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The vertical heat transfer in B\'enard-Marangoni convection of a fluid layerwith infinite Prandtl number is studied by means of upper bounds on the Nusseltnumber $Nu$ as a function of the Marangoni number $Ma$. Using the backgroundmethod for the temperature field, it has recently been proven by Hagstrom &Doering that $ Nu\leq 0.838\,Ma^{2/7}$. In this work we extend previousbackground method analysis to include balance parameters and derive avariational principle for the bound on $Nu$, expressed in terms of a scaledbackground field, that yields a better bound than Hagstrom & Doering'sformulation at a given $Ma$. Using a piecewise-linear, monotonically decreasingprofile we then show that $Nu \leq 0.803\,Ma^{2/7}$, lowering the previousprefactor by 4.2%. However, we also demonstrate that optimisation of thebalance parameters does not affect the asymptotic scaling of the optimal boundachievable with Hagstrom & Doering's original formulation. We subsequentlyutilise convex optimisation to optimise the bound on $Nu$ over all admissiblebackground fields, as well as over two smaller families of profiles constrainedby monotonicity and convexity. The results show that $Nu \leq O(Ma^{2/7}(\lnMa)^{-1/2})$ when the background field has a non-monotonic boundary layer nearthe surface, while a power-law bound with exponent 2/7 is optimal within theclass of monotonic background fields. Further analysis of our upper-boundingprinciple reveals the role of non-monotonicity, and how it may be exploited ina rigorous mathematical argument.

Bounds on heat transfer for B\'enard-Marangoni convection at infinite Prandtl number