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.
The Simplex Algorithm is a popular method for solving linear programming problems by iteratively moving along the edges of the feasible region to find the optimal vertex. It efficiently navigates through feasible solutions in a systematic way, making it a cornerstone technique in operations research and optimization.
Interior Point Methods are a class of algorithms used to solve linear and nonlinear convex optimization problems by traversing the interior of the feasible region. They are known for their polynomial-time complexity and efficiency in handling large-scale problems compared to traditional simplex methods.
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.