ISYEl 6669l Finall Examl (Latestl 2025/l 2026l Update)l Review|l Q/Al |l Gradel A|l 100%l Correctl (Verifiedl Answers)
Q:l Whatl isl thel optimalityl conditionl forl thel leastl squaresl probleml inl matrixl form?l Whatl doesl thel optimalityl conditionl telll us?l Whatl isl anotherl namel forl thel optimalityl condition?
Answer:
min(Ax-b)^t(Ax-b)l =l x^tA^TAxl -l 2(A^tb)^tx+b^tbl andl thenl takingl thel gradientl ofl thatl functionl wel getl A^tAxl =l A^tbl sol anyl solutionl tol thatl optimalityl conditionl isl anl optimall solutionl tol thel leastl squaresl problem.l Thisl isl calledl thel normall equation.
Q:l Whyl dol wel needl tol solvel thel normall equationl forl Ax=b?
Answer:
Becausel Al hasl morel rowsl thanl columns,l wel can'tl findl al uniquel solutionl forl thel systeml ofl equations.l Becausel Al hasl fulll columnl rankl (justl notl fulll rowl rank),l A^tAl isl squarel andl invertible.l Sol thenl A^tAxl =l A^tbl canl bel rewrittenl asl xl =l (A^tA)^-1A^tb
Q:l Whatl isl (A^tA)^-1A^tl called?
Answer:
Itl isl calledl thel Moore-Penrosel pseudoinversel andl isl givenl byl A^+.l Itl isl notl quitel thel inversel ofl Al butl measurablel close.l Thenl thel solutionl tol thel leastl squaresl probleml isl xl =l A^+bl (ifl Al isl notl invertible).l Al isl notl invertiblel whenl therel arel morel rowsl thanl columns.
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Q:l Whatl isl thel motivatingl factorl forl usingl SVD?
Answer:
Matrixl inversionl isl veryl computationallyl expensive.l A^tAl =l V(Sigma)^2V^tl andl invertingl thisl isl muchl easier.l Sigmal isl al diagonall matrixl withl eigenvaluesl ofl Al sol inversionl ofl thatl isl easyl =l V(Sigma)^-2V^t
Q:l Whatl isl thel meaningfull partl aboutl thel LPl standardl form?
Answer:
Thel non-negativityl constraintsl forml thel polyhedronl andl allowl solutionsl throughl thel simplexl method.
Q:l Whatl isl thel definitionl ofl al convexl cone?
Answer:
axl isl inl Kl forl alll al >=l 0l wheneverl xl isl inl K.l Thel non-negativel orthantl isl al convexl cone.
Q:l Whatl isl anl orderl andl howl doesl itl relatel tol inequalitiesl andl LP?
Answer:
al >=l bl isl a-bl >=l 0.l Thisl isl statingl thatl a-bl isl inl thel nonl negativel orthant,l whichl isl al convexl cone.l Sol inl thel non-negativel orthantl cone,l a-bl >=0.l Thisl canl bel generalizedl tol differentl conesl forl LP.
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