What is meant by weak duality?

In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. That means the solution to the dual (minimization) problem is always greater than or equal to the solution to an associated primal problem.

What is duality theory?

In general, duality theory addresses itself to the study of the connection between two related linear programming problems, where one of them, the primal, is a maximization problem and the other, the dual, is a minimization problem. It focuses on the fundamental theorems of linear programming.

How do you use the weak duality theorem?

Theorem 4.1 (Weak Duality Theorem) If x 2 Rn is feasible for P and y 2 Rm is feasible for P, then cT x  yT Ax  bT y. Thus, if P is unbounded, then P is necessarily infeasible, and if P is unbounded, then P is necessarily infeasible.

Does strong duality hold for SVM?

If Slater’s condition is satisfied, strong duality holds, and furthermore for the optimal value x⋆ , λ⋆ and μ⋆ , the Karush-Kuhn-Tucker (KKT) conditions also holds.

What is strong weak duality?

Strong duality is a condition in mathematical optimization in which the primal optimal objective and the dual optimal objective are equal. This is as opposed to weak duality (the primal problem has optimal value larger than or equal to the dual problem, in other words the duality gap is greater than or equal to zero).

What is dual and primal?

18. General Rules for Constructing Dual 1. The number of variables in the dual problem is equal to the number of constraints in the original (primal) problem. The number of constraints in the dual problem is equal to the number of variables in the original problem.

What is the difference between dual problem and primal problem?

In the primal problem, the objective function is a linear combination of n variables. In the dual problem, the objective function is a linear combination of the m values that are the limits in the m constraints from the primal problem.

What is an example of strong duality?

For example: If is quadratic convex, and the functions are all affine, then the duality gap is always zero, provided one of the primal or dual problems is feasible. In particular, strong duality holds for any feasible linear optimization problem.

What is the weak duality theorem?

The weak duality theorem says that the z value for x in the primal is always less than or equal to the v value of y in the dual. The difference between (v value for y) and (z value for x) is called the optimal duality gap, which is always nonnegative. [3]

When does strong duality hold for the primal problem?

We say that strong duality holds for the above problem if the duality gap is zero: . We can replace the above by a weak form of Slater’s condition, where strict feasibility is not required whenever the function is affine. If the primal problem is convex, and satisfies the weak Slater’s condition, then strong duality holds, that is, .

When does strong duality hold for linear optimization problems?

In particular, strong duality holds for any feasible linear optimization problem. Assume that there is only one inequality constraint in the primal problem ( ), and let The problem is feasible if and only if intersects the left-half plane. Furthermore, we have