Optimization Problems, Duality, and $n$-ality
Optimization Problems, Duality, and
n-ality
If we consider an optimization problem in $\mathbb R^n$ with a nonlinear objective function $f:\mathbb R^n \to \mathbb R$ and a constraint set $\mathcal C_c := \{x\in\mathbb R^n:g(x)=c\}$ for a nonlinear function $g:\mathbb R^n\to\mathbb R$, then we have a particularly clear form of duality
If we consider an optimization problem in
Rn with a nonlinear objective function
f:Rn→R and a constraint set
Cc:={x∈Rn:g(x)=c} for a nonlinear function
g:Rn→R, then we have a particularly clear form of duality
AI Individuality, and One-Time Signatures
AI Individuality, and One-Time Signatures
Extremal Distance and Duality
Extremal Distance and Duality
If we just look at the level lines of $f$ and the level lines of $g$, then it is clear (in a non-degenerate situation) that any minimum of $f$ on $\mathcal C_c$ corresponds to an intersection of a level line of $f$ and a level of $g$
If we just look at the level lines of
f and the level lines of
g, then it is clear (in a non-degenerate situation) that any minimum of
f on
Cc corresponds to an intersection of a level line of
f and a level of
gAnd basically the Lagrange multipliers situation is very clear
And basically the Lagrange multipliers situation is very clear
A natural question in my opinion is whether if we have a constraint set made of two constraints $\mathcal C_{c_1,c_2}:=\{x\in \mathbb R^n:g_1(x)=c_1,g_2(x)=c_2\}$,
A natural question in my opinion is whether if we have a constraint set made of two constraints
Cc1,c2:={x∈Rn:g1(x)=c1,g2(x)=c2},
If we were to consider perhaps sublevel sets $\mathcal S_c:=\{x\in\mathbb R^n:g(x)\leq c\}$, then we can really say that we can either try to fix a maximum level that we accept for $f$ and then try to lower $c$ as much as we can, or fix $c$ and find the minimum for $f$; and there is a (simple) form of duality here
If we were to consider perhaps sublevel sets
Sc:={x∈Rn:g(x)≤c}, then we can really say that we can either try to fix a maximum level that we accept for
f and then try to lower
c as much as we can, or fix
c and find the minimum for
f; and there is a (simple) form of duality here
But it is not entirely clear that this is the same as the standard duality, which exchanges variables and constraints; in the case of standard duality, the variable of the problem will be associated with the constraint associated with $g$... is the duality as nicely about exchanging $f$ and $g$?
But it is not entirely clear that this is the same as the standard duality, which exchanges variables and constraints; in the case of standard duality, the variable of the problem will be associated with the constraint associated with
g... is the duality as nicely about exchanging
f and
g?
This becomes more obvious in the case of several constraints:
This becomes more obvious in the case of several constraints:
The symmetry of $f$ and $g$ is much cleaner if we say we just consider the set of solutions to all the optimization problems with all the constraints $c$ that can be; that set is in clear correspondence with
The symmetry of
f and
g is much cleaner if we say we just consider the set of solutions to all the optimization problems with all the constraints
c that can be; that set is in clear correspondence with
.