How to determine the convexity of multiple matrix variables function?
$begingroup$
This formula is :
$$f(W,V,B) =|XW-V|^2_F +|Y-VB|^2_F +operatorname{tr}(V'LV) +2operatorname{tr}(W'DW),$$
where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a constant diagonal matrix.
After @ A.Γ.'s suggestion, I modified the question. The original question is as follows:
Is the convexity of a function with a saddle point multivariate function plus a multivariate convex function necessarily non-convex?
convex-analysis machine-learning operations-research non-convex-optimization
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
$endgroup$
add a comment |
$begingroup$
This formula is :
$$f(W,V,B) =|XW-V|^2_F +|Y-VB|^2_F +operatorname{tr}(V'LV) +2operatorname{tr}(W'DW),$$
where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a constant diagonal matrix.
After @ A.Γ.'s suggestion, I modified the question. The original question is as follows:
Is the convexity of a function with a saddle point multivariate function plus a multivariate convex function necessarily non-convex?
convex-analysis machine-learning operations-research non-convex-optimization
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
$endgroup$
$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52
add a comment |
$begingroup$
This formula is :
$$f(W,V,B) =|XW-V|^2_F +|Y-VB|^2_F +operatorname{tr}(V'LV) +2operatorname{tr}(W'DW),$$
where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a constant diagonal matrix.
After @ A.Γ.'s suggestion, I modified the question. The original question is as follows:
Is the convexity of a function with a saddle point multivariate function plus a multivariate convex function necessarily non-convex?
convex-analysis machine-learning operations-research non-convex-optimization
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
$endgroup$
This formula is :
$$f(W,V,B) =|XW-V|^2_F +|Y-VB|^2_F +operatorname{tr}(V'LV) +2operatorname{tr}(W'DW),$$
where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a constant diagonal matrix.
After @ A.Γ.'s suggestion, I modified the question. The original question is as follows:
Is the convexity of a function with a saddle point multivariate function plus a multivariate convex function necessarily non-convex?
convex-analysis machine-learning operations-research non-convex-optimization
convex-analysis machine-learning operations-research non-convex-optimization
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
edited Dec 9 '18 at 13:18
learn_truth
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
asked Dec 9 '18 at 2:44
learn_truthlearn_truth
12
12
$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52
add a comment |
$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52
$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52
add a comment |
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$begingroup$
The question is unclear. In your formula, the second term is neither saddle point nor convex function. Answering the question in the title: no, it may be convex, for example, $f(x,y)=x^2-y^2$ and $g(x,y)=y^2$ where $f+g$ is convex.
$endgroup$
– A.Γ.
Dec 9 '18 at 12:25
$begingroup$
@ A.Γ. Sorry, I don't know about this knowledge about convex optimization, so the question is not rigorous enough. why is this formula non-convex?
$endgroup$
– learn_truth
Dec 9 '18 at 12:52