Abstract
<p><span>The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.<span> </span>In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. <span> </span>Thus the main concepts of duality will be explored by the solution of simple optimization problem.</span></p>
Highlights
The duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem
The objective function is a linear combination of n variables
Any feasible solution to the dual problem gives a bound on the optimal objective function value in the primal problem
Summary
The duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem. The objective function is a linear combination of the m values that are the limits in the m constraints from the primal problem. Any feasible solution to the dual problem gives a bound on the optimal objective function value in the primal problem. Changing the right-hand side constraint vector of the primal or adding a new constraint to it can make the original primal optimal solution infeasible This operation changes only the objective function or adds a new variable to the dual respectively. If there exist a profit maximization problem with a resource constraint i, the value yi of the corresponding dual variable in the optimal solution indicates an increase of yi in the maximum profit for each unit increase in the amount of resource i (absent degeneracy and for small increases in resource i). It indicates a fairly close relationship existing between linear programming and the theory of games
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