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The constraints in both cases are inherently non-convex, so it's unlikely that there's a convex formulation. In fact, though I believe your formulation in the second case to be conceptually correct, it is no longer convex. For an indication of why, you can investigate the Hessian of $f(x,\epsilon,\xi) = x^2 +\epsilon\xi$, which is a simpler objective in the ...


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Consider the problem in standard format \begin{equation}\notag \begin{split} \min & -(x^2+y^2) \\ \mathrm{s.t.} & (x^2-1) \leq 0 \\ & -y \leq 0 \\ & (y-2) \leq 0 \end{split} \end{equation} Now the Lagrange Dual will be \begin{equation} L(x,y,\lambda)=-x^2-y^2+\lambda_1 (x^2-1) - \lambda_2 y + \lambda_3 (y-2) \hspace{2ex} \lambda_1, \...


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