How to solve a non-convex programming problem?
Let $A$ and $C $ be $ntimes n$ symmetric matrices, and $Abullet C=Tr(A^TC)$. Let $Ssubseteq [n]times [n]times [n]$. Define a non-convex programming problem as follows.
begin{equation}
begin{split}
&max Cbullet X\
&text{ s.t.} Abullet X=1,\
&qquad quad X_{i,j}X_{i,k}=0, forall (i,j,k)in S,\
&qquad quad Xsucceq 0, Xgeq 0,
end{split}
end{equation}
where $X$ is a variable ranging over the set of $ntimes n $ symmetric matrices over $mathbb{R}$, $ Xsucceq 0$ means $X$ is positive semi-definite, $ Xgeq 0$ means each entry of $X$ is nonnegative, and $X_{i,j}$ is the $(i,j)$-entry of $X$.
Could you please tell me how to solve this problem? And how to consider its dual problem? Thank you.
combinatorics nonlinear-optimization non-convex-optimization programming semidefinite-programming
asked Nov 29 '18 at 13:54
XiupingXiuping
214
|
show 1 more comment
Let $A$ and $C $ be $ntimes n$ symmetric matrices, and $Abullet C=Tr(A^TC)$. Let $Ssubseteq [n]times [n]times [n]$. Define a non-convex programming problem as follows.
begin{equation}
begin{split}
&max Cbullet X\
&text{ s.t.} Abullet X=1,\
&qquad quad X_{i,j}X_{i,k}=0, forall (i,j,k)in S,\
&qquad quad Xsucceq 0, Xgeq 0,
end{split}
end{equation}
where $X$ is a variable ranging over the set of $ntimes n $ symmetric matrices over $mathbb{R}$, $ Xsucceq 0$ means $X$ is positive semi-definite, $ Xgeq 0$ means each entry of $X$ is nonnegative, and $X_{i,j}$ is the $(i,j)$-entry of $X$.
Could you please tell me how to solve this problem? And how to consider its dual problem? Thank you.
combinatorics nonlinear-optimization non-convex-optimization programming semidefinite-programming
asked Nov 29 '18 at 13:54
XiupingXiuping
214
You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29
|
show 1 more comment
Let $A$ and $C $ be $ntimes n$ symmetric matrices, and $Abullet C=Tr(A^TC)$. Let $Ssubseteq [n]times [n]times [n]$. Define a non-convex programming problem as follows.
begin{equation}
begin{split}
&max Cbullet X\
&text{ s.t.} Abullet X=1,\
&qquad quad X_{i,j}X_{i,k}=0, forall (i,j,k)in S,\
&qquad quad Xsucceq 0, Xgeq 0,
end{split}
end{equation}
where $X$ is a variable ranging over the set of $ntimes n $ symmetric matrices over $mathbb{R}$, $ Xsucceq 0$ means $X$ is positive semi-definite, $ Xgeq 0$ means each entry of $X$ is nonnegative, and $X_{i,j}$ is the $(i,j)$-entry of $X$.
Could you please tell me how to solve this problem? And how to consider its dual problem? Thank you.
combinatorics nonlinear-optimization non-convex-optimization programming semidefinite-programming
asked Nov 29 '18 at 13:54
XiupingXiuping
214
Let $A$ and $C $ be $ntimes n$ symmetric matrices, and $Abullet C=Tr(A^TC)$. Let $Ssubseteq [n]times [n]times [n]$. Define a non-convex programming problem as follows.
begin{equation}
begin{split}
&max Cbullet X\
&text{ s.t.} Abullet X=1,\
&qquad quad X_{i,j}X_{i,k}=0, forall (i,j,k)in S,\
&qquad quad Xsucceq 0, Xgeq 0,
end{split}
end{equation}
where $X$ is a variable ranging over the set of $ntimes n $ symmetric matrices over $mathbb{R}$, $ Xsucceq 0$ means $X$ is positive semi-definite, $ Xgeq 0$ means each entry of $X$ is nonnegative, and $X_{i,j}$ is the $(i,j)$-entry of $X$.
Could you please tell me how to solve this problem? And how to consider its dual problem? Thank you.
combinatorics nonlinear-optimization non-convex-optimization programming semidefinite-programming
combinatorics nonlinear-optimization non-convex-optimization programming semidefinite-programming
asked Nov 29 '18 at 13:54
XiupingXiuping
214
asked Nov 29 '18 at 13:54
XiupingXiuping
214
edited Nov 29 '18 at 14:00
Xiuping
asked Nov 29 '18 at 13:54
XiupingXiuping
214
asked Nov 29 '18 at 13:54
XiupingXiuping
214
asked Nov 29 '18 at 13:54
XiupingXiuping
214
214
You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29
|
show 1 more comment
You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29
You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29
|
show 1 more comment
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You could model the product constraint as a mixed-integer SDP.
– Michal Adamaszek
Nov 29 '18 at 15:03
@MichalAdamaszek Could you please tell me how to do it? Thanks very much.
– Xiuping
Nov 30 '18 at 9:50
Well $xy=0, x,ygeq 0$ is the same as $0leq xleq Mz$, $0leq yleq M(1-z)$ where $z$ is a binary variable and $M$ is an upper bound on $x,y$. See docs.mosek.com/modeling-cookbook/mio.html#boolean-operators But there is not a load of MISDP solvers, if any. YALMIP can model it I think. But then it is possible that YALMIP can model your whole problem directly.
– Michal Adamaszek
Nov 30 '18 at 9:59
@MichalAdamaszek Thank you so much. If I write the problem as a MISDP, is it the case that the optimal value of MISDP is the same with the optimal value of its dual MISDD? Thanks.
– Xiuping
Nov 30 '18 at 10:13
I'm not sure I know what a dual of a mixed-integer problem is.
– Michal Adamaszek
Nov 30 '18 at 12:29