A constrained maximization problem for a general function $f:left(a,bright)^n to mathbb{R}$
$begingroup$
Let $a,b in mathbb{R}$ with $a<b$ and $f:left(a,bright)^n to mathbb{R}$ be a (possibly non-linear) function. I want to maximize $f$ on the region :
$$S=left{left(x_1,cdots,x_nright) in left(a,bright)^n : x_1 leq cdots leq x_nright}$$
i.e., I want to find :
$$max_{left(x_1,cdots,x_nright) in S}{fleft(x_1,cdots,x_nright)}$$
Question : What are some general methods to work with such maximization problems?
What I know : In my multivariable calculus course, I have learnt about unconstrained maxima problems (with the second derivative tests, i.e. taking gradient of $f$ to be $0$ and showing that the hessian matrix at the critical point to be negative definite and so on). Also, I have learnt about constrained maxima problems with lagrange multiplier method.
I think that one needs some other methods to solve such irregular ($S$ looks pretty irregular to me!) optimization problem which I have not learnt yet. If anyone knows any method to tackle this and can share that, it would be great. Thank you.
real-analysis multivariable-calculus optimization maxima-minima
asked Dec 27 '18 at 9:59
DragonDragon
598215
$endgroup$
add a comment |
$begingroup$
Let $a,b in mathbb{R}$ with $a<b$ and $f:left(a,bright)^n to mathbb{R}$ be a (possibly non-linear) function. I want to maximize $f$ on the region :
$$S=left{left(x_1,cdots,x_nright) in left(a,bright)^n : x_1 leq cdots leq x_nright}$$
i.e., I want to find :
$$max_{left(x_1,cdots,x_nright) in S}{fleft(x_1,cdots,x_nright)}$$
Question : What are some general methods to work with such maximization problems?
What I know : In my multivariable calculus course, I have learnt about unconstrained maxima problems (with the second derivative tests, i.e. taking gradient of $f$ to be $0$ and showing that the hessian matrix at the critical point to be negative definite and so on). Also, I have learnt about constrained maxima problems with lagrange multiplier method.
I think that one needs some other methods to solve such irregular ($S$ looks pretty irregular to me!) optimization problem which I have not learnt yet. If anyone knows any method to tackle this and can share that, it would be great. Thank you.
real-analysis multivariable-calculus optimization maxima-minima
asked Dec 27 '18 at 9:59
DragonDragon
598215
$endgroup$
$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38
add a comment |
$begingroup$
Let $a,b in mathbb{R}$ with $a<b$ and $f:left(a,bright)^n to mathbb{R}$ be a (possibly non-linear) function. I want to maximize $f$ on the region :
$$S=left{left(x_1,cdots,x_nright) in left(a,bright)^n : x_1 leq cdots leq x_nright}$$
i.e., I want to find :
$$max_{left(x_1,cdots,x_nright) in S}{fleft(x_1,cdots,x_nright)}$$
Question : What are some general methods to work with such maximization problems?
What I know : In my multivariable calculus course, I have learnt about unconstrained maxima problems (with the second derivative tests, i.e. taking gradient of $f$ to be $0$ and showing that the hessian matrix at the critical point to be negative definite and so on). Also, I have learnt about constrained maxima problems with lagrange multiplier method.
I think that one needs some other methods to solve such irregular ($S$ looks pretty irregular to me!) optimization problem which I have not learnt yet. If anyone knows any method to tackle this and can share that, it would be great. Thank you.
real-analysis multivariable-calculus optimization maxima-minima
asked Dec 27 '18 at 9:59
DragonDragon
598215
$endgroup$
Let $a,b in mathbb{R}$ with $a<b$ and $f:left(a,bright)^n to mathbb{R}$ be a (possibly non-linear) function. I want to maximize $f$ on the region :
$$S=left{left(x_1,cdots,x_nright) in left(a,bright)^n : x_1 leq cdots leq x_nright}$$
i.e., I want to find :
$$max_{left(x_1,cdots,x_nright) in S}{fleft(x_1,cdots,x_nright)}$$
Question : What are some general methods to work with such maximization problems?
What I know : In my multivariable calculus course, I have learnt about unconstrained maxima problems (with the second derivative tests, i.e. taking gradient of $f$ to be $0$ and showing that the hessian matrix at the critical point to be negative definite and so on). Also, I have learnt about constrained maxima problems with lagrange multiplier method.
I think that one needs some other methods to solve such irregular ($S$ looks pretty irregular to me!) optimization problem which I have not learnt yet. If anyone knows any method to tackle this and can share that, it would be great. Thank you.
real-analysis multivariable-calculus optimization maxima-minima
real-analysis multivariable-calculus optimization maxima-minima
asked Dec 27 '18 at 9:59
DragonDragon
598215
asked Dec 27 '18 at 9:59
DragonDragon
598215
asked Dec 27 '18 at 9:59
DragonDragon
598215
asked Dec 27 '18 at 9:59
DragonDragon
598215
asked Dec 27 '18 at 9:59
DragonDragon
598215
598215
$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38
add a comment |
$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38
$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38
add a comment |
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$begingroup$
Without further assumptions on $f$ you should not expect any general method to work. Just imagine $f$ being a constant function almost everywhere. You need some property such as concavity or Lipschitz continuity.
$endgroup$
– LinAlg
Dec 27 '18 at 16:40
$begingroup$
@LinAlg : What are some results with added conditions on f?
$endgroup$
– Dragon
Dec 27 '18 at 18:20
$begingroup$
Concavity: first-order methods, (quasi)newton-based methods, ADMM. Lipschitz continuity: global optimization.
$endgroup$
– LinAlg
Dec 27 '18 at 18:38