We study a linear programming problem whose number of constraints is much higher than the number of variables. Under some restrictions on the problem data, we show that the problem is equivalent to the maximization of a concave nonsmooth penalty function under nonnegativity constraints with a penalty parameter that can be easily determined. Moreover, after transforming the latter program into an unconstrained maximization problem, we suggest the use of supergradient algorithms with space transformation. By means of randomly generated test problems with a very large number of constraints, the viability and efficiency of this approach is demonstrated.