Introduction to Optimization Theory

Announcements

★★★ 2026/03/20 ★★★ Your Pi Day artwork has been uploaded to this webpage. Please scroll to the bottom of the page to find your creation :)

★★★ 2026/03/10 ★★★ Homework Assignment 1 has been announced (see below or eCourse2); please submit your work before due date and late submission is not allowed! 

★★★ 2026/02/12 ★★★ This course will be delivered as an EMI (English-medium instruction) course, in line with the government’s bilingual education policy. This offering will focus specifically on Convex Optimization (different from the previous years)! For undergraduates: if you are taking this course with the intention of applying for credit exemption at other top-tier universities, please note that our depth and coverage may not fully match theirs, so credit exemption may not be possible. 

Course Information

Instructor: Jian-Jia Weng

Time: Tuesdays and Thursdays, 10:15-11:30

Location: R103, Innovation Building

Office Hour: Upon Request

Teaching Method: Chalkboard Teaching with Video Recording Supplementary

Textbook: Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2024.

You can download the textbook and other materials here. Prof. Boyd also provides full lecture videos on YouTube (2023 version), which makes self-study feasible. That said, I strongly encourage you to attend class: convex optimization is an advanced subject and requires a solid foundation, and the in-person lectures will help you build the key concepts and avoid common pitfalls.

 Grade Evaluation: 4 Homework Assignments (each 10%) + Midterm Exam (20%) + Final Exam (20%) + Notes&In-Class Participation (20%) 

Contact Information

Office: R428, Innovation Building  (please make an appointment before coming)

Campus Internal Phone Number：33528

Email: jjweng AT ccu.edu.tw

If you have any questions regarding the course, you can email me from your school email account with:  

• Subject: [OPT 2026] Inquiry - Your name and Student ID number (example: [OPT 2026] Inquiry - 周杰倫 1234567)

• Contents: (1) topics you want to discuss and (2) your preferred time to meet in person (please specify at least 3 time slots)

for a special accommodation. I should reply to your email within 24hours; if not, please send the email again.

Tentative Schedule (To be updated)

• Week 1 (02/24, 02/26): Section 1

  1. Mathematical Definition of Optimization Problems

  2. Six Applications (Communications, Control System, Patter Recognition, etc) 

• Week 2 (03/03, 03/05): Section 2.1-2.2

  1. Affine and Convex Sets and Cone

  2. Affine, Convex, Conice Combinations

  3. Affine, Convex, and Conic Hulls

  4. Hyperplane and Half Spaces

  5. Norm Balls, Ellipsoids, and Norm Cone

• Week 3 (03/10, 03/12): No classes

• Week 4 (03/17, 03/19): Section 2.3-2.4

  1. Polyhedra (Simplexes)

  2. Positive Semidefinite Cone

  3. Operations Preserving Convexity

  4. Perspective and Linear-Fractional Functions

  5. Proper Cone and Genearlized Inequalities

• Week 5 (03/24, 03/26): Section 2.5-2.6

  1. Minimum and Minimal Elements

  2. Separation and Supporting Hyperplane Theorems

  3. Dual Cone and the Associated Generalized Inequalities

  4. Characterizations of Minimum and Minimal Elements

  5. Pareto Optimal Solutions

• Week 6 (03/31, 04/02): Section 3.1

  1. Five Definitions of Convexity

  2. Epi-graph

• Week 7 (04/07, 04/09): Sections 3.2-3.3

  1. Operations that Preserving Convexity

  2. Conjugate Function

• Week 8 (04/14, 04/16):  Sections 3.3-3.4

  1. Quasi-convex Function

  2. Why are quasi-convex functions important? Sub-level set structure matters

  3. Solving an optimization problem via feasibility problem with bisection method 

• Week 9 (04/21, 04/23): Sections 3.3-3.4

  1. Log-Concave/Log-Convex Functions

  2. Moment Generating Function (MGF) and Cumulant Generating Function (CGF)

  3. Finding the Tightest Bound in Chernoff Bound

  4. Rate Function in Large Deviation Theory (is conjugate function of MGF)

• Week 10 (04/28, 04/30): Section 4.1 & Midterm Exam

  1. Standard form of Optimization Problems

  2. (An Opt. Problem can be) Unbounded Below, Solvable

  3. Optimal Values and Achievable

  4. Global Minimizers, and Local Minimizers 

• Week 11 (05/05, 05/07): Sections 4.1-4.2

  1. Equivalent Problems

  2. Convex Optimizations

  3. Optimality Conditions

• Week 12 (05/12, 05/14): Sections 4.1-4.2

  1. Linaer Programming (LP)

  2. Qudratic Programming (QP)

  3. Qudactically Constrained QP (QCQP)

  4. Secon-Order Cone Programming (SOCP)

  5. The Relationships of Those Programming

• Week 13 (05/19, 05/21): 

  1. Semi-Definite Programming (SDP)

  2. Scalarization of Multi-objective Optimization

• Week 14 (05/26, 05/28): Section 5.1-5.2

  1. Lagrangian and Dual Function

  2. Primal and Dual Problems

  3. Weak and Stong Dualities

  4. Slater's Constraint Qualificaion

• Week 15 (06/02, 06/04):

• Week 16: (06/09, 06/11) Final Exam

Selected Homework Problems

You need to submit your answers on eCourse2, although they are not marked anyway. You are encouraged to work them together with your friends. Discussion is a good way for learning a new subject.   

• HW1: 2.2, 2.7, 2.8, 2.10, 2.12, 2.15, 2.17, 2.25, 2.28, 2.29 (Due: 3/30, 9PM)

• HW2: 3.1, 3.2, 3.6, 3.10, 3.12, 3.16, 3.32, 3.35, 3.36, 3.43, 3.49 (Due: 4/27, 9PM)

• HW3: 4.1, 4.2, 4.7, 4.8, 4.18, 4.20, 4.22, 4.42, 4.50, 4.57 (Due: 5/25, 9PM)

• HW4: 5.1, 5.2, 5.3, 5.4, 5.7, 5.13, 5.20, 5.22, 5.26, 5.27 (Due: 6/17, 9PM)

References

1. E. K. P. Chong, W.-S. Lu, and S. H. Zak, An Introduction to Optimization with Applications to Machine Learning (5ed) 

2. D. G. Luenberger, Optimization by Vector Space Methods. Available here

3. S. Bubeck, Convex Optimization: Algorithms and Complexity. Available: arXiv1405.4980v2

Additional Materials

1. Polytope and Polyhedron [Wiki]; also see this lecture note from Cornell Univ. and this one from Illinois. 

In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a polyhedron is the generic object in any dimension (referred to as polytope in this article) and polytope means a bounded polyhedron. This terminology is typically confined to polytopes and polyhedra that are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and is defined by its vertices.

2. A Reference for Differentiability in Higher Dimension

3. A Reference for Orthogonally Diagonalizable

4. Change-of-Basis [Wiki]

5. Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain Edition 1 1/4, 1994.

Numerical Linear Algebra

As we saw in class an equivalence between solving Ax=b and minimizing 1/2 x^TAx - b^Tx, many iterative algorithms developed to solve linear systems can be transformed into procedures to find the minimizers of the quadratic function. The following materials covers fundamental results on how to solve Ax=b numerically, particularly the Krylov subspace methods. Such methods are highly related to the conjugate gradient (CG) methods in Section 10 of our textbook.  

1. Henk A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, Cambridge, 2003. 

2. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, PA, 1994.

3. T. Sogabe, Krylov Subspace Methods for Linear Systems: Principles of Algorithms, Springer, 2022 

4. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, SIAM, 2003.

5. W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Springer, 2016.

6. Krylov Subspaces and Conjugate Gradients from MIT

7. A Brief Introduction to Krylov Space Methods for Solving Linear Systems from ETHZ

8. Gerand Sleijpen, A Course on Numerical Linear Algebra

9. Marc Bonnet, Lecture Notes in Numerical Linear Algebra

Fun Readings

1. G. Foschini, R. Gitlin and S. Weinstein, "Optimization of two-dimensional signal constellations in the presence of Gaussian noise," IEEE Trans. Commun., vol. 22, no. 1, pp. 28-38, Jan. 1974, doi: 10.1109/TCOM.1974.1092061.

2. M. Beko and R. Dinis, "Designing good multi-dimensional constellations," IEEE Wireless Commun. Lett., vol. 1, no. 3, pp. 221-224, Jun. 2012, doi: 10.1109/WCL.2012.032312.120203

3. J. Feldman, M. J. Wainwright and D. R. Karger, "Using linear programming to Decode Binary linear codes," IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 954-972, March 2005, doi: 10.1109/TIT.2004.842696.

4. T. Cui, T. Ho and C. Tellambura, "Linear Programming Detection and Decoding for MIMO Systems," in Proc. IEEE ISIT, Seattle, WA, USA, 2006, pp. 1783-1787, doi: 10.1109/ISIT.2006.261741.

5. T. Wadayama and M. Hagiwara, "LP-Decodable Permutation Codes Based on Linearly Constrained Permutation Matrices," IEEE Trans. on Inf. Theory, vol. 58, no. 8, pp. 5454-5470, Aug. 2012, doi: 10.1109/TIT.2012.2196253.

課程安排、教學建議與修課心得分享

1. 只能說emi是我的問題，不是教授的問題，但我有時候上課教授在問問題的時候我一臉茫然是因為我根本聽不懂教授的問題🤣。 要改進的地方硬要講的話，我會覺得每次上課的速度跟時間都不太一樣對我來說有點難適應，有時候前一堂課步調緩下來，下一堂課又突然暴衝，腦袋會大當機，然後有時候一口氣上個1小時，喔哇，那感覺就跟憋了一口氣憋了一小時一樣酸爽。 然後最後，我有個小建議，這堂課我個人認為開在下午會比較好，開在早上主要問題是剛睡醒或是精神不好，開在下午就沒有這個問題了 ( 不過這是我個人，可能不適用於每個人 ) 。不過考慮到教授實驗室的metting都在下午，我覺得有點難。 最後，還是感謝教授一個學期的教導，真的是受益良多。

老師喜歡把教學工作在一天的開端完成，這樣後續工作流程比較不容易中斷；換到下午時段是好的想法，未來會考慮！速度部分，有時候講的比較久是因為看到同學們的表情只好慢下來，畢竟是第一次用這課本，每節課都有其進度，比較沒辦法隨機應變，再教幾次應該就會變好了！

2. 最佳化這門課本身就不輕鬆，這個課本也是很難讀。我覺得自己看肯定很難看懂，或是需要花很久才能知道作者想要表達的意思，畢竟他很多時候都是講幾句話就過去了，不多花一點時間思考背後的意思可能沒有辦法學到東西。所以感謝老師選擇這個課本，並盡量用簡單的方式來讓我們了解課本的各種內容

謝謝同學有理解到老師選材的目的之一，在學校裡站在老師的肩膀上把一些相對困難的學問學起來可以省掉未來很多時間。

3. 老師上課到最後幾分鐘有時候會比較急，有點跟不上老師的速度，不過透過課本閱讀還是能掌握住內容。

我知道，但下節課通常老師也會再複習一下下；回頭看課本（本來就得做的事情）能解決就好，不能解決老師也很樂意回答同學的疑問。

4. 老師在課堂上會從不同角度，或透過一些例子帶大家了解課程內容，也幫助我們從幾何上理解。老師也常常補充課本推導中省略的細節，讓很多原本不容易理解的地方變得清楚許多。謝謝老師這一學期的用心教學，辛苦老師了。

謝謝同學的理解！

5. 非常感謝老師竭盡心力的教學！老師在課堂上經常帶領我們連結前面的章節，讓我更清楚看懂理論間的邏輯關聯。有了這層脈絡，我自己看書時更能有效補足細節。這跟我大四時獨自看影片、啃這本教科書的煎熬然後最後放棄的經歷，簡直是天壤之別。老師的課總是能多多少少激起學生對學習的熱情，讓人對這 一期一會 有深刻的印象。

這本書要自學有難度，不過看來同學有撐過去，以後再找些時間自己補足後續內容，有好的前五章基礎應該都過得去。如果還過不去，那就前五章內容再拿出來多複習幾次！

6. 1. 上課時長：就算老師偶爾會請假，但總是會再找時間補課，這樣學習到的內容是真的很多，但一次一個半小時完全是我的極限···只是延長上課時長好像也是補課最好的方法了。 2. 建議方向：有別於其他課程，老師是用手寫的板書，非常的能感覺到老師的熱誠，也會更加的集中於上課內容。只是這樣很高度的憑藉聽力在理解課程內容，像是在證明 SOCP -> SP 的時候不知道為甚麼突然冒出一個 Schur complement，後來整理筆記才知道是憑藉這個工具在做證明，我想說的是由於老師不可能將每一句話都寫在黑板上，動機、結論通常會是用口述的，黑板上留下的通常是證明過程，如果稍有恍神就會失去方向。所以或許在上課一開始時寫下流程會是不錯的方向嗎？因為感覺以投影片上課會讓人怠惰，也會失去上老師課的醍醐味 3. 謝謝老師在忙碌之中還是帶給我們這麼好的一堂課，這真的是我學到最多內容覺得最紮實的課程

上課時長的部分真的很抱歉，這學期外務確實比較多，時間掌控上有些地方做得不好，之後如果需要延長上課時間，老師會更確實安排並落實中間的休息時段。至於課程流程的部分，如果是證明細節，我自己覺得邊推導、邊說明會比較流暢；等證明完成後，再回頭審視整體結構，也會比較有感。不過在教學內容安排上，老師確實比較常用口述方式說明當天流程，之後會再多加留意，讓同學能更清楚掌握課程進行的脈絡。

7. 課程循序漸進，我認為這樣的上課方式已經很好

謝謝同學的回饋～

8. The design of this course is remarkably elegant. The progression from the foundational geometry of convex sets to the mechanics of duality, followed by practical numerical algorithms, provides a complete and cohesive intellectual arc. I deeply appreciated how the lectures balanced rigorous mathematical theory with diverse practical applications, spanning machine learning, signal processing, and network economics. The emphasis on Duality and Sensitivity Analysis was a highlight for me. Learning how to derive the dual problem completely transformed how I interpret bounded resources, shadow prices, and the mathematical boundaries of feasibility. The homework assignments were intensely challenging but serves as excellent vehicles for internalizing the text. While the theoretical foundation is robust, it could be highly beneficial to introduce a brief, dedicated recitation or seminar on Disciplined Convex Programming (DCP) rule-checking. Students sometimes struggle not with the math, but with understanding why a mathematically convex problem is rejected by modern solvers. A short deep-dive into how parsers decompose expressions into atom libraries would smooth out the learning curve for the computational assignments without diluting the theoretical rigor of the lectures. Generally, this has been one of the most intellectually transformative courses. It has permanently changed the way I formulate engineering and mathematical problems.

9. 私心覺得有課程ppt會更好

如果同學真的想要ppt，可以直接到Boyd的網站上下載課程投影片～那裡有幫你整理好的重點。老師的課程用板書有其理由，行之有年。老師的板書應該還沒有醜到沒辦法記錄的地步，而資訊的梳理必須要靠自已，大家各負責一些學習任務，這樣才能有效學習！

10. 以下純個人上課後觀感(建議)，沒有抱怨老師的上課方式，希望老師不要走心 >o< 1.感覺下半學期有些證明可以不用講，如SDP general form/standard form關聯性那邊，印象中這裏講蠻久的，但這章應該是要著重在怎麼化簡(技巧)、歸納，而且這是數學定義，一般來說不是都不會去證明定義嗎? 2.其實我整學期都蠻期待老師對我的筆記去做出回覆的，如哪裡定義這樣寫怪怪的，或是給一些筆記建議，這對我幫助才比較大，也可以更快釐清觀念，但老師好像都只回覆我對老師上課方式的反饋，有點小失望 3.為啥我們沒拍最後一堂的照片阿阿阿阿阿阿阿阿阿阿，大學回憶-1 by 時不時關注BIT LAB 網站最新動態的我 4.下半學期改成2周一次我覺得蠻好的，不然每周都要寫suggestion 有時候真的不知道要寫啥，不過後面是不是都沒出問答題了，這明明是很好幫我快速抓章節重點的東西啊... 5.希望畢業後還可以用老師的解憂雜貨店，應該遇不到這麼熱心想幫助人的老師了 T_T 6.很感謝能在大四的最後一個學期上這門課，這學期學到非常多東西，感謝每一個為這堂課付出的人，也感謝有撐到現在沒有退選的自己，終於可以大聲說出一拳超人經典名言，忍了一個學期。。。。。。

LP、QP、QCQP、SOP、SDP 都有其基本定義，但如果要證明它們萬本歸宗到 SDP，那就不是單靠定義可以解決的問題。那些證明過程其實就是在告訴你如何將 LP、QP、QCQP、SOP 轉換成 SDP，而這也正是你所說的技巧之一。 一般來說，老師會閱讀同學的筆記並給予回饋，但這學期真的有點忙。再者，老師原本要求的是「反思」，但很多同學其實是直接上傳上課筆記，這會大幅增加我的負擔。看過幾次、溝通幾次後，我發現真的不太行，因為太花時間、同學也沒有想改變，不過我記得我還是有看大約三週左右，也有給一些回饋。 沒有拍照是因為我不確定同學你的想法是什麼，但老師上到最後，其實真心覺得還願意理我的人可能不到三分之一，實在是有點心累。而且最後一堂課有不少同學缺課，課程也超時比較久，我看很多同學似乎都想離開了，所以最後就打消拍照的念頭。很抱歉，沒有能夠替同學們多留下一些回憶。 至於兩週一次的教學建議，我覺得是可以的；但如果是課程內容的確認，老師還是覺得每週一次會比較合適。解憂雜貨店隨時都可以來信。這世界上沒有過不去的坎，有時候把話說出口，心裡就會舒坦一些。哈，同學辛苦了！也謝謝你撐下來。你應該會感覺到自己的耐受度真的增加了不少惹～

11. 我覺得沒什麼好說的,只能怪我自己程度不好

課程一開始老師有說這門課不容易，不曉得同學是基於什麽情況在看了課本內容加上被老師「恐嚇」後還是想選這門課。你說的程度不好可能是老師的教學方式不適合你、可能是老師的解釋不易讓你明白，很多同學也都是努力撐著，撐過去就成長了！有問題多問才能減輕負擔啦。

12. 講得很好，課程安排上也很不錯，如果有問題應該是我自身的問題而不是老師的問題。

老師也是課堂上的一份子，如果課堂上沒有讓大家感到足夠自在、願意提出問題，老師需要負起比較多的責任。不過，學習和互動畢竟是雙向的，也希望同學們日後在其他課程中，遇到問題時不要害怕，仍然可以多多發問唷～

13. 老師上課非常認真與熱情，雖然是全英文但語言不會成為學習障礙，也會在課堂中拋出問題讓我們思考，讓上課不是一昧地被灌輸知識，而是跟著老斯的思維理解理念。課後透過feedback與繳交筆記，老師會適時給予回饋幫助大家。

謝謝同學的肯定，也很開心知道全英文授課並沒有成為你學習上的障礙。老師一直希望課堂不只是單向地傳遞知識，而是能夠讓大家一起思考、一起理解每個觀念背後的動機與脈絡。看到你感受到這一點，對老師來說是很大的鼓勵。

14. (1) Course Design: Course design is good. The course is well designed and covers important optimization concepts that are useful for both academic research and practical engineering applications. (2) Teaching Suggestions: Your teaching style is excellent. You teach with enthusiasm, energy, and dedication, which helps maintain students' interest and motivation throughout the semester. (3) Thoughts: The content of this course is mathematically intensive, and some concepts require time to fully understand. Therefore, I would highly appreciate it if the lectures could be recorded (video and audio). Recorded lectures would allow students to review difficult topics multiple times and learn at their own pace after class (4) Improvement: It would be helpful to include more practical examples and applications after each major topic. Real-world examples and step-by-step problem-solving demonstrations would help students better understand the theory and connect it to research and engineering applications. (5) It would also be beneficial to provide a list of prerequisite topics and recommended learning materials before the course begins. This would help students prepare themselves and gain the necessary background knowledge in advance.

15. 首先想感謝老師對這堂課的付出，感覺的到老師很用心並真心的希望我們都能在這堂課學到東西，有幾點是我這學期上完後的感想 : 1.老師會利用課後延長半小時來維持所預定的進度，但隨這課程後期逐漸變複雜可能比較難維持專注度，因此後面可能會走神之類的，所以我想建議老師如果在您時間允許下可以考慮用預錄的方式補課。 2.因為我習慣在課堂專心聽課，所以會先將課堂筆記拍起來回家整理，在整理的過程中也會藉由textbook及其他資料將筆記做的更完善，導致筆記會有些課程外的部分，每周的feedback雖然老師有明確表達希望我們提交對於本週課程的反思，但我還是期待老師能給一些關於筆記的建議，因此我都會交兩份檔案(一份筆記，一份類似反思的心得)，但可能老師這學期也比較忙沒時間執一回覆，這是我覺得小可惜的地方。 3.還有一個小小建議是希望老師能檢討考試題目及作業題目。 最後再次謝謝老師這學期的用心教導

對於延長時間和筆記的回饋部分可以看老師在第10位同學留言的回覆，老師日後會特別注意。檢討考試題目可以列入考慮，作業題目老師傾向同學們彼此多討論，然後有問題再來和老師商討（這是有原因的，不是老師偷懶...

16. 對初學者來說有點太深入

從上課的過程找到自己不理解的點往回追也是重要的事，很多時候並不是課程本身難，而是基礎薄弱，但關關難過關關過，補就是了！課程進行中多問多討論絕對是節省時間的不二法門。

17. 我盡力了，希望能通過QQ

雖然老師也是希望大家都能通過這門課，但（無意冒犯）你要試著接受這世界上有很多盡力也沒辦法如願的事情，這是事實（嘆），不過課程的配分設計應該還...可以吧...還是同學是研究生？如果是研究生，確實會辛苦一點 Orz

18. 也許是自己天份不夠也或許是不夠努力，但我覺得在準備期末時已盡了最大努力了，但考卷下來是還是一片空白，總覺得沒有看到我有讀過的題目，甚至在考前一週教授有提到會考第四單元60%的內容，因此我大概花了9成的心力去準備第四單元，甚至把原文書都看過去了，但考試看到題目時一直懷疑是自己有漏掉第四什麼內容嗎怎麼會幾乎沒有題目是能下手的，但總而言之，即便最後無法pass這堂課，但還是很感謝教授的認真教學，雖然在考試時幾乎都沒有會寫的題目，但至少對之後看論文會有幫助，謝謝這學期教授的用心！

不確定同學說的看過去是什麽情況，但這門課一定得透過平常的長時間積累去熟悉、深化內容，要求比較深的思考。老師知道這門課的負擔不輕，也一直鼓勵同學要發問來緩解壓力，不過看起來效果還是不太好 T_T。總之，老師確定你一定會帶走些什麽，而且保證很快你就會在論文上看到需要的內容，到時候需要時再回來翻翻課本，凡走過並留下痕跡的！第四章：Problems 1, 2 (10pts+10pts=20pts) +Problem 3 (12pts)+Problem 4 (25pts)=57pts，不算Bonus額外40分的話，這幾乎六成惹。

19. 我覺得這堂課受限於老師常常要因公請假的緣故，變成了每周四小時的課比較有點吃不消。因為主要都是板書為主，常常一不小心寫太慢就錯過口述上可能很重要的內容，回去看課本會卡卡，也要多花一些時間在補筆記上。然後考試部分就是很慘哪，一看到考卷題目就知道自己回去讀的都是寫特，對不起老師的苦心。但還是謝謝老師這學期的教導，雖然我可能學得很糟，好像要去掉可能，但透過這門課我也能對最佳化有基本認識，就看以後能不能有機會在研究上應用到。

課程時長的部分老師日後會再注意。考試只是為了測試自己還有哪些地方不理解，有空的時候打鐵趁熱，把課本和筆記再拿出來翻翻，會有意想不到的效果的，老師相信最佳化日後用到的機率非常高！

後記

謝謝同學們這學期給予的回饋。未來課程安排上，老師會更加注意上課節奏與時間控制，避免進度忽快忽慢，若因補課或進度需求需要延長上課時間，也會更確實安排中間休息，減少大家長時間維持高強度專注的負擔。由於這門課本身理論較重，加上採用EMI與板書教學，未來我也會在課堂開始時更明確說明當天的流程與重點，並在較長的證明或推導前後補充整體脈絡，讓大家比較不容易在過程中迷失方向。課後回饋部分，也會再調整反思與筆記繳交的方式，希望讓同學能更清楚提出自己不理解的地方，也讓老師能給予更有效的回應。此外，老師會考慮增加部分實作說明、章節重點問題、考試或作業檢討，以及先備知識與補充資源，幫助大家在理論、應用與自學之間建立更好的連結。整體而言，這門課仍會維持原本重視數學基礎、幾何直覺與工程應用的方向，但未來會努力在課程結構、學習節奏與回饋機制上做得更清楚、更穩定，讓同學在面對困難內容時能更知道自己正在學什麼、卡在哪裡，以及該如何繼續往前走。