Maths WorkshopMaths problems with full solutions

Optimisation in several variables

30 problems · Extrema and Lagrange multipliers.

Ex.435Critical points of a polynomial
#critical points #polynomial #Hessian
Optimisation in several variables
L1
L1
Ex.436Summary of constrained optimization
#summary #constrained optimization
Optimisation in several variables
High school
High school
Ex.733Lagrange multipliers on a circle
#Lagrange #multipliers #equality constraint
Optimisation in several variables
L2
L2
Ex.734Optimisation under a linear constraint
#Lagrange #linear constraint #minimum
Optimisation in several variables
L2
L2
Ex.857Boundary classification
#boundary #classification #circle
Optimisation in several variables
L1
L1
Ex.858Distance from point to plane
#Lagrange #distance #plane
Optimisation in several variables
L1
L1
Ex.859Two critical points and their nature
#critical points #saddle #maximum
Optimisation in several variables
L1
L1
Ex.860Lagrange multipliers on a circle
#Lagrange #circle #extrema
Optimisation in several variables
L1
L1
Ex.861Optimization under a linear constraint
#Lagrange #linear constraint #minimum
Optimisation in several variables
L1
L1
Ex.862Maximizing a product
#Lagrange #product #constraint
Optimisation in several variables
L1
L1
Ex.863A unique critical point that is a saddle
#saddle #unique critical #Hessian
Optimisation in several variables
L1
L1
Ex.1081Saddle point
#saddle point #Hessian #classification
Optimisation in several variables
L1
L1
Ex.1475Cauchy-Schwarz inequality via optimisation
#Cauchy-Schwarz #optimisation #proof
Optimisation in several variables
Olympiad
Olympiad
Ex.1476Constrained extrema and a compactness argument
#Lagrange #compactness #global maximum
Optimisation in several variables
L3
L3
Ex.1477Maximizing a product under a linear constraint
#Lagrange #product #linear constraint
Optimisation in several variables
L3
L3
Ex.1478Least squares as an optimisation problem
#least squares #optimisation #regression
Optimisation in several variables
L3
L3
Ex.1591Cauchy–Schwarz via optimization
#Cauchy-Schwarz #optimization #inequality
Optimisation in several variables
L1
L1
Ex.1592Optimization under inequality constraints
#inequality constraint #KKT #boundary
Optimisation in several variables
L1
L1
Ex.1593Classifying a 3D critical point
#Hessian #3D #saddle
Optimisation in several variables
M1
M1
Ex.1594Three-variable saddle point
#three variables #saddle #Hessian
Optimisation in several variables
L1
L1
Ex.1595Shortest distance from a point to a line
#Lagrange #distance #line
Optimisation in several variables
L1
L1
Ex.1596Distance from origin to an ellipsoid
#Lagrange #ellipsoid #distance
Optimisation in several variables
L1
L1
Ex.1597Lagrange with exponential objective
#Lagrange #exponential #constrained maximum
Optimisation in several variables
L1
L1
Ex.1598Global extrema on a compact set
#compactness #global extrema #boundary
Optimisation in several variables
L1
L1
Ex.1599Maximizing a volume with fixed perimeter
#Lagrange #geometric optimization #volume
Optimisation in several variables
L1
L1
Ex.1600Constrained quadratic form
#quadratic form #Rayleigh quotient #eigenvalues
Optimisation in several variables
M1
M1
Ex.1796The Hessian matrix
#Hessian #second derivatives #quadratic form
Optimisation in several variables
L1
L1
Ex.2070Rayleigh quotient and eigenvalues
#Rayleigh quotient #eigenvalues #optimisation on the sphere
Optimisation in several variables
Olympiad
Olympiad
Ex.2071The AM-GM inequality via optimisation
#inequality #AM-GM #optimisation #proof
Optimisation in several variables
Olympiad
Olympiad
Ex.2115Two equality constraints
#Lagrange #multiple constraints #KKT
Optimisation in several variables
M1
M1