Foundations — Lagrange multipliers — one and two constraints

4.4.15 · D1 · Maths › Multivariable Calculus › Lagrange multipliers — one and two constraints

Is page pe koi assumption nahi hai. $\nabla f=\lambda \nabla g$ padhne se pehle, tumhe yeh jaanna zaroori hai ki har squiggle kya hai aur kaisa dikhta hai. Hum unhe ek-ek karke, ek ke upar doosra, banate hain.

1. Points, functions, aur "ek landscape"

Picture: graph paper pe ek dot (2D) ya ek room mein floating (3D). Bas itna hi.

Picture: $f(x,y)$ ko floor position $(x,y)$ ke upar ek hill ki altitude samjho. Har floor spot ko ek height milti hai. Yahi woh "landscape" hai jiska hum baar baar zikr karte hain.

Topic ko yeh kyun chahiye: poora game hai "$f$ ko jitna ho sake bada ya chhota karo." Agar value assign karne wala function hi nahi hoga toh optimize karne ke liye kuch hoga hi nahi.

2. Level sets — hill ko bina 3D ke padhna

Ek 3D hill ke andar khadhe rehna draw karna mushkil hai. Toh hum use flatten karte hain.

Picture: ek hilltop ke around rings. Har ring hai "exactly 500 m wali saari jagahein," agla ring "exactly 600 m," aur aise aage. Rings paas paas = steep; rings door door = gentle.

Topic ko yeh kyun chahiye: parent page ka climax yeh hai ki optimum pe constraint path, $f$ ke ek level curve ke saath tangent hoti hai. Jab tak level curve kya hoti hai yeh nahi pata, yeh nahi dikh sakta. Dekho Level sets and contours.

3. Gradient $\nabla f$ — steepest climb ka arrow

Yeh star symbol hai.

Yeh tool kyun, plain derivative kyun nahi? Plain derivatives ek input ke liye hoti hain. Hamara landscape kai inputs $(x,y,z,\dots )$ wala hai; partial derivative hamen alag alag poochne deti hai "is ek direction mein change ki rate kya hai," jo hum phir bundle karte hain.

Do baatein jo tumhe sirf padhni nahi, feel karni hain:

1. $\nabla f$ steepest ascent ki direction mein point karta hai — seedha uphill. Iski length batati hai kitna steep hai.
2. $\nabla f$ us point ke through level curve ke perpendicular hai. (Uphill, rings ke 90° across hai, kabhi unke saath nahi.)

Topic ko yeh kyun chahiye: $\nabla f$ literally hai "kaunsi taraf $f$ sabse tezi se badhta hai aur kitni tezi se." Optimization ka matlab hai us increase ko allowed directions mein khatam karna. Poori detail Gradient and directional derivative mein hai.

4. Constraints $g=0$ — woh fence jis pe rehna zaroori hai

Picture: hill ke floor pe ek fence khinchi hui. Tum sirf fence pe chal sakte ho. $g(x,y)=x^2+y^2-1$ ke saath $g=0$ ka matlab hai "radius 1 ke circle pe raho."

Topic ko yeh kyun chahiye: "constrained optimization" = fence pe sabse acchi height, kahin bhi sabse acchi height nahi. Fence hi poora reason hai kyun ordinary calculus (wahan dhundho jahan $\nabla f=0$) kaafi nahi hai.

5. Tangent aur normal — "path ke saath" vs "path ke across"

Yeh kehne ke liye ki "$f$ path ke saath badhna band kar deta hai," humein do directions clearly alag karne hain.

Picture: fence pe khade ho. "Fence ke saath aage" = tangent $\mathbf{T}$. "Sideways fence ke upar step karo" = normal $\nabla g$. Details: Tangent planes and normal vectors.

Topic ko yeh kyun chahiye: poori derivation yeh hai "best point pe, $\nabla f$ mein koi tangent part nahi hota — woh poori tarah normal $\nabla g$ ke saath rehta hai." Yeh sentence tab tak meaningless hai jab tak tangent aur normal clearly define na ho jaayein.

6. Dot product — "along" ko precisely measure karna

Yeh tool kyun, koi aur kyun nahi? Hum baar baar poochte hain "kya $\nabla f$ ka path direction ${\mathbf{r}}^{\prime }$ ke saath koi component hai?" Dot product $\nabla f\cdot {\mathbf{r}}^{\prime }$ precisely hai "kitna $\nabla f$, ${\mathbf{r}}^{\prime }$ ki taraf point karta hai." Ise zero set karna kehta hai "bilkul nahi." Isliye parent ka Step 2 hai $\nabla f\cdot {\mathbf{r}}^{\prime }=0$.

7. Chain rule — chalte waqt height kaise change hoti hai

Topic ko yeh kyun chahiye: yeh bridge hai. "$f$ path ke saath change karna band kar deta hai" hai $\frac{d}{dt}f(\mathbf{r}(t))=0$; chain rule ise $\nabla f\cdot {\mathbf{r}}^{\prime }=0$ mein badal deti hai, jo dot product padhta hai as "$\nabla f\perp$ har tangent." Yahi woh tarika hai jisse hum conclude karte hain ki $\nabla f$ normal hona chahiye.

8. Multiplier $\lambda$ — "kitna parallel" ke liye ek single knob

Picture: $\nabla f$ aur $\nabla g$ do arrows hain jo optimum pe planted hain, ek hi line pe lie karte hain. $\lambda$ hai "kitne $\nabla g$ milake ek $\nabla f$ banta hai" — unki lengths ka ratio (ek sign ke saath agar woh opposite taraf point karein).

• $\lambda >0$: $\nabla f$ aur $\nabla g$ ek hi taraf point karte hain.
• $\lambda <0$: woh opposite taraf point karte hain (phir bhi ek hi line).
• $\lambda =0$: $\nabla f=0$ — constraint koi kaam nahi kar raha; yeh ek unconstrained critical point hai. Dekho Unconstrained optimization — critical points.

Topic ko yeh kyun chahiye: yeh geometric statement "parallel" ko ek solvable equation mein convert karta hai. Aur, jaise Dual problem and shadow prices dikhata hai, $\lambda =d{f}^{\ast }/dc$ measure karta hai ki fence ko nudge karne se best value kitni improve hoti hai.

9. Do constraints — doosra multiplier $\mu$ kyun aata hai

Do fences $g=0$ aur $h=0$ ke saath, tum sirf wahan khade ho sakte ho jahan woh cross karti hain: intersection curve. Us single curve ke saath ek tangent $\mathbf{T}$ hoti hai, lekin do normals $\nabla g$ aur $\nabla h$ poora "sideways" plane span karte hain. $\nabla f$ mein koi along-curve part na ho, iske liye usse us plane mein rehna hoga — dono normals ka blend: $\nabla f=\lambda \nabla g+\mu \nabla h.$ Doosra scalar $\mu$ doosra multiplier hai — ek fence ke liye ek knob. Yeh KKT conditions mein inequality versions se aage connect karta hai.

Prerequisite map

Equipment checklist

Khud ko test karo — parent derivation padhne se pehle har sawaal ka jawab dena aana chahiye.