A simplex algorithm is a linear programming method for solving optimization problems, especially those involving functions and other constraints expressed as inequalities. In most instances, the inequalities describe a polygonal region, and the solution is always displayed in one of the vertices. Therefore, the simplex algorithm can be defined as a technique for testing vertices for possible solutions. It is remarkably fast compared to other algorithms that perform a similar function, which is why it has been used to solve problems in linear programming for years.
Lagrange multiplier is a method for optimizing a function based on a given set of constraints. It enables us to find both the local maxima and local minima of a specific function by applying a series of statistical conditions. One of the conditions to be met is that the selected variables satisfy one or multiple equations. The whole idea behind Lagrange multipliers is to transform constrained problems in a manner that the derivative tests of unconstrained problems can still be implemented.
A saddle point (also known as minimax point) is a spot on a graph’s surface where the derivatives (slopes) in orthogonal directions are all equal to zero. A good example of a minimax point is when a graph has a critical point along one axial direction and crossing axis. A saddle point in two dimensions forms a trace or contour graph in which the values in the contours appear to intersect themselves. Since a saddle point is actually a point of inflection, it’s not considered a local extremum.
An optimal solution is a special type of solution in which the objective function acquires its maximum (and sometimes minimum) value, for instance, the highest profit or the lowest cost. It can be either local or global. A locally optimal solution is a solution whereby no feasible solutions are available in the vicinity. A globally optimal solution, on the other hand, is one where no feasible solutions with a better objective value are available.
Duality in linear programming simply means that every linear program has a related linear program from which it can be derived. The derived linear program is commonly referred to as the dual while the one from which it is derived is called the primal. Before solving quality linear programs, the original programs must be formulated in their standard forms. The standard form in this case means that the variables contained in the original linear program should be non-negative. The greater or equal to sign is used in maximization cases and less than or equal to sign is used in minimization cases.