= Solution
With inequalities written as $c_i\geq0$, use the <optimization Lagrangian> $L(x,\lambda)=f(x)-\sum_i\lambda_ic_i(x)$. The <Karush-Kuhn-Tucker conditions> at $(x_*,\lambda_*)$ are
$$
c_i(x_*)\geq0,\quad\lambda_{*,i}\geq0,\quad\lambda_{*,i}c_i(x_*)=0,\quad\nabla f(x_*)=\sum_i\lambda_{*,i}\nabla c_i(x_*).
$$
For any <feasible point> $z$, the first-order inequalities for a <convex function> and a <concave function> give
$$
\begin{aligned}
f(z)-f(x_*)&\geq\nabla f(x_*)^T(z-x_*)\\
&=\sum_i\lambda_{*,i}\nabla c_i(x_*)^T(z-x_*)\\
&\geq\sum_i\lambda_{*,i}\bigl(c_i(z)-c_i(x_*)\bigr)\\
&=\sum_i\lambda_{*,i}c_i(z)\geq0.
\end{aligned}
$$
The last equality uses <complementary slackness>. Thus $x_*$ is a <global minimizer>. <Slater's condition> is useful for necessity and existence of <Lagrange multipliers>, but is not required for this sufficiency argument once a <KKT point> is supplied.
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