How to write Frank-Wolfe algorithm in two steps optimization problems?
$begingroup$
Let $g(w)$ be a differentiable convex function. Frank-Wolfe algorithm over a convex set $C in mathbb{R}^n$ is defined so as to find the local minimum of the function:
$$
s_{t+1}=argmin_{s in C} langle s, nabla g(w_t) rangle tag{1}
$$
$$
w_{t+1} = (1-eta_t)w_{t}+eta_ts_t tag{2}
$$
where $w_0 in C$ is the starting point, $eta_t$ is the step size in $[0,1]$, and $t=1,2,cdots,T$.
We are done for $(1)$ because it is posed as an optimization problem.
How can we solve for $(2)$.
optimization convex-optimization numerical-optimization
asked Jan 18 at 0:53
SaeedSaeed
1,149310
$endgroup$
add a comment |
$begingroup$
Let $g(w)$ be a differentiable convex function. Frank-Wolfe algorithm over a convex set $C in mathbb{R}^n$ is defined so as to find the local minimum of the function:
$$
s_{t+1}=argmin_{s in C} langle s, nabla g(w_t) rangle tag{1}
$$
$$
w_{t+1} = (1-eta_t)w_{t}+eta_ts_t tag{2}
$$
where $w_0 in C$ is the starting point, $eta_t$ is the step size in $[0,1]$, and $t=1,2,cdots,T$.
We are done for $(1)$ because it is posed as an optimization problem.
How can we solve for $(2)$.
optimization convex-optimization numerical-optimization
asked Jan 18 at 0:53
SaeedSaeed
1,149310
$endgroup$
add a comment |
$begingroup$
Let $g(w)$ be a differentiable convex function. Frank-Wolfe algorithm over a convex set $C in mathbb{R}^n$ is defined so as to find the local minimum of the function:
$$
s_{t+1}=argmin_{s in C} langle s, nabla g(w_t) rangle tag{1}
$$
$$
w_{t+1} = (1-eta_t)w_{t}+eta_ts_t tag{2}
$$
where $w_0 in C$ is the starting point, $eta_t$ is the step size in $[0,1]$, and $t=1,2,cdots,T$.
We are done for $(1)$ because it is posed as an optimization problem.
How can we solve for $(2)$.
optimization convex-optimization numerical-optimization
asked Jan 18 at 0:53
SaeedSaeed
1,149310
$endgroup$
Let $g(w)$ be a differentiable convex function. Frank-Wolfe algorithm over a convex set $C in mathbb{R}^n$ is defined so as to find the local minimum of the function:
$$
s_{t+1}=argmin_{s in C} langle s, nabla g(w_t) rangle tag{1}
$$
$$
w_{t+1} = (1-eta_t)w_{t}+eta_ts_t tag{2}
$$
where $w_0 in C$ is the starting point, $eta_t$ is the step size in $[0,1]$, and $t=1,2,cdots,T$.
We are done for $(1)$ because it is posed as an optimization problem.
How can we solve for $(2)$.
optimization convex-optimization numerical-optimization
optimization convex-optimization numerical-optimization
asked Jan 18 at 0:53
SaeedSaeed
1,149310
asked Jan 18 at 0:53
SaeedSaeed
1,149310
edited Feb 1 at 18:54
Saeed
asked Jan 18 at 0:53
SaeedSaeed
1,149310
asked Jan 18 at 0:53
SaeedSaeed
1,149310
asked Jan 18 at 0:53
SaeedSaeed
1,149310
1,149310
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Try the following optimization:
$$
w_{t+1}=argmin_{w in C} -beta_tlangle w,s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2 tag{3}
$$
where $beta_t$ and $gamma_t$ are positive.
Since the objective, i.e., $f(w) = -beta_tlangle w,(s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2$ is convex, the necessary and sufficient condition in order $w_{t+1}$ be the minimizer of $(3)$ is the following:
$$
langle -beta_t (s_t -w_t) + gamma_t (w-w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
$$
langle w - ( frac{beta_t}{gamma_t}s_t + (1-frac{beta_t}{gamma_t})w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
answered Feb 1 at 18:56
SepideSepide
5048
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Try the following optimization:
$$
w_{t+1}=argmin_{w in C} -beta_tlangle w,s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2 tag{3}
$$
where $beta_t$ and $gamma_t$ are positive.
Since the objective, i.e., $f(w) = -beta_tlangle w,(s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2$ is convex, the necessary and sufficient condition in order $w_{t+1}$ be the minimizer of $(3)$ is the following:
$$
langle -beta_t (s_t -w_t) + gamma_t (w-w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
$$
langle w - ( frac{beta_t}{gamma_t}s_t + (1-frac{beta_t}{gamma_t})w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
answered Feb 1 at 18:56
SepideSepide
5048
$endgroup$
add a comment |
$begingroup$
Try the following optimization:
$$
w_{t+1}=argmin_{w in C} -beta_tlangle w,s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2 tag{3}
$$
where $beta_t$ and $gamma_t$ are positive.
Since the objective, i.e., $f(w) = -beta_tlangle w,(s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2$ is convex, the necessary and sufficient condition in order $w_{t+1}$ be the minimizer of $(3)$ is the following:
$$
langle -beta_t (s_t -w_t) + gamma_t (w-w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
$$
langle w - ( frac{beta_t}{gamma_t}s_t + (1-frac{beta_t}{gamma_t})w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
answered Feb 1 at 18:56
SepideSepide
5048
$endgroup$
add a comment |
$begingroup$
Try the following optimization:
$$
w_{t+1}=argmin_{w in C} -beta_tlangle w,s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2 tag{3}
$$
where $beta_t$ and $gamma_t$ are positive.
Since the objective, i.e., $f(w) = -beta_tlangle w,(s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2$ is convex, the necessary and sufficient condition in order $w_{t+1}$ be the minimizer of $(3)$ is the following:
$$
langle -beta_t (s_t -w_t) + gamma_t (w-w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
$$
langle w - ( frac{beta_t}{gamma_t}s_t + (1-frac{beta_t}{gamma_t})w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
answered Feb 1 at 18:56
SepideSepide
5048
$endgroup$
Try the following optimization:
$$
w_{t+1}=argmin_{w in C} -beta_tlangle w,s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2 tag{3}
$$
where $beta_t$ and $gamma_t$ are positive.
Since the objective, i.e., $f(w) = -beta_tlangle w,(s_t -w_t rangle + frac{gamma_t}{2} |w - w_t|_2^2$ is convex, the necessary and sufficient condition in order $w_{t+1}$ be the minimizer of $(3)$ is the following:
$$
langle -beta_t (s_t -w_t) + gamma_t (w-w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
$$
langle w - ( frac{beta_t}{gamma_t}s_t + (1-frac{beta_t}{gamma_t})w_t), w - w_{t+1} rangle geq 0 ,,,, forall w in C
$$
answered Feb 1 at 18:56
SepideSepide
5048
answered Feb 1 at 18:56
SepideSepide
5048
answered Feb 1 at 18:56
SepideSepide
5048
answered Feb 1 at 18:56
SepideSepide
5048
5048
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