DUAL PROBLEM Meaning and
Definition
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The term "dual problem" refers to a concept often encountered in mathematics and optimization theory. Specifically, in the context of linear programming and convex optimization, the dual problem is a mathematical formulation that accompanies the original or primal problem. The original problem seeks to maximize or minimize an objective function under certain constraints.
The dual problem is constructed to be associated with the primal problem and is formulated to provide important information about its properties and solutions. It arises by considering the optimization problem from a different perspective, exchanging the roles of the objective function and the constraints.
In the dual problem, the objective function becomes a set of constraints and the primal problem's constraints become the objective function. These constraints in the dual problem are derived based on the original problem's coefficients and constants.
The dual problem serves a crucial purpose by offering insights into the dual relationship between the primal and dual variables, providing a lower bound for the optimal value of the primal problem and helping to demonstrate the duality gap.
Solving the dual problem is beneficial as it allows for the establishment of strong duality, indicating that the optimal solutions of both the primal and dual problems are achieved simultaneously. Additionally, the dual problem aids in determining the shadow prices (also known as dual variables or Lagrange multipliers) associated with the primal problem's constraints, which can provide valuable information regarding the sensitivity of the optimal solution with respect to changes in the constraints.
Etymology of DUAL PROBLEM
The word "dual" originates from the Latin word "dualis", which means "two" or "pair". It is derived from the Latin word "duo", meaning "two". The term "problem" comes from the Latin word "problema", which stems from the Greek word "problema" meaning "task" or "obstacle". Therefore, the term "dual problem" refers to a pair of related tasks or obstacles.
