4.10.18 · D1First-order optimality conditions — gradient = 0
Kisi smooth landscape par pahad ki choti ya ghati ka sabse nichla point dhundhne ke liye, aap woh ek jagah dhundhte hain jahan zameen har direction mein bilkul flat ho. Poora topic bas yahi machinery hai jo measure karta hai "yahan zameen kitni tilted hai?" — aur woh measuring tool hai gradient , jo kisi bhi interior peak ya pit par zero ke barabar hona chahiye.
Parent note padhne se pehle, aapko uske symbols padhne aane chahiye. Neeche har ek notation ka tukda aur har ek idea hai jo woh silently assume karta hai, ek ek karke banaya gaya hai — har ek earn kiya gaya hai iska use hone se pehle. Yahan kuch bhi yeh assume nahi karta ki aapne pehle calculus notation dekhi hai.
f f rule hai jo numbers khata hai aur ek number ugalta hai . Hum likhte hain f ( x ) x f
Ek landscape ki picture banao. Aap kahi khade ho, aur machine aapko samundar ki surface se aapki height batati hai. Ek variable mein, input ek path par aapki position hai; do variables mein, ek map par aapki position.
Parent note ko f
Definition Number line aur uske bade cousins
R tamam real numbers ka set — ek infinite number line par har point, fractions aur decimals samete.R n n n-dimensional space mein ek point . R 2 x , y R 3
Arrow notation f : R n → R f n
Intuition Topic yeh is tarah kyun likhta haiParent statement ko kisi bhi number of dimensions mein ek valley ek saath cover karni hai. R n → R
Reveal check:
f : R 2 → R ek point ( x , y )
Definition Star ka matlab hai "jawab"Plain x x ∗ specific, distinguished input hai — woh location jahan optimum hai jise hum describe karne ki koshish kar rahe hain. Star bas ek name-tag hai jo kehta hai "yeh wala special hai."
Ek valley ki picture mein, x ∗ floor hai: sabse nichle point ke seedha neeche horizontal position.
Definition Teen location words
Local minimum : ek aisa point jo sirf apne nearby neighbours se compare karte hue sabse nichla ho — poori duniya mein nahi. Ek hillside par ek dimple bhi count hota hai.Interior point : ek aisa point jahan har direction mein move karne ki jagah ho — aap kisi bhi axis par thoda + − Boundary point : allowed region ki edge par ek point , jahan kam se kam ek direction blocked ho.
Intuition Topic itni parwah kyun karta haiPoora proof dono taraf step lene par nirbhar karta hai. Yeh sirf ek interior point par hi possible hai. Yahi wajah hai ki parent ka "I forgot the boundary" wali warning exist karti hai — flat-ground argument silently assume karta hai ki aap kisi wall ke khilaaf dabaye nahi ho.
Reveal check:
x ∗ Aapko + −
h h tiny nudge hai kisi point se door: x ∗ x ∗ + h h
Definition Limit kya hai, words meinlim h → 0 ( expression ) h par value nahi (jo shayad undefined ho).
Intuition Shrinking ke liye ek tool kyun chahiyeCurve ka slope badhta rehta hai. Slope ko bilkul ek point par pin down karne ke liye, hum ek tiny segment ka slope dekhte hain aur segment ko zero tak shrink karte hain. Limit woh machine hai jo us shrink ko cleanly perform karti hai.
Chota superscript matter karta hai:
h → 0 + h positive side se shrink karo" (right se approach karo).h → 0 − negative side se " (left se approach karo).
Parent ka proof dono sides se alag alag approach karta hai, isliye yeh + / −
Definition Derivative = slopef ′ ( x ) x f slope hai — height kitni steeply rise ya fall karti hai agar aap right ki taraf ek infinitesimal step lo.
f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) .
Fraction ko tod do, kyunki har piece ek picture hai:
Yeh tool kyun, koi aur kyun nahi?
Hum jaanna chahte hain "downhill kaun si direction hai aur kitna steep?" Derivative bilkul wahi jawaab hai: positive f ′ negative ka matlab right ki taraf downhill, aur zero ka matlab flat. Yahi flatness hai jiske peeche yeh topic bhaag raha hai.
Sign meaning (teeno cases):
f ′ ( x ) > 0 f ′ ( x ) < 0 f ′ ( x ) = 0 x 3
Definition Ek ko chod kar sab freeze karoJab f ∂ x i ∂ f x i ∂ d
Intuition Humein kayi kyun chahiye2-D landscape par "the slope" ambiguous hai — north-to-south steep, maybe east-to-west flat. Har axis ke liye ek partial mein split karne se hum ek clean direction mein tilt measure kar sakte hain, phir reassemble kar sakte hain.
Reveal check:
Curly ∂ d Ki doosre input variables constant rakhe ja rahe hain jabki hum ek axis ke saath differentiate kar rahe hain.
Definition Ek vector, seedha seedhaVector ek arrow hai jiske paas length aur direction hoti hai . Coordinates mein hum iske parts ko column mein stack karte hain, e.g. ( 3 − 2 )
Gradient ∇ f f f ∇ f = ∂ x 1 ∂ f ⋮ ∂ x n ∂ f . arrow hai jo steepest increase ki direction mein point karta hai , aur uski length batati hai woh steepest direction kitni steep hai.
Intuition Gradient poore topic ka star kyun haiAgar koi bhi direction uphill hai, toh gradient arrow us taraf point karta hai aur uski non-zero length hai — isliye aap uske khilaaf chalkaar improve kar sakte ho. Ek maatra tarika hai ki koi direction uphill na ho, woh yeh hai ki arrow ki length zero ho: ∇ f = 0
Bold zero 0 vector == — zero length ka arrow, kahi point nahi karta — plain number 0 ∇ f ( x ∗ ) = 0 = 0
Poori construction ke liye dekho Gradient and directional derivatives .
Unit vector u bilkul 1 length ho. Yeh ek pure direction carry karta hai bina magnitude ke, isliye yeh jawaab deta hai "kaun si taraf?" bina yeh bhi jawaab diye "kitna door?"
Dot product a ⋅ b woh kitni same direction mein point karte hain . Same direction → bada positive; perpendicular → zero; opposite → negative.
Definition Directional derivativeD u f ( x ∗ ) = ∇ f ( x ∗ ) ⋅ u u
Intuition Yeh sabse gehri proof kyun deta haiDirection u khilaaf point kare: u = − ∇ f / ∥ ∇ f ∥ ∥ ⋅ ∥ vector ki length . Tab D u f = − ∥ ∇ f ∥
Reveal check:
D u f ( x ∗ ) ∇ f u Dot product ∇ f ( x ∗ ) ⋅ u
Definition Stationary (critical) pointKoi bhi input jahan ∇ f = 0 candidate hai — flat ground, wajah unknown.
Definition Hessian (yahan naam diya, baad mein padhenge)Hessian second derivatives ki table hai — surface ki "curvature." Yeh max ko min se aur saddle se alag karta hai baad mein jab gradient ne ek flat point shortlist kar liya ho. Poori detail Hessian matrix and second-order conditions mein; jis trap ko yeh resolve karta hai woh Saddle points mein hai.
Function f: input to height
Derivative f prime: slope in 1D
Partial derivative: slope along one axis
Vector: arrow with length and direction
Gradient: all slopes in one arrow
Unit vector: pure direction
Interior point: room both ways
Gradient = zero condition
Hessian: classify the candidate
Upar har ek foundation ek prerequisite arrow hai jo parent mein feed karta hai: the topic note .
Har line ko ek question ki tarah padho; parent note par sirf tabhi jao jab aap inhe sab answer kar sako.
Main f : R n → R f n
Main 0 0 0 0 vector hai — zero length ka arrow.
Main ek interior point describe kar sakta/sakti hoon aur kyun topic ko iska zaroorat hai. Ek aisa point jahan har direction mein step lene ki jagah ho; proof ko + h − h
Main lim h → 0 + lim h → 0 − Zero ko positive (right) side se vs negative (left) side se approach karo.
Main bata sakta/sakti hoon ki derivative kya measure karta hai. Ek point par curve ka slope — rise over run jaise run zero tak shrink ho.
Main jaanta/jaanti hoon curly ∂ Ek partial derivative — ek axis ke saath slope jabki doosre variables fixed rakhe gaye hain.
Main bata sakta/sakti hoon gradient ∇ f Woh arrow jo steepest increase ki direction mein point karta hai; uski length woh steepness hai.
Main dot product ko ek sentence mein define kar sakta/sakti hoon. Ek number jo measure karta hai ki do vectors kitni same direction mein point karte hain.
Main directional derivative ko formula mein likh sakta/sakti hoon. D u f ( x ∗ ) = ∇ f ( x ∗ ) ⋅ u
Main jaanta/jaanti hoon stationary point kya hota hai aur ki woh sirf ek candidate hai. Ek point jahan ∇ f = 0
