  This work frames the problem of constructing non-oscillatory entropy stable
fluxes as a least square optimization problem. A flux sign stability condition
is defined for a pair of entropy conservative flux ($F^*$) and a
non-oscillatory flux ($F^s$). This novel approach paves a way to construct
non-oscillatory entropy stable flux ($\hat{F}$) as a simple combination of
$(F^*$ and $F^s)$ which inherently optimize the numerical diffusion in the
entropy stable flux ($\hat{F}$) such that it reduces to the underlying
non-oscillatory flux ($F^s$) in the flux sign stable region. This robust
approach is (i) agnostic to the choice of flux pair $(F^*,F^s)$, (ii) does not
require the computation of costly dissipation operator and high order
reconstruction of scaled entropy variable to construct the diffusion term.
Various non-oscillatory entropy stable fluxes are constructed and exhaustive
computational results for standard test problems are given which show that
these entropy stable schemes completely remove spurious oscillations in
approximating the discontinuities compared to the non-oscillatory schemes using
underlying fluxes ($F^s$) only. Moreover, these entropy stable schemes maintain
the formal order of accuracy of the lower order flux in the pair.
