! Misplaced \omit. \multispan ->\omit

It appears \multicolumn command must appear first in its cell. Your code (completed into a MWE) compiles with {}\multicolumn replaced by \multicolumn and the extra \left removed.

The {} group should go inside the \multicolumn; you're forgetting @{} too.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
\begin{array} {l r@{}l@{} r@{}l@{}}
\text{Max}  \quad   &   R   &\multicolumn{3}{@{}l}{{}=\log\left(...............\right)}    \\
\text{s.t.} \quad   &   A   &{}=    B.C.D   &   E   &{}=    F.G.H   \\
                    &   I   &{}=    J.K.L   &   M   &{}=    N.O.P   \\
\end{array}
\]

\end{document}

See Why is \[ ... \] preferable to $$ ... $$? for reasons why $$ should never be used in LaTeX.

A different trick:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\multialign}[2]{%
  \multispan{#1}\mbox{$\displaystyle#2$}%
}

\begin{document}

\begin{alignat*}{3}
&\textnormal{Max}  \quad & R &\multialign{3}{{}=\log\left(...............\right)} \\
&\textnormal{s.t.} \quad & A &= B.C.D \quad&  E &= F.G.H \\
&&                         I &= J.K.L \quad&  M &= N.O.P
\end{alignat*}

\end{document}