Feasible solutions refer to the set of solutions that satisfy all constraints in an optimization problem. In the context of mathematical programming and operations research, these solutions are critical as they define the boundaries within which the optimal solution must be found.
Karmarkar's Algorithm revolutionized linear programming by introducing a polynomial-time method that improved the efficiency of solving large-scale optimization problems. By operating within the simplex of feasible solutions, it paved the way for new, more feasible numerical approaches compared to previous methods like the simplex algorithm.