Robust Geometric Programming

Consider the robust geometric program

\[\begin{array}{ll} \mbox{minimize} & c^T \mathbin{\diamond} x \\ \mbox{subject to} & \sup_{U^{(1)},\ldots,U^{(k)} \in \mathcal{U}}\sum_{\ell=1}^k (A^{(\ell)} + U^{(\ell)}) \mathbin{\diamond} x \leq b. \end{array}\]

The operator \(\mathbin{\diamond}\) denotes geometric matrix multiplication. If \(\mathcal{U} = [-\delta, \delta]^{m \times n}\), then the problem above is equivalent to

\[\begin{array}{ll} \mbox{minimize} & c^T \mathbin{\diamond} x \\ \mbox{subject to} & \sum_{\ell=1}^k (a^{(\ell)}_i \mathbin{\diamond} x) (\prod_{i=1}^n \max\{1/x,x\})^\delta \leq b_i, \quad i=1,\ldots,m, \end{array}\]

where \(a^{(\ell)}_i\) is the \(i\)th row of \(A^{(\ell)}\).

Written on August 26, 2020