# Looking for an equivalent formulation of an optimization problem in such a way that the CVX solver understands and solves it.

 0 I need to solve the following problem: $\left\{\begin{array}{lll} {\displaystyle \inf_{(\beta,\lambda)\in\mathbb{R}^{2},s\in\mathbb{R}^{N}} } & {\displaystyle \lambda \varepsilon +\frac{1}{N}\sum_{i=1}^{N}s_{i}} &\\ \mbox{subject to} &\beta^{2}+4a_{i}\lambda\beta+4a_{i}^{2}\lambda-4\lambda s_{i}+4s_{i}\leq 0 & \forall i\leq N \\ &\lambda >1& \\ & \beta\geq 0.& \end{array}\right.\tag{$$\bigstar$$}$ where $$\varepsilon, a_{i}\in \mathbb{R}$$ are fixed for all $$i=1,2,\ldots,N$$. My intention is to use CVX (see this link), but I have problems, as it is stated ( $$\bigstar$$ ) the solver does not interpret it. Remark: I do not know if the following is useful, but note that the constrain $\beta^{2}+4a_{i}\lambda\beta+4a_{i}^{2}\lambda-4\lambda s_{i}+4s_{i}\leq 0 \quad \forall i\leq N$ can be changed by $\frac{\beta^{2}}{4(\lambda-1)}+\frac{\lambda}{\lambda-1}a_{i}(\beta+a_{i})-s_{i}\leq 0 \quad \forall i\leq N$ I showed that the function $$f:\mathbb{R}^{3}\rightarrow \mathbb{R}$$ given by $$f(\beta,\lambda,s):=\frac{\beta^{2}}{4(\lambda-1)}+\frac{\lambda}{\lambda-1}a(\beta+a)-s$$ is convex in $$\mathbb{R}\times\mathbb{R}_{\geq 1}\times\mathbb{R}$$ for any $$a\in \mathbb{R}$$. My attempt: CVX can solve geometric programs (GP), the problem is that ( $$\bigstar$$ ) is not a geometric program because $$a_{i}$$ can be negative, my idea was to express ( $$\bigstar$$ ) as a geometric program but what arrived was the following equivalent formulation: $$\left\{ \begin{array}{lll} {\displaystyle \inf_{\beta,\lambda,s_{i},r_{i}} } & r_{0} & \\ \mbox{subject to} & \varepsilon\lambda r_{0}^{-1}+\frac{1}{N}\sum_{i=1}^{N}s_{i}r_{i}\leq 1 &\\ &\left.\begin{array}{l}\beta^{2}r_{i}^{-1}+4a_{i}\lambda \beta r_{i}^{-1}+4a_{i}^{2}\lambda r_{i}^{-1}+4s_{i}r_{i}^{-1}+r_{i}^{-1}\leq 1 \\ 4\lambda s_{i}r_{i}^{-1}+r_{i}^{-1}\geq 1 \end{array}\right\} &\forall i\in I_{1} \\ &\left.\begin{array}{l}\beta^{2}r_{i}^{-1}+4a_{i}^{2}\lambda r_{i}^{-1}+4s_{i}r_{i}^{-1}+r_{i}^{-1}\leq 1 \\ 4\lambda s_{i}r_{i}^{-1}+4(-a_{i})\lambda\beta r_{i}^{-1}+r_{i}^{-1}\geq 1 \end{array}\right\} &\forall i\in I_{2} \\ & \lambda >1 & \\ & \beta\geq 0. \end{array}\right.\tag{\(\blacktriangle$$}\) where \( I_{1}:=\left\{i \:|\: 0= a monomial (for your special case, the constant 1), is non-convex. That is why CVX rejects it. (12 Dec '17, 19:12) Mark L Stone 1 After reading http://cvxr.com/cvx/doc/ and http://ask.cvxr.com/t/why-isnt-cvx-accepting-my-model-read-this-first/570 , you can feel free to post an appropriate question at http://ask.cvxr.com/ , which is a specialized forum for CVX. (12 Dec '17, 19:15) Mark L Stone
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