Convex optimization zero duality gap

  • Can duality gap be negative?

    Note: By weak duality we know that duality gap is always non-negative..

  • How do you show that duality gap is zero?

    If f0 is quadratic convex, and the functions f1,,fm,h1,,hp 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. with optimal value d⋆ = 0.Feb 9, 2012.

  • How do you show that duality gap is zero?

    If f0 is quadratic convex, and the functions f1,,fm,h1,,hp 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. with optimal value d⋆ = 0..

  • What is the duality gap in convex problems?

    This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the convex relaxation of the primal problem: The convex .

  • What is the duality gap of a linear program?

    The duality theorem states that the duality gap between the two LP problems is at least zero.
    Economically, it means that if the first factory is given an offer to buy its entire stock of raw material, at a per-item price of y, such that ATy ≥ c, y ≥ 0, then it should take the offer..

  • The main difference between the primal problem and the dual problem is that the primal problem seeks to minimize or maximize a certain objective function subject to a set of constraints, while the dual problem seeks to find the best set of constraint multipliers that satisfy certain conditions.
  • This theory provides the idea that the dual of a standard maximum problem is defined to be the standard minimum problem.
    This technique allows for every feasible solution for one side of the optimization problem to give a bound on the optimal objective function value for the other.
For any convex optimization problem with a differentiable objective and constraint functions, any points that satisfy the KKT conditions are primal and dual optimal and have zero duality gap.
In general, 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. This fact is called strong duality.

Does strong duality hold for a convex optimization problem?

Duality gap and strong duality

We have seen how weak duality allows to form a convex optimization problem that provides a lower bound on the original (primal) problem, even when the latter is non-convex

The duality gap is the non-negative number p d

We say that strong duality holds for problem (8

1) if the duality gap is zero: p = d

Is there a zero duality gap if f0 is quadratic convex?

Note that there are many other similar results that guarantee a zero duality gap

For example: Theorem 2 (Quadratic convex optimization problems)

If f0 is quadratic convex, and the functions f1; : : : ; fm; h1; : : : ; hp are all a ne, then the duality gap is always zero, provided one of the primal or dual problems is feasible

For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality.

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