McCormick envelope of two variables which are also defined in terms of an envelope
I have a equation which is defined as $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as envelope of cosine function.
The McCormick envelope is defined as
begin{align*}
langle x_ix_jrangle^M &= { alpha ge x_i^lx_j+x_j^lx_i-x_i^lx_j^l \
&qquad alpha ge x_i^ux_j+x_j^ux_i-x_i^ux_j^u \
&qquad alpha le x_i^lx_j+x_j^ux_i-x_i^lx_j^u \
&qquad alpha le x_i^ux_j+x_j^lx_i-x_i^ux_j^l},
end{align*}
where $x^l,x^u$ are constants for both $x_i,x_j$ and known.
The envelope for cosine function is defined as
begin{align*}
langle cos(theta)rangle^C &= {beta le 1-frac{1-cos(theta^m)}{(theta^m)^2}theta^2\
&qquad betagefrac{cos(theta^l)-cos(theta^u)}{(theta^l-theta^u)}(theta-theta^l)+cos(theta^l)},
end{align*}
where $theta^l,theta^u,theta^m$ are constants and known.
Now, I have to define the McCormick envelope of $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$. Is it equivalent/true to write $langlelangle x_ix_jrangle^Mlangle cos(theta rangle^Crangle^M$ as
begin{align*}
langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M = langle alpha beta rangle^M &= { gamma ge alpha^lbeta+beta^lalpha-alpha^lbeta^l \
&qquad gamma ge alpha^ubeta+beta^ualpha-alpha^ubeta^u \
&qquad gamma le alpha^lbeta+beta^ualpha-alpha^lbeta^u \
&qquad gamma le alpha^ubeta+beta^lalpha-alpha^ubeta^l}?
end{align*}
If this is true, then how to define $alpha^l,alpha^u,beta^l,beta^u$? Do the first two inequalities of $langle x_ix_jrangle^M$ becomes the lower limit for $alpha$ and the last two ineqaulites of $langle x_ix_jrangle^M$ becomes upper limit of $alpha$? Or is there any other/better way to define McCormick envelope for such equation?
real-analysis convex-optimization constraints
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
add a comment |
I have a equation which is defined as $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as envelope of cosine function.
The McCormick envelope is defined as
begin{align*}
langle x_ix_jrangle^M &= { alpha ge x_i^lx_j+x_j^lx_i-x_i^lx_j^l \
&qquad alpha ge x_i^ux_j+x_j^ux_i-x_i^ux_j^u \
&qquad alpha le x_i^lx_j+x_j^ux_i-x_i^lx_j^u \
&qquad alpha le x_i^ux_j+x_j^lx_i-x_i^ux_j^l},
end{align*}
where $x^l,x^u$ are constants for both $x_i,x_j$ and known.
The envelope for cosine function is defined as
begin{align*}
langle cos(theta)rangle^C &= {beta le 1-frac{1-cos(theta^m)}{(theta^m)^2}theta^2\
&qquad betagefrac{cos(theta^l)-cos(theta^u)}{(theta^l-theta^u)}(theta-theta^l)+cos(theta^l)},
end{align*}
where $theta^l,theta^u,theta^m$ are constants and known.
Now, I have to define the McCormick envelope of $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$. Is it equivalent/true to write $langlelangle x_ix_jrangle^Mlangle cos(theta rangle^Crangle^M$ as
begin{align*}
langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M = langle alpha beta rangle^M &= { gamma ge alpha^lbeta+beta^lalpha-alpha^lbeta^l \
&qquad gamma ge alpha^ubeta+beta^ualpha-alpha^ubeta^u \
&qquad gamma le alpha^lbeta+beta^ualpha-alpha^lbeta^u \
&qquad gamma le alpha^ubeta+beta^lalpha-alpha^ubeta^l}?
end{align*}
If this is true, then how to define $alpha^l,alpha^u,beta^l,beta^u$? Do the first two inequalities of $langle x_ix_jrangle^M$ becomes the lower limit for $alpha$ and the last two ineqaulites of $langle x_ix_jrangle^M$ becomes upper limit of $alpha$? Or is there any other/better way to define McCormick envelope for such equation?
real-analysis convex-optimization constraints
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
add a comment |
I have a equation which is defined as $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as envelope of cosine function.
The McCormick envelope is defined as
begin{align*}
langle x_ix_jrangle^M &= { alpha ge x_i^lx_j+x_j^lx_i-x_i^lx_j^l \
&qquad alpha ge x_i^ux_j+x_j^ux_i-x_i^ux_j^u \
&qquad alpha le x_i^lx_j+x_j^ux_i-x_i^lx_j^u \
&qquad alpha le x_i^ux_j+x_j^lx_i-x_i^ux_j^l},
end{align*}
where $x^l,x^u$ are constants for both $x_i,x_j$ and known.
The envelope for cosine function is defined as
begin{align*}
langle cos(theta)rangle^C &= {beta le 1-frac{1-cos(theta^m)}{(theta^m)^2}theta^2\
&qquad betagefrac{cos(theta^l)-cos(theta^u)}{(theta^l-theta^u)}(theta-theta^l)+cos(theta^l)},
end{align*}
where $theta^l,theta^u,theta^m$ are constants and known.
Now, I have to define the McCormick envelope of $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$. Is it equivalent/true to write $langlelangle x_ix_jrangle^Mlangle cos(theta rangle^Crangle^M$ as
begin{align*}
langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M = langle alpha beta rangle^M &= { gamma ge alpha^lbeta+beta^lalpha-alpha^lbeta^l \
&qquad gamma ge alpha^ubeta+beta^ualpha-alpha^ubeta^u \
&qquad gamma le alpha^lbeta+beta^ualpha-alpha^lbeta^u \
&qquad gamma le alpha^ubeta+beta^lalpha-alpha^ubeta^l}?
end{align*}
If this is true, then how to define $alpha^l,alpha^u,beta^l,beta^u$? Do the first two inequalities of $langle x_ix_jrangle^M$ becomes the lower limit for $alpha$ and the last two ineqaulites of $langle x_ix_jrangle^M$ becomes upper limit of $alpha$? Or is there any other/better way to define McCormick envelope for such equation?
real-analysis convex-optimization constraints
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
I have a equation which is defined as $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as envelope of cosine function.
The McCormick envelope is defined as
begin{align*}
langle x_ix_jrangle^M &= { alpha ge x_i^lx_j+x_j^lx_i-x_i^lx_j^l \
&qquad alpha ge x_i^ux_j+x_j^ux_i-x_i^ux_j^u \
&qquad alpha le x_i^lx_j+x_j^ux_i-x_i^lx_j^u \
&qquad alpha le x_i^ux_j+x_j^lx_i-x_i^ux_j^l},
end{align*}
where $x^l,x^u$ are constants for both $x_i,x_j$ and known.
The envelope for cosine function is defined as
begin{align*}
langle cos(theta)rangle^C &= {beta le 1-frac{1-cos(theta^m)}{(theta^m)^2}theta^2\
&qquad betagefrac{cos(theta^l)-cos(theta^u)}{(theta^l-theta^u)}(theta-theta^l)+cos(theta^l)},
end{align*}
where $theta^l,theta^u,theta^m$ are constants and known.
Now, I have to define the McCormick envelope of $langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M$. Is it equivalent/true to write $langlelangle x_ix_jrangle^Mlangle cos(theta rangle^Crangle^M$ as
begin{align*}
langlelangle x_ix_jrangle^Mlangle cos(theta)rangle^Crangle^M = langle alpha beta rangle^M &= { gamma ge alpha^lbeta+beta^lalpha-alpha^lbeta^l \
&qquad gamma ge alpha^ubeta+beta^ualpha-alpha^ubeta^u \
&qquad gamma le alpha^lbeta+beta^ualpha-alpha^lbeta^u \
&qquad gamma le alpha^ubeta+beta^lalpha-alpha^ubeta^l}?
end{align*}
If this is true, then how to define $alpha^l,alpha^u,beta^l,beta^u$? Do the first two inequalities of $langle x_ix_jrangle^M$ becomes the lower limit for $alpha$ and the last two ineqaulites of $langle x_ix_jrangle^M$ becomes upper limit of $alpha$? Or is there any other/better way to define McCormick envelope for such equation?
real-analysis convex-optimization constraints
real-analysis convex-optimization constraints
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
edited Dec 12 '18 at 15:54
AryanSonwatikar
31312
31312
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
asked Dec 12 '18 at 15:24
Muhammad UsmanMuhammad Usman
136
136
add a comment |
add a comment |
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