化工具，而非僅抄定義。
• problem 2, 3 require working code — verify plots are generated from their own implementation, not copied. HW2/HW3 需繳交可執行程式碼——確認圖表由自己實作產生。
• problem 3 requires statistical rigor (50 runs, multiple parameter settings). Check for proper experimental design. HW3 需具備統計嚴謹度（50次試驗、多組參數），檢查實驗設計是否完整。