A copositive Farkas lemma and minimally exact conic relaxations for robust quadratic optimization with binary and quadratic constraints (Q2294373)

A robust generalized Farkas' lemma for a finite system of quadratic inequalities over the intersection of a closed convex cone \(\mathcal{K\subset }\mathbb{R}^{n}\) with \(\left\{ 0,1\right\} ^{n}\) is obtained under a generalization of the convexifiability condition introduced by \textit{N. H. Chieu} et al. [Eur. J. Oper. Res. 280, No. 2, 441--452 (2020; Zbl 1430.90467)] for the case when \(\mathcal{K=}\mathbb{R}^{n}\). This result is used to provide a minimally exact conic relaxation for a robust binary conic quadratic program. When \(\mathcal{K=}\mathbb{R}^{n}\), the relaxation turns out to be a semi-definite programming problem. For problems with linear equality constraints but no quadratic constraint, a sufficient condition for the convexifiability condition to hold is proved to be the inclusion in \(\left[0,1\right] ^{n}\) of the intersection of \(\mathcal{K}\) with the solution set of the linear equality constraints (in other words, the feasible set obtained when one removes the binary constraints).