Mixing
Nonlinear optimization with parametric constraint
$begingroup$
Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$min f(x_1,dots,x_n)$ subject to $c(x_1,dots,x_n,t)le0$, $tin[t_{min},t_{max}]$
Here $f:mathbb{R}^nrightarrowmathbb{R}$ is a convex function. The constraint function $c:mathbb{R}^{n+1} rightarrow mathbb{R}$ is nonlinear and varies smoothly with the parameter $t$. A solution $(x^*_1,dots,x^*_n)$ must satisfy the constraint $cle0$ for all $tin[t_{min},t_{max}]$.
EDIT:
I believe it can be assumed that the constraint function $c$ is convex for each $t$.
EDIT 2:
I want to design a propeller for an axial pump with a prescribed "sweep" $s(r)$, where $r$ is the radial coordinate in its cylindrical coordinate system $(r,theta,z)$. The sweep is defined as $pi/2-phi(r)$, where $phi$ is the angle between the tangent to the leading edge stacking curve $gamma$ and the tangent to the meanline at the leading edge, which i call $u=u(r)$ from now on. The sweep $s$ must satisfy $sge a_0 + frac{r}{r_{max}}a_1$.
If I constrain the shape of the stacking curve $gamma$ to be polynomial in the $r-theta$ and the $r-z$ planes respectively, e.g. $theta(r)=p_1(r)$ and $z(r)=p_2(r)$ with coefficients $x_1,dots,x_k$ for $p_1$ and $x_{k+1},dots,x_n$ for $p_2$, I will have a tangent $v=v(x_1,dots,x_n)$ to $gamma$.
Thus, since $cosphi=frac{(ucdot v)}{|u|cdot|v|}$, we have
$frac{(ucdot v)}{|u|cdot|v|}lecos(pi/2 -a_0 - frac{r}{r_{max}}a_1)$, which, in essence, constitutes $c(x_1,dots,x_n,r)$. Note that $u=u(r)$ and $v=v(x_1,dots,x_n)$.
The objective function $f$ is the arc length of $gamma$, i.e. $f=int sqrt{|v|^2+1}dr$.
Hope this made sense...
optimization matlab nonlinear-optimization parametric numerical-optimization
asked Jan 22 at 21:45
BobbyBobby
468
$endgroup$
add a comment |
$begingroup$
Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$min f(x_1,dots,x_n)$ subject to $c(x_1,dots,x_n,t)le0$, $tin[t_{min},t_{max}]$
Here $f:mathbb{R}^nrightarrowmathbb{R}$ is a convex function. The constraint function $c:mathbb{R}^{n+1} rightarrow mathbb{R}$ is nonlinear and varies smoothly with the parameter $t$. A solution $(x^*_1,dots,x^*_n)$ must satisfy the constraint $cle0$ for all $tin[t_{min},t_{max}]$.
EDIT:
I believe it can be assumed that the constraint function $c$ is convex for each $t$.
EDIT 2:
I want to design a propeller for an axial pump with a prescribed "sweep" $s(r)$, where $r$ is the radial coordinate in its cylindrical coordinate system $(r,theta,z)$. The sweep is defined as $pi/2-phi(r)$, where $phi$ is the angle between the tangent to the leading edge stacking curve $gamma$ and the tangent to the meanline at the leading edge, which i call $u=u(r)$ from now on. The sweep $s$ must satisfy $sge a_0 + frac{r}{r_{max}}a_1$.
If I constrain the shape of the stacking curve $gamma$ to be polynomial in the $r-theta$ and the $r-z$ planes respectively, e.g. $theta(r)=p_1(r)$ and $z(r)=p_2(r)$ with coefficients $x_1,dots,x_k$ for $p_1$ and $x_{k+1},dots,x_n$ for $p_2$, I will have a tangent $v=v(x_1,dots,x_n)$ to $gamma$.
Thus, since $cosphi=frac{(ucdot v)}{|u|cdot|v|}$, we have
$frac{(ucdot v)}{|u|cdot|v|}lecos(pi/2 -a_0 - frac{r}{r_{max}}a_1)$, which, in essence, constitutes $c(x_1,dots,x_n,r)$. Note that $u=u(r)$ and $v=v(x_1,dots,x_n)$.
The objective function $f$ is the arc length of $gamma$, i.e. $f=int sqrt{|v|^2+1}dr$.
Hope this made sense...
optimization matlab nonlinear-optimization parametric numerical-optimization
asked Jan 22 at 21:45
BobbyBobby
468
$endgroup$
1
$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
1
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22
add a comment |
$begingroup$
Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$min f(x_1,dots,x_n)$ subject to $c(x_1,dots,x_n,t)le0$, $tin[t_{min},t_{max}]$
Here $f:mathbb{R}^nrightarrowmathbb{R}$ is a convex function. The constraint function $c:mathbb{R}^{n+1} rightarrow mathbb{R}$ is nonlinear and varies smoothly with the parameter $t$. A solution $(x^*_1,dots,x^*_n)$ must satisfy the constraint $cle0$ for all $tin[t_{min},t_{max}]$.
EDIT:
I believe it can be assumed that the constraint function $c$ is convex for each $t$.
EDIT 2:
I want to design a propeller for an axial pump with a prescribed "sweep" $s(r)$, where $r$ is the radial coordinate in its cylindrical coordinate system $(r,theta,z)$. The sweep is defined as $pi/2-phi(r)$, where $phi$ is the angle between the tangent to the leading edge stacking curve $gamma$ and the tangent to the meanline at the leading edge, which i call $u=u(r)$ from now on. The sweep $s$ must satisfy $sge a_0 + frac{r}{r_{max}}a_1$.
If I constrain the shape of the stacking curve $gamma$ to be polynomial in the $r-theta$ and the $r-z$ planes respectively, e.g. $theta(r)=p_1(r)$ and $z(r)=p_2(r)$ with coefficients $x_1,dots,x_k$ for $p_1$ and $x_{k+1},dots,x_n$ for $p_2$, I will have a tangent $v=v(x_1,dots,x_n)$ to $gamma$.
Thus, since $cosphi=frac{(ucdot v)}{|u|cdot|v|}$, we have
$frac{(ucdot v)}{|u|cdot|v|}lecos(pi/2 -a_0 - frac{r}{r_{max}}a_1)$, which, in essence, constitutes $c(x_1,dots,x_n,r)$. Note that $u=u(r)$ and $v=v(x_1,dots,x_n)$.
The objective function $f$ is the arc length of $gamma$, i.e. $f=int sqrt{|v|^2+1}dr$.
Hope this made sense...
optimization matlab nonlinear-optimization parametric numerical-optimization
asked Jan 22 at 21:45
BobbyBobby
468
$endgroup$
Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$min f(x_1,dots,x_n)$ subject to $c(x_1,dots,x_n,t)le0$, $tin[t_{min},t_{max}]$
Here $f:mathbb{R}^nrightarrowmathbb{R}$ is a convex function. The constraint function $c:mathbb{R}^{n+1} rightarrow mathbb{R}$ is nonlinear and varies smoothly with the parameter $t$. A solution $(x^*_1,dots,x^*_n)$ must satisfy the constraint $cle0$ for all $tin[t_{min},t_{max}]$.
EDIT:
I believe it can be assumed that the constraint function $c$ is convex for each $t$.
EDIT 2:
I want to design a propeller for an axial pump with a prescribed "sweep" $s(r)$, where $r$ is the radial coordinate in its cylindrical coordinate system $(r,theta,z)$. The sweep is defined as $pi/2-phi(r)$, where $phi$ is the angle between the tangent to the leading edge stacking curve $gamma$ and the tangent to the meanline at the leading edge, which i call $u=u(r)$ from now on. The sweep $s$ must satisfy $sge a_0 + frac{r}{r_{max}}a_1$.
If I constrain the shape of the stacking curve $gamma$ to be polynomial in the $r-theta$ and the $r-z$ planes respectively, e.g. $theta(r)=p_1(r)$ and $z(r)=p_2(r)$ with coefficients $x_1,dots,x_k$ for $p_1$ and $x_{k+1},dots,x_n$ for $p_2$, I will have a tangent $v=v(x_1,dots,x_n)$ to $gamma$.
Thus, since $cosphi=frac{(ucdot v)}{|u|cdot|v|}$, we have
$frac{(ucdot v)}{|u|cdot|v|}lecos(pi/2 -a_0 - frac{r}{r_{max}}a_1)$, which, in essence, constitutes $c(x_1,dots,x_n,r)$. Note that $u=u(r)$ and $v=v(x_1,dots,x_n)$.
The objective function $f$ is the arc length of $gamma$, i.e. $f=int sqrt{|v|^2+1}dr$.
Hope this made sense...
optimization matlab nonlinear-optimization parametric numerical-optimization
optimization matlab nonlinear-optimization parametric numerical-optimization
asked Jan 22 at 21:45
BobbyBobby
468
asked Jan 22 at 21:45
BobbyBobby
468
edited Jan 22 at 22:46
Bobby
asked Jan 22 at 21:45
BobbyBobby
468
asked Jan 22 at 21:45
BobbyBobby
468
asked Jan 22 at 21:45
BobbyBobby
468
468
1
$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
1
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22
add a comment |
1
$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
1
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22
1
1
$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
1
1
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22
add a comment |
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$begingroup$
Such problems can (without restrictions on constrain't function $c$) involve a lot of local minima. I take it you are primarily interested in a global minimum, so any additional information about $c$ would help.
$endgroup$
– hardmath
Jan 22 at 21:51
$begingroup$
This is straightforward to code in Mathematica. Whether or not a solution can be computed depends, however, on the specific nature of $f$ and $c$ (of course).
$endgroup$
– David G. Stork
Jan 22 at 21:57
$begingroup$
could you post the function $c$?
$endgroup$
– LinAlg
Jan 22 at 22:11
$begingroup$
A naive approach would just be to discretize $t$ and solve the problem with each of the convex $c$ constraints at each $t$.
$endgroup$
– nathan.j.mcdougall
Jan 22 at 22:27
1
$begingroup$
What you have is a convex semi-infinite program. See this review for some general approaches for solving such problems.
$endgroup$
– madnessweasley
Jan 22 at 23:22