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For a convex optimization problem with equality constraints:

Max $f(\boldsymbol{x},a)$ subject to $g(\boldsymbol{x})=0$.

$a$ is a parameter and the function $f(\boldsymbol{x},a)$ is convex in $x_i$. Note the parameter $a$ is not in the constraint.

Question is: if $f(\boldsymbol{x},a)$ is linear in $a$ is the value function convex?

Ashley
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1 Answers1

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If i understand you correct, then:

  1. $f(\mathbf{x}, a)=af_1(\mathbf{x})$
  2. $f_1(t\mathbf{x_1} + (1-t)\mathbf{x_2}) \leq tf_1(\mathbf{x_1}) + (1-t)f_1(\mathbf{x_2})$ since $f(\mathbf{x}, a)$ is convex in $x_i$

We then have, for $a\geq 0$:

$f(t\mathbf{x_1} + (1-t)\mathbf{x_2}, a) = af_1(t\mathbf{x_1} + (1-t)\mathbf{x_2}) \leq taf_1(\mathbf{x_1}) + (1-t)af_1(\mathbf{x_2}) = tf(\mathbf{x_1}, a) + (1-t)f(\mathbf{x_2},a)$

which imply that $f(\mathbf{x},a)$ is convex for $a\geq 0$. If $a<0$ then $f(\mathbf{x}, a)$ would be concave.

rook
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