The authors' aim is to establish an implementable algorithm for finite minimax problems of the form \[ \min_{x\in \mathbb{R}^ n} \Phi(x),\quad\text{where } \Phi(x):= \max_{i=1,\dots,m} f_ i(x)\tag{1} \] and each \(f_ i: \mathbb{R}^ n\to \mathbb{R}\) is twice continuously differentiable. Problem (1) is equivalent to the nonlinear programming problem \[ \text{minimize } z\text{ subject to } f_ i(x)- z\leq 0,\;x\in \mathbb{R}^ n,\;z\in\mathbb{R},\;i=1,\dots,m.\tag{2} \] The authors construct a suitable continuously differentiable exact penalty function \(P\) for problem (2) by defining a continuously differentiable multiplier function that yields an estimate of the multiplier vector associated with (2). The stationary points of \(P\) are related to the critical points of (1), where \(x^*\in \mathbb{R}^ n\) is a critical point of (1) if \[ \max_{i\in I_ A(x^*)} \nabla f_ i(x^*)' d\geq 0\quad\text{for all }d\in \mathbb{R}^ n,\quad I_ A(x^*):= \{i: f_ i(x^*)= \Phi(x^*)\}. \] Moreover, a correspondence is established between (global and local) minimizers of \(P\) and of \(\Phi\). The authors then succeed in constructing an implementable algorithm which is globally convergent towards critical points of problem (1). Finally, some numerical results obtained for a set of well-known test problems are discussed.