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How to minimize $sum_i |a_{i1}|x_1|^2 + a_{i2}x_1overline{x_2} + a_{i3}overline{x_1}x_2 + a_{i4}|x_2|^2 -...
$begingroup$
Consider the following nonlinear minimization problem
begin{align} tag{P1}
min_{x_1, x_2 in mathbb C} sum_{i=1}^m big|a_{i1}|x_1|^2 + a_{i2}x_1overline{x_2} + a_{i3}overline{x_1}x_2 + a_{i4}|x_2|^2 - b_ibig|^2
end{align}
where $a_{i1}, a_{i2}, a_{i3}, a_{i4}, b_i$ are nonzero constants in $mathbb C$, $i=1,ldots,m$.
The first thing that occurred to me was to reformulate it with a change of variables
begin{align} tag{P2}
min_{y_1, y_2, y_3, y_4 in mathbb C} & sum_{i=1}^m left|a_{i1}y_1 + a_{i2}y_2 + a_{i3}y_3 + a_{i4}y_4 - b_iright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
so that the objective function can be written more compactly in matrix form (e.g., $|Ax-b|_2^2$). The equivalence can be established by the result kindly proved by Batominovski here.
My questions are for example:
- Is there any algorithm which can effectively solve (P1)?
- If there is, is it guaranteed to converge to a global or a local minimum?
- Can (P2) be recast as a convex problem?
- If it cannot, is there any similar (convex) problem which can be considered?
The following attempt concerns the special case $m=1$. In this case (P2) becomes
begin{align} tag{P3}
min_{y_1, y_2, y_3, y_4 in mathbb C} & left|a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4 - bright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
where $a_1, a_2, a_3, a_4, b$ are nonzero constants in $mathbb C$.
The idea is to linearize the bilinear constraint by taking the logarithm.
Assume $y_j neq 0$, let $z_j := ln y_j + ln a_j$, $j=1,ldots,4$. In order to take advantage of the convexity of the log-sum-exp function, consider a similar problem
begin{align} tag{P4}
min_{z_1, z_2, z_3, z_4 in mathbb C} & left|ln(e^{z_1}+e^{z_2}+e^{z_3}+e^{z_4}) - ln bright|^2 \
text{subject to} quad & z_1 - ln a_1 in mathbb R, y_4 - ln a_4 in mathbb R \
& z_2 - overline{z_3} = ln a_2 - ln overline{a_3} \
& z_1 - z_2 - z_3 + z_4 = ln a_1 - ln a_2 - ln a_3 + ln a_4
end{align}
The rationale is to take the logarithm of both sides of $b approx a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4$. Obviously, this trick cannot be applied to the general case due to the varying $a$'s. Despite the extra assumption, I am also uncomfortable about $ln$ being multivalued in $mathbb C$.
nonlinear-optimization nonlinear-system several-complex-variables
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
$endgroup$
add a comment |
$begingroup$
Consider the following nonlinear minimization problem
begin{align} tag{P1}
min_{x_1, x_2 in mathbb C} sum_{i=1}^m big|a_{i1}|x_1|^2 + a_{i2}x_1overline{x_2} + a_{i3}overline{x_1}x_2 + a_{i4}|x_2|^2 - b_ibig|^2
end{align}
where $a_{i1}, a_{i2}, a_{i3}, a_{i4}, b_i$ are nonzero constants in $mathbb C$, $i=1,ldots,m$.
The first thing that occurred to me was to reformulate it with a change of variables
begin{align} tag{P2}
min_{y_1, y_2, y_3, y_4 in mathbb C} & sum_{i=1}^m left|a_{i1}y_1 + a_{i2}y_2 + a_{i3}y_3 + a_{i4}y_4 - b_iright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
so that the objective function can be written more compactly in matrix form (e.g., $|Ax-b|_2^2$). The equivalence can be established by the result kindly proved by Batominovski here.
My questions are for example:
- Is there any algorithm which can effectively solve (P1)?
- If there is, is it guaranteed to converge to a global or a local minimum?
- Can (P2) be recast as a convex problem?
- If it cannot, is there any similar (convex) problem which can be considered?
The following attempt concerns the special case $m=1$. In this case (P2) becomes
begin{align} tag{P3}
min_{y_1, y_2, y_3, y_4 in mathbb C} & left|a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4 - bright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
where $a_1, a_2, a_3, a_4, b$ are nonzero constants in $mathbb C$.
The idea is to linearize the bilinear constraint by taking the logarithm.
Assume $y_j neq 0$, let $z_j := ln y_j + ln a_j$, $j=1,ldots,4$. In order to take advantage of the convexity of the log-sum-exp function, consider a similar problem
begin{align} tag{P4}
min_{z_1, z_2, z_3, z_4 in mathbb C} & left|ln(e^{z_1}+e^{z_2}+e^{z_3}+e^{z_4}) - ln bright|^2 \
text{subject to} quad & z_1 - ln a_1 in mathbb R, y_4 - ln a_4 in mathbb R \
& z_2 - overline{z_3} = ln a_2 - ln overline{a_3} \
& z_1 - z_2 - z_3 + z_4 = ln a_1 - ln a_2 - ln a_3 + ln a_4
end{align}
The rationale is to take the logarithm of both sides of $b approx a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4$. Obviously, this trick cannot be applied to the general case due to the varying $a$'s. Despite the extra assumption, I am also uncomfortable about $ln$ being multivalued in $mathbb C$.
nonlinear-optimization nonlinear-system several-complex-variables
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
$endgroup$
add a comment |
$begingroup$
Consider the following nonlinear minimization problem
begin{align} tag{P1}
min_{x_1, x_2 in mathbb C} sum_{i=1}^m big|a_{i1}|x_1|^2 + a_{i2}x_1overline{x_2} + a_{i3}overline{x_1}x_2 + a_{i4}|x_2|^2 - b_ibig|^2
end{align}
where $a_{i1}, a_{i2}, a_{i3}, a_{i4}, b_i$ are nonzero constants in $mathbb C$, $i=1,ldots,m$.
The first thing that occurred to me was to reformulate it with a change of variables
begin{align} tag{P2}
min_{y_1, y_2, y_3, y_4 in mathbb C} & sum_{i=1}^m left|a_{i1}y_1 + a_{i2}y_2 + a_{i3}y_3 + a_{i4}y_4 - b_iright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
so that the objective function can be written more compactly in matrix form (e.g., $|Ax-b|_2^2$). The equivalence can be established by the result kindly proved by Batominovski here.
My questions are for example:
- Is there any algorithm which can effectively solve (P1)?
- If there is, is it guaranteed to converge to a global or a local minimum?
- Can (P2) be recast as a convex problem?
- If it cannot, is there any similar (convex) problem which can be considered?
The following attempt concerns the special case $m=1$. In this case (P2) becomes
begin{align} tag{P3}
min_{y_1, y_2, y_3, y_4 in mathbb C} & left|a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4 - bright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
where $a_1, a_2, a_3, a_4, b$ are nonzero constants in $mathbb C$.
The idea is to linearize the bilinear constraint by taking the logarithm.
Assume $y_j neq 0$, let $z_j := ln y_j + ln a_j$, $j=1,ldots,4$. In order to take advantage of the convexity of the log-sum-exp function, consider a similar problem
begin{align} tag{P4}
min_{z_1, z_2, z_3, z_4 in mathbb C} & left|ln(e^{z_1}+e^{z_2}+e^{z_3}+e^{z_4}) - ln bright|^2 \
text{subject to} quad & z_1 - ln a_1 in mathbb R, y_4 - ln a_4 in mathbb R \
& z_2 - overline{z_3} = ln a_2 - ln overline{a_3} \
& z_1 - z_2 - z_3 + z_4 = ln a_1 - ln a_2 - ln a_3 + ln a_4
end{align}
The rationale is to take the logarithm of both sides of $b approx a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4$. Obviously, this trick cannot be applied to the general case due to the varying $a$'s. Despite the extra assumption, I am also uncomfortable about $ln$ being multivalued in $mathbb C$.
nonlinear-optimization nonlinear-system several-complex-variables
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
$endgroup$
Consider the following nonlinear minimization problem
begin{align} tag{P1}
min_{x_1, x_2 in mathbb C} sum_{i=1}^m big|a_{i1}|x_1|^2 + a_{i2}x_1overline{x_2} + a_{i3}overline{x_1}x_2 + a_{i4}|x_2|^2 - b_ibig|^2
end{align}
where $a_{i1}, a_{i2}, a_{i3}, a_{i4}, b_i$ are nonzero constants in $mathbb C$, $i=1,ldots,m$.
The first thing that occurred to me was to reformulate it with a change of variables
begin{align} tag{P2}
min_{y_1, y_2, y_3, y_4 in mathbb C} & sum_{i=1}^m left|a_{i1}y_1 + a_{i2}y_2 + a_{i3}y_3 + a_{i4}y_4 - b_iright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
so that the objective function can be written more compactly in matrix form (e.g., $|Ax-b|_2^2$). The equivalence can be established by the result kindly proved by Batominovski here.
My questions are for example:
- Is there any algorithm which can effectively solve (P1)?
- If there is, is it guaranteed to converge to a global or a local minimum?
- Can (P2) be recast as a convex problem?
- If it cannot, is there any similar (convex) problem which can be considered?
The following attempt concerns the special case $m=1$. In this case (P2) becomes
begin{align} tag{P3}
min_{y_1, y_2, y_3, y_4 in mathbb C} & left|a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4 - bright|^2 \
text{subject to} quad & y_1, y_4 in mathbb R_{geq 0} \
& y_2 = overline{y_3} \
& y_1y_4 = y_2y_3
end{align}
where $a_1, a_2, a_3, a_4, b$ are nonzero constants in $mathbb C$.
The idea is to linearize the bilinear constraint by taking the logarithm.
Assume $y_j neq 0$, let $z_j := ln y_j + ln a_j$, $j=1,ldots,4$. In order to take advantage of the convexity of the log-sum-exp function, consider a similar problem
begin{align} tag{P4}
min_{z_1, z_2, z_3, z_4 in mathbb C} & left|ln(e^{z_1}+e^{z_2}+e^{z_3}+e^{z_4}) - ln bright|^2 \
text{subject to} quad & z_1 - ln a_1 in mathbb R, y_4 - ln a_4 in mathbb R \
& z_2 - overline{z_3} = ln a_2 - ln overline{a_3} \
& z_1 - z_2 - z_3 + z_4 = ln a_1 - ln a_2 - ln a_3 + ln a_4
end{align}
The rationale is to take the logarithm of both sides of $b approx a_1y_1 + a_2y_2 + a_3y_3 + a_4y_4$. Obviously, this trick cannot be applied to the general case due to the varying $a$'s. Despite the extra assumption, I am also uncomfortable about $ln$ being multivalued in $mathbb C$.
nonlinear-optimization nonlinear-system several-complex-variables
nonlinear-optimization nonlinear-system several-complex-variables
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
edited Jan 8 at 21:05
B. Groeger
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
asked Jan 8 at 20:33
B. GroegerB. Groeger
205
205
add a comment |
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