This paper concerns the (nondifferentiable) optimization problem (1) \(\lambda^* = \inf_{x \in \mathbb{R}^ m} \lambda_ 1 (x)\), where \(\lambda_ 1 (x)\) are the largest eigenvalues of \(A(x) = A_ 0 + \sum^ m_{j=1} x_ jA_ j\) and the \(A_ j\) are symmetric. It gives a sufficient condition on \(x\) to satisfy \(\lambda_ 1 (x) - \varepsilon \leq \lambda^* < \lambda_ 1 (x)\) \((\varepsilon\) given) and a construction of descent directions for \(\lambda_ 1 (x)\) when the sufficient condition fails to hold, furthermore a line search rule for (1) when a descent condition is given, and an algorithm for solving (1) under the assumption that the multiplicity of \(\lambda_ 1 (x)\) at the solution is known. Numerical experiments on the proposed algorithm concern six randomly generated \(5 \times 5\) matrices, and eleven \(20 \times 20\) matrices for which that multiplicity is 3; the latter resulted from work by \textit{M. L. Overton} [SIAM J. Matrix Anal. Appl. 9, No. 2, 256-268 (1988; Zbl 0647.65044)].