The present paper addresses the multi-objective aerodynamic shape optimization of the two-dimensional LS-89 turbine cascade. The objective is to minimize the entropy generation at subsonic and transonic flow conditions while maintaining the same flow turning. Nineteen design variables are used to parametrize the geometry. The optimization problem is used to compare two major classes of optimization algorithms and at the same time deduce if this problem has multiple local solutions or one global optimum. A first optimization strategy uses a gradient-based Sequential Quadratic Programming (SQP) algorithm. This SQP algorithm allows to directly handle the non-linear constraints during the optimization process. An adjoint solver is used for computing the sensitivities of the flow quantities with respect to the design variables, such that the additional gradient computational cost is nearly independent of the number of design variables. In addition, the same optimization problem is performed with a gradient-free-metamodel assisted-evolutionary algorithm. A comparison of the two Pareto-fronts obtained with both methods shows that the gradient-based approach allows to find the same optimum at a reduced computational cost. Moreover, the results suggest that the considered optimization problem is uni-modal. In other terms, it is characterized by a single optimal solution.