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I am looking for conditions on parameters $a$ and $b$ and on non-convex function $g$ such that the scalar function

$$f(x) = a g(x) + \frac b2 x^2$$

is convex. It would be a great help if someone can guide me on this problem.

Rodrigo de Azevedo
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user571754
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    is $g$ concave? What can you tell us about $g$? – Rodrigo de Azevedo Jun 23 '18 at 13:02
  • The first-order characterization of convexity requires that $f(y)-f(x)\ge f’(x)(y-x)$ for all $x,y$ in the domain. Playing with this might help (assuming $g$ is differentiable) – David M. Jun 23 '18 at 13:02
  • Dear Rodrigo de Azevedo, the function g(x) is non-convex. – user571754 Jun 24 '18 at 07:20
  • @user571754 you're missing Rodrigo's point. If the only thing you can say about $g$ is that it is non-convex, then there are no conditions at all you can place on $a$ and $b$, except perhaps $a=0, b>0$. You need to offer something more: e.g., $g$ is continuous and twice-differentiable. – Michael Grant Jun 25 '18 at 21:35
  • Thanks Michael! You are exactly right. We may assume that g is continuous and twice-differentiable. – user571754 Jul 02 '18 at 12:41

1 Answers1

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If you are willing to (or can always) choose $a>0$, and $g$ is twice differentiable on an open interval, then you obtain your result on that interval if $$g''>-b.$$

Allawonder
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